Finding your dream job

Career is the result of a conscious position and behavior of an individual in the sphere of labor related to official or professional growth (Porfeli 47). Career as a trajectory of motion is constructed by an individual in accordance with the peculiarities of internal and external organizational reality and above all, in accordance with personal goals, desires and attitudes. The activities of people are often judged by their careers. At the beginning of the professional cycle, human efforts are usually aimed at preparing for a future career – the development of skills, values, views and other aspects necessary for the acquisition of professional identity. At the end of the professional cycle, a person typically tends to concentrate forces at determining the degree of personal career success. Both stages include the analysis of the correctness to the chosen career.

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The study, which has become a classic one (Feldt 238), shows that certain personality factors and pressure situations influence the fact whether people change their careers or try their best to hold on to the one acquired at the very beginning of their careers, believing that it is the right choice. The way an individual likes his job, and therefore, the extent to which his career will be successful, depends on certain factors. The following ones could be listed among them: a) knowledge of the profession; b) correspondence between the characteristics of a person and environment; c) good professional role models; d) stimulating, but not threatening demands of work and associated expectations; e) reduced concern for prestige; f) correspondence between personal and professional values; g) the context of working environment in which the socialization takes place (Feldt 235-45). In addition to these variables, researchers have identified two basic types of focus – on earnings and on job satisfaction, – expressed by workers and affecting career choice (Verbruggen 3-15).
Thus, people are choosing their own “right” job basing on a variety of reasons: money, status, prestige, communication, satisfaction, etc., and when choosing a career, they often take into account if not all, but at least some of those reasons. Person’s perception of certain careers and reasons why he should do this activity is largely determined by previous experiences and prevailing social attitudes (Porfeli 46-58).
To assess personality traits and optimize the choice of the professional activity, it is important to consider the type of personality selecting this or that activity. One of the most operational typologies for this purpose is the personality typology by Holland (Primé 179-80), predetermining the content of career activities and including: realistic type (focus on manipulations with tools and machinery), research type (focus on search), artistic (emotional expression, self-presentation), social (interactions with people), business (focus on the impact on people), and conventional type (manipulations with data and information).
Although the concept of Holland assumes that one of the types is always dominating, people can adapt to the conditions, using the strategy of two or more types (Primé 181). The closer are the orientations of the dominant type and the second (or third) orientations, the closer the personality types are. Taking into account the nature of dominant and non-dominant orientations, one can choose the activities that are closer to one’s own nature and where one will be more successful. If the dominant and the following orientations are far from each other, it is much more difficult to make a right career choice (Primé 185-86).
Thus, the formation of a career is a continuous process, during which the person is using the information about oneself and about the world, chooses the sphere of activity, and then – a specific profession. When choosing the direction of a professional career, one must take into account three basic conditions for a successful career: the profession should be in the sphere of one’s interests; the profession should correspond to personal abilities; and the profession should be in demand in the labor market (Perrone 291-94). Any person has an access to several ways to get acquainted with the basic terms of career choices and strategies on making career choice decisions.
One of the most wide spread resources for that is coming through psychological testing to identify professional inclinations, select the first higher education institution or education institution for reeducation or specialization (Perrone 295-97). In addition, one can rely on statistical information on payment rates and schedules in different careers. After deciding what one prefers more – a 6-digit salary or a flexible schedule, a person can examine the statistics of suitable jobs.
Most of the information can be found on the Internet. Every modern establishment or company has its own website, where one can get all the latest information. It is also possible to send resumes to employers or find several HR agencies; both methods will include prior assessment of a candidate. In addition, the international professional networks could help in getting acquainted with people who work in the same area. Communicating with them, one can find out everything about the career of one’s dream (Porfeli 46-58).
Finally, it is acceptable to use the services of the employment centre, the specialists of which can provide all necessary information and test candidates on the professional suitability. In addition to state agencies of career choices which are governmental organization that provide advice to the population on education and career choices, there are private consulting companies providing consultations with a specialist on career choices in the immediate customer service centers (Verbruggen 3-5).
Contemporary career consulting is a process of evaluating opportunities, potential and real (not imaginary, imposed by society or influence of friends and parents) wishes by professional consultants, possessing information on the labor market and demand occupations. Such consultations usually do not involve testing, but only free dialogue between the specialist and the client. The procedure lasts from 1,5 to 2,5 hours. Finally, the customer receives the conclusion of a specialist with recommendations on career development, given information on skills that need to be acquired for achieving success. This service is relevant not only for students but also for those who have already graduated from university or other educational institutions, and cannot decide on the choice of their dream career (Verbruggen 3-13).
Having made a mistake in choosing a career, people often suffer in the future. Doing something that does not bring joy may harm both the health and success in personal life. All areas of life are intertwined, so it is difficult to underestimate the importance of correct choice.
About 50-80% of people make mistakes when choosing their careers (Primé 178). And they usually make this choice consciously. Society imposes the understanding of the proper career; therefore people often follow the established stereotypes. Instead of choosing what one likes, one chooses what is considered prestigious; common sense becomes a victim of dictate of the public opinion. Because of this, the actual percentage increases to 95-99.9% (Feldt 240). Indeed, there really a few of those who have made the right choice combining one’s career with one’s passion. These people do not trust their future to the fate; they deliberately choose such a life.
Thus, the choice of career is one of the most important decisions man makes in life. Everyone wants the job to meet one’s interests and capabilities, bringing joy and money. To create a dynamic career it is necessary to realize one’s own interests, abilities and labor market requirements. Taking everything into account, it is possible to say that gradually the career choice becomes easier. More search methods, more alternatives emerge every day. Realizing one’s own weak and strong sides, interests and preferences, one can make the right choice.
Works Cited:
Feldt, Ronald C., Ferry, Ashley, Bullock, Melinda, Camarotti-Carvalho, Ana, Collingwood, Melinda, Eilers, Scott, Meyer, Luke, Nurre, Emily, and Cheryl Woelfel. “Factorial Structure of the Career Decision Scale: Incremental Validity of the Five-Factor Domains.” Measurement and Evaluation in Counseling and Development 42 (2010): 235-245. Print.
Perrone, Kristin M., Tschopp., Molly K., Snyder, Erin R., Boo, Jenelle N., and Claudine Hyatt. “A Longitudinal Examination of Career Expectations and Outcomes of Academically Talented Students 10 and 20 Years Post—High School Graduation.” Journal of Career Development 36 (2010): 291-309. Print.
Primé, Dominic R, and Terence J. G. Tracey. “Psychometric Properties of the Career Clusters Interest Survey.” Journal of Career Assessment 18.2 (2010): 177-188. Print.
Porfeli, Erik J., and Vladimir B. Skorikov. “Specific and Diversive Career Exploration During Late Adolescence.” Journal of Career Assessment 18.1 (2010): 46-58. Print.
Verbruggen, Marijke, and Luc Sels. “Social-Cognitive Factors Affecting Clients’ Career and Life Satisfaction After Counseling.” Journal of Career Assessment 18.1 (2010): 3-15. Print.
 

An Analysis of Chiyo’s Journey Finding her Identity In “Memoirs of a Geisha”

Finding One’s True Self:

 An Analysis of Chiyo’s Journey Finding her Identity In “Memoirs of a Geisha” by Arthur Golden

Most people wear makeup to enhance, modify, or obscure their natural look; that is the purpose of makeup. The Geisha population in “Memoirs of a Geisha” have more than a surface level relationship with makeup. Makeup to them is crucial and not only externally changes them but internally as well. A geisha putting on makeup is comparable to Peter Parker transforming into his alter superhero ego except it takes hours. When they do it, they assume a whole new identity. Their makeup is almost impenetrable, like a shield. The girl under the makeup conceals her true identity and creates a deception as people will perceive them from what they see but what they are seeing is a facade made up of white makeup and red lipstick. “Memoirs of a Geisha” by Arthur Golden is a story about a young girl named Chiyo who grows up in a poor village only to find herself becoming one of the most popular geisha’s. She serves as a servant in an okiya (a geisha home) only to soon be taken under the wing of Mameha who trains her to be a successful geisha like Mameha is herself. Chiyo faces many hardships and obstacles, one of them being her own identity crisis. Through her long journey, she finds herself questioning her place in the world and who she really is. In “Memoirs of a Geisha”, Arthur Golden demonstrates the significance of identity through Chiyo’s journey as she embarks becoming a Geisha. Throughout the novel, Golden uses the imagery and symbolism of water and makeup, which allows readers to identify Chiyo’s personal struggles as she endures through the hardships of transforming from Chiyo, the poor girl into Sayuri, the geisha.

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    Golden frequently uses the imagery of water to exemplify the complexity of Chiyo’s identity. Chiyo is noticed for her unusual blue-grey eyes which many believe signifies she has a lot of water in her personality as she is constantly told she “has a great deal of water” (32) in her personality. This represents her honest and truthful nature which contrasts with the artificial and deceptive world of the Geisha. In the Japanese Buddhist tradition, it is said water is one of the five elements that make up the fabric of the universe and personalities of every person. Like water shifting to fit the shape of a container, people with a lot of water in their personality have a tendency towards adaptability and flexibility. Therefore, when the characters in the book comment on the amount of water Sayuri has in her personality, they link her to these traits. Chiyo recognizes these attributes she attains: “Those of us with water in our personalities don’t pick where we’ll flow to. All we can do is flow where the landscape of our lives carries us,” (89). She notes that she cannot keep her life under control, who she is and where she goes will change under the given circumstances and all she can do is let it flow. Through the hardships Chiyo endures like being targeted by Hatsumomo, she does not take action but stays quiet as it is her nature to believe things will work out, “I knew I was trapped in the web Hatsumomo had spun for me. I could do nothing but wait,” (91). Golden using the imagery of water to present Chiyo’s personality enables the readers to understand Chiyo better as it will be easier to comprehend her adaptability and reaction towards given situations. As a little girl, Chiyo is known to have a lot of water but as she matures and becomes a geisha, her name changing to Sayuri does not just signify the birth of another Geisha but a rebirth of a young girl. As Chiyo becomes Sayuri, she learns to control the water and balance her personality: “My new name came from ‘sa’ meaning ‘together’, ‘yu’ from the zodiac sign for the hen-in order to balance other elements in my personality- and ‘ri’ meaning understanding,” (160).

Through the symbolism of makeup, Golden portrays Chiyo’s transformation into Sayuri as the makeup creates a facade for the young girl. Makeup is a crucial part of a Geisha’s life. It enables them to present themselves perfectly as a piece of art as they entertain the rich men. The white face mimics porcelain skin and the red lips symbolize everything a man wants. That is why Chiyo becomes a different person when she is transforming into her Geisha self. She is not Chiyo anymore but Sayuri, “Only when she sits before her mirror to apply her makeup with care does she become a geisha. And I don’t mean that this is when she begins to look like one. This is one she begins to think like one too,” (115). This passage enables readers to identify the change between Chiyo and Sayuri as Sayuri comes to life once the makeup is all set. Along with the appearance, the thoughts change as well. This young girl is not just a normal girl anymore but an elegant and poise entertainer, she should be thinking like one too. While these beautiful artifices conceal the geishas’ actual appearances, geisha must also conceal their desires, true feelings, and inner self so that they can shift their personalities in order to please or amuse their male clients: “Makeup alone won’t be enough to change Chiyo into something beautiful,” (62). From this quote, it can be seen that the speaker is referring to an internal change. Makeup is the first stop to the deception of beauty but along with the makeup, the willingness to change and to conceal their true feelings is essential in a successful geisha’s life. Through the use of makeup, Golden encourages readers to understand how the geisha’s really feel under their white masks and to understand the duality they attain as they have two different personalities: one with makeup and one without. The facade makeup can create for one’s self can be seen in Hatsumomo’s character. Hatsumomo’s beauty paved the way for her success however she was hiding her cruelness under all the makeup until her inner ugliness slowly becomes obvious even to her clients: “A tree may look as beautiful as ever; but when you notice the insects infesting it, and the tips of the branches that are brown from disease, even the trunk seems to lose some of its magnificence,” (324).

In “Memoirs of a Geisha”, Arthur Golden portrays Chiyo’s journey to finding her true self through the imagery of water and the symbolism of makeup. Through the imagery of water, Golden displays Chiyo’s personality as she is often compared to water. Water is adaptable, always flows and will take the shape of any area it is put in, alike Chiyo who is a young girl that will excel and fight through any situation she is put through. Golden also uses the symbolism of makeup to present the facade all geisha’s – not only Chiyo – must be able to put on in order to be a successful geisha. They have to conceal their desires and hide their true feelings under a painted face and how they handle this can expose their true nature as Hatsumomo’s beauty did not overpower her cruel nature as it is later on recognized by everyone including her clients. Through these devices the readers should be able to identify that Chiyo experiences many hardships that test her will and strength, altering the person she may be. Through the struggles and through the painted faces, she holds onto her true self and only matures who she always has been: the poor girl from the poor village. Her humble roots enable her to grow into a successful geisha and a person who is liked by many. In a world like today’s, people can be excused for their inner ugliness if they are beautiful enough or if they generally meet the standards in all aspects including personality. One can never be too much of something: too kind, too nice, too shy – there always has to be a perfect balance. This is not true. What brings people together and initiates diversity amongst communities is the ability to be themselves even if it means they are not a perfect balance. Pretending to be someone you are not is not going to last long as one can not pretend forever without causing a little damage.

Works Cited:

Golden, Arthur. Memoirs of a Geisha: a Novel / M. Alfred A. Knopf, 1998.

 

Strategies for Finding the Right Employee

Finding the Right Employee

 It has always been a challenge for companies to find the right person for the right job; therefore, having a HR department or an employee working as HR will definitely help the multiple process of finding and hiring the right person.

Analysis

 Human Resources primary functions are work design & work planning, managing employee competencies, managing employee attitudes and behaviour (Lepak & Gowan, 2010, p. 8). There are of course challenges that come with the job such as organizational demands, environmental influences and regulatory issues. To avoid these issues, HR needs to pay more attention to earlier process starting from job analysis, work design, workforce planning, recruitment and selection.

Managing Employees For Competitive Advantage

Managing employees can be a difficult task because different person has different personality and even though most workers have the same goal (to earn money), they are usually motivated by different things. Some workers are motivated by incentives, some by benefits, and some other likes to travel as part of their job. Keeping employees happy or satisfied is a way to keep the morale or motivation high so the employees’ performance are benefiting the company.

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Giving feedbacks are a must to review and to keep the employees in check. Receiving good feedback can boost the employee’s morale and negative feedback can also motivate employee to do better or improve at their job. Incentives are given to achieve goals that will take longer to achieve otherwise. Employees are more motivated and sales/ performance will also increase. Healthcare benefits are one of the reason why employees are attracted to the job in the first place. Increasing benefits will get the employees excited and also boost their morale. Employees who are more attracted to travelling will love their job better and are more motivated to do their job.

Manage Organizational Demands and Environmental Influences

 A Company needs to understand market demands in their industry and plan their strategy based on that knowledge. There are strategies that fits one company but not necessary the others because no two companies are the same (Lepak & Gowan, 2010, 28). A few of the most common strategies are cost leadership strategy and differentiation strategy. Cost leadership strategy focus on providing customer with the cheapest price in the market; therefore, generating more revenue to the company. Usually company that runs with this strategy have less-skilled-low-cost employees as their majority and it has been proven to be a major success by introducing revolutionary business models built on a single base – the lowest possible prices for a given perceived value (“Competitive Strategies – Cost Strategy VS. Differentiation Strategy”). Differentiation strategy is usually used by company that promotes quality over quantity and the one that provides products to specific high-end markets. Ariel Rodriguez mentioned in his interview with Worchester Business Journal that this business model results in “a higher quality product engineered specifically for use in the target application.” (Finaldi, 2017).

 Aside from business strategies, companies also have to consider environmental influences such as labor force trends, technology, globalization and ethics & social responsibility. Changes in labor force trends influences the options for companies to hire specific demographic in order to realize their strategy. Aging population is one of the reason there are changes in labor force trends as suggested by an article published in Bureau of Labor Statistics website (Frazis, 2017). Technology definitely plays a part in influencing companies decision to hire more workers or moving towards machine automation to do the job. With the current rates of technology advancement, by 2030, it is expected for robots to replace almost a third of the U.S. workforce (Paquette, 2017). Companies can utilize the current technology system and save a big portion of labor cost to improve other sections of the company. With the help of technology these days, it is not as hard as it used to be for companies to enter the global market. Globalization helps company to have wider pool of employees/ candidates to choose from. It helps reduce cost as living cost and wages are different from one country to another. Outsourcing is one of the main cost-reducing action that companies take to expand their business. “Job outsourcing helps U.S. companies be more competitive in the global marketplace. It allows them to sell to foreign markets with overseas branches” (Amadeo, 2018). Despite all the cost-reducing strategy that company can implement, it still has to adhere to ethics and social responsibility as it will also help their brand and reputation as a company to gain additional consumer support (Lepak & Gowan, 2010, p. 45).

Manage Regulatory Issues

 Equal opportunity of employment are expected and regulated in every company. Companies have to follow the regulation to accommodate several protected classifications to ensure that equal opportunity is enforce at their workplace. The protected classifications are race, sex, religion, color, national origin, age, disability and veteran status. If an individual of the protected class is treated differently due to the description of that class, the company has a case of disparate treatment (Lepak & Gowan, 2010, p. 54). This regulation do not just apply during hiring process or the obvious discrimination but also when promotions are available but that opportunity is not presented to the protected class due to membership of the class.

In order to avoid that, company has to choose and manage their employees accordingly.

Manage Job Design and Job Analysis

 Choosing an employee starts from systematically identifying tasks, duties and responsibilities that needs to be done (Lepak & Gowan, 2010, p. 87). Job analysis is used for HR or hiring department to compile the list of responsibilities the candidates need to do in order to contributes to the needs of the company. After identifying the test, company needs to design the work that are expected to be done. There are several approaches to design a job; efficiency approach and motivational approach. Efficiency approach simplifies the job by removing decision-making and it requires less training but this approach increase production and standardize the production process. However, since this approach lacks complexity and require repetition, it leads to boredom and lack of motivation in doing the job.

Motivational approach is used to maximize the employees drive and focus on making the job as interesting and challenging as it can be (Lepak & Gowan, 2010, p. 90). These approach works well with employee who has high growth need strength. Job enlargement, rotation, enrichment and empowerment can keep employees motivated and feel less bored in doing their job.

Workforce Planning

After designing and analyzing the required position, HR has to plan the workforce needed in order to decide whether recruitment is necessary or simply give the task to an existing employee and offer over time benefits. First, HR needs to determine the labor demand and labor supply. Labor demand are the data needed to know how many employees are needed to do the required tasks and labor supply refers to the number of workers available. If labor demand are higher than labor supply, the company has a labor shortage and it can be resolved by several actions. Employee overtime and outsourcing are the biggest 2 answers to labor shortage. Outsourcing is becoming the sought-out resolution to this matter. When a particular type of work is outsourced, then the company does not require hiring skilled people for it (“How Does Outsourcing Reduce Cost? (Benefits of Outsourcing)”).

  If labor supply are larger than labor demand, the company needs to take some measures to reduce their labor force. There are several way of reducing the gap between the large labor supply and labor demand. Early retirement, attrition and layoffs are the biggest contributor in reducing the labor surplus. As proven in the automobile company Ford, Attrition and early retirement help them from their negatives revenue back to acquiring profits (Bunkley, 2010).

Recruitment

 Recruitment process can sometimes takes longer than necessary due to the need of finding the perfect fit for the designed job. Recruitments can be done in several ways such as  internal recruitment, advertising, college and employment agencies. Sometimes, internal recruitment is the best way to find the right person for the position as the candidate already knows the company well and there is not a lot of difference in the company culture. The employer knows what skills the candidate possessed and also what the candidate is capable of doing/ handling. Advertising is used to recruit external candidates and usually concrete job description is needed in order to attract future candidates to apply for the position. The job description will later be posted on newspaper, bulletin boards and of course the vast audience of the internet.

College recruitment allows company to get freshly graduates with fresh ideas to contribute to the company’s benefits. Hiring younger generation will also help with updating the company’s technological system to the latest version in order to maximize performance. Employment agencies will help the company to look for available candidates who are eligible according to the skill required and the needs of the company. They usually charges fee for this service; however, in Britain, it is proven to be the most important source of job matching (Gregg & Wadsworth, 1998).

Selection

 Selection process is the responsibility of the hiring manager. After having suitable candidates for the job, hiring manager needs to decide which candidate is the best fit for the position. There are several methods in selecting a candidate: initial screenings and final screenings. There are of course rating errors during the screenings such as bias, candidate’s personality, contrast effect, halo effect, devil’s horn effect and impression management (Lepak & Gowan, 2010, p. 204). Despite these rating errors, these selection process are one of the best indicators whether a candidate is a fit to the position and the company as a whole.

Summary and Conclusion

 In conclusion, company needs to analyze and design a job, look at their employee inventory, decide on recruitment or other labor shortage strategy, find the right person in order to advance the company to the next level.

References

Amadeo, K. “How Outsourcing Jobs Affects the U.S. Economy” (2018, March 19). The Balance. Retrieved from https://www.thebalance.com/how-outsourcing-jobs-affects-the-u-s-economy-3306279

Competitive Strategies – Cost Strategy VS. Differentiation Strategy. Retrieved from http://consilue.com/en/competitive-strategies-cost-strategy-vs-differentiation-strategy/

Frazis, H. “Employed workers leaving the labor force: an analysis of recent trends” (May 2017). Monthly Labor Review, U.S. Bureau of Labor Statistics. Retrieved from  https://doi.org/10.21916/mlr.2017.16.

How does outsourcing reduce cost? (Benefits of outsourcing). Educba. Retrieved from https://www.educba.com/how-does-outsourcing-reduce-cost/

Gregg, P. & Wadsworth, J. “HOW EFFECTIVE ARE STATE EMPLOYMENT AGENCIES? JOBCENTRE USE AND JOB MATCHING IN BRITAIN” (August 1998). Oxford Bulletin of Economics and Statistics. Retrieved from https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1468-0084.1996.mp58003002.x

Lepak, D. & Gowan, M. (2010).  Human resource management: Managing employees for competitive advantage. Upper Saddle River, NJ: Pearson Prentice Hall.

Mercury Wire advocates for U.S. manufacturing. (2017, May 23). Worchester Business Journal. Retrieved from http://www.wbjournal.com/article/20170523/NEWS01/170529987/mercury-wire-advocates-for-us-manufacturing

Paquette, D. “Robots could replace nearly a third of the U.S. workforce by 2030”. (2017, November 30). The Washington Post. Retrieved from https://www.washingtonpost.com/news/wonk/wp/2017/11/30/robots-could-soon-replace-nearly-a-third-of-the-u-s-workforce/?noredirect=on&utm_term=.831707efcea3

 

Finding the Equilibrium of Iron & the Thiocyanate

Finding the Equilibrium of
FeSCN2+
Introduction
Background
Thiocyanate (SCN-) is natural occurring in the human body that is secreted in the salivary glands. ¹ It is produced with the digestion of food and drugs used specifically to treat thyroid disorders or hypertension. High thiocyanate levels are indicative of cyanide poisoning² but could also be used to assess smoke exposure. Spectrophotometers are used in order to view the concentration of SCN- in a solution.
Theory
Metal ions can form bonds with ligands; however, they often become complex and each have individual equilibria. The equilibrium constant will correlate with the binding affinity of the metal ion and ligand, which is in this case iron (Fe+3) and thiocyanate (SCN-) respectively.  The iron and the thiocyanate should create a complex
FeSCN2+.
Its concentration could then be found by a spectrophotometer set at a wavelength of 447 m. Through the calculated concentrations and the absorbance found through the spectrophotometers, the calibration curve could be created with this data. ICE tables can be used to find the equilibrium constant of five different concentrations of
FeSCN2+.
Hypothesis
If a solution has a high amount of thiocyanate then it will have a greater bonding affinity..
Objective
The objective is to find the equilibrium constant of
FeSCN2+. 
Methods
Part 1 Methods
In order to create 0.5 M of 250 mL HNO3, 1 M of 125 mL HNO3 and 125 mL of DI water were placed in a 250 mL volumetric flask and mixed together. In order to create Fe(NO3)3, 1.21 g of Fe2(NO3)2 was mixed with 25 mL 0.5 M HNO3 in a 25 mL volumetric flask. In order to create KSCN, 0.010 g of KSCN was mixed with 50 mL of 0.5 HNO3 in a 50 mL volumetric flask.

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2.5 mL of Fe(NO3)3 was added in each of the five test tubes using a 5 mL pipette. In test tube #1, 0.5 mL KSCN and 7 mL HNO3 were added. In test tube #2, 0.75 mL KSCN and 6.75 mL HNO3 were added. In test tube #3, 1 mL KSCN and 6.5 mL HNO3 were added. In test tube #4, 1.25 mL KSCN and 6.25 mL HNO3 were added. In test tube #5, 1.5 mL KSCN and 6 mL HNO3 were added. All five of the test tubes should have a total volume of 10 mL each.
A spectrophotometer was then used to find the absorbance at the wavelength 447 m. Each of the solutions needed to be diluted with 2 mL solution and 2 mL HNO3 when adding into the capsule for spectrophotometer testing. A calibration curve was created using the data obtained (absorbency) and the calculated concentrations.
Part 2 Methods
In order to create 0.5 M of 250 mL HNO3, 1 M of 125 mL HNO3 and 125 mL of DI water were placed in a 250 mL volumetric flask and mixed together. In order to create 0.002M solution of Fe(NO3)3, 0.2 M of 0.02 g Fe(NO3)2 was mixed with 50 mL 0.5 M HNO3 in a 50 mL volumetric flask. In order to create 0.002 M KSCN, 0.01 g of KSCN was mixed with 50 mL of 0.5 HNO3 in a 50 mL volumetric flask.
5 mL of Fe(NO3)3 was added in each of the five test tubes using a 5 mL pipette. In test tube #1, 1 mL KSCN and 4 mL HNO3 were added. In test tube #2, 2 mL KSCN and 3 mL HNO3 were added. In test tube #3, 3 mL KSCN and 2 mL HNO3 were added. In test tube #4, 4 mL KSCN and 1 mL HNO3 were added. In test tube #5, 5 mL KSCN was added. All five of the test tubes should have a total volume of 10 mL each.
A spectrophotometer was then used to find the absorbance at the wavelength 447 m. Each of the solutions needed to be diluted with 2 mL solution and 2 mL HNO3 when adding into the capsule for spectrophotometer testing. A calibration curve was created using the data obtained: absorbency and the concentrations. The Keq was found using ICE tables.
Safety

Chemical Name

Chemical Formula

Molecular Weight

Potential Hazards

Safety Equipment Needed

Precautions used

Nitric Acid

HNO3

63.01 g/mol

Oxidizing liquid, corrosive to metal, acute inhalation toxicity, skin and eye damage/irritation

Wear safety goggles, gloves, and lab coat

Use under fume hood. Do not breathe in vapors or mist.

Iron (II) Nitrate

Fe(NO3)3

241.86 g/mol

Skin and eye irritation; dangerous to certain organs

Wear safety goggles, gloves, and lab coat

Avoid contact with eyes, skin, or clothing. Do not store with oxidizer

Potassium thiocyanate

KSCN

97.181 g/mol

Eye, skin, inhalation, and ingestion

Wear safety goggles, gloves, and lab coat

Use only in chemical fume hood.

Results
Part 1 Results

Test Tubes

Concentration of
FeSCN2+
 

Absorbance

1

0.00005

0.25

2

0.000075

0.36

3

0.0001

0.58

4

0.000125

0.64

5

0.00015

0.79

Table 2:
 

Graph 1: Results from table 2 were plotted. The
FeSCN2+
concentration was found using calculations whereas the absorbance was found using the spectrophotometer.
Part 2 Results

Test Tubes

Concentration of
FeSCN2+
 

Absorbance

1

0.000022509

0.1

2

0.00003125

0.15

3

0.000027574

0.13

4

0.000042279

0.21

5

0.000047794

0.24

Table 3
 

 
Graph 2: Results from table 3 were plotted. The
FeSCN2+
concentration was found using calculations whereas the absorbance was found using the spectrophotometer.

Test Tube

Keq

1

166.07

2

87.479

3

101.96

4

58.261

5

52.7

 
Table 4: The equilibrium was found using the M1V1=M2V2 equation then using ICE tables.
Calculations
Example of how concentration was found in part 1:
M1V1=M2V2 (M1= molarity of KSCN V1= volume of KSCN M2= unknown volume of
the solution V2= total volume in test tube)
 0.002*0.5=M2*10
M2=0.0001/2 (to account for dilution)
M2= 0.00005
Example of how Keq was found in part 2:

 

Fe+3

SCN-

FeSCN2+

I

0.0005

0.0006

0

C

-x

-x

+x

E

0.0005-x

0.0006-x

x

Finding molarity of Fe+3
M1V1=M2V2
(0.001)(5)=M2(10)
M2=0.0005 M Fe+3
Finding molarity of SCN-
M1V1=M2V2
(0.002)(3)=M2(10)
M2=0.00006 M SCN-
Equilibrium expression:
Keq=[FeSCN2+]Fe3+[SCN–]
Keq= (0.000027574) / (0.0005-0.000027574)(0.0006-0.000027574)
Keq= 101.96
Discussion
Part 1 Discussion
When the solutions were created in the test tubes, there seemed to be a gradient from light to dark on test tubes 1 to 5 indicating the increasing quantities of KSCN in each of the test tubes. When the group was performing the spectrophotometer analysis for absorbency, it showed numbers greater than 1 for test tube #1. This may be attributed to the fact that the nitric acid was prepared by another group, and there is no way to account for their mistakes. In order to fix this, 2 mL solution was diluted in 2 mL nitric acid. Since it was diluted twice, the concentrations found using calculations were divided by half. As a result, the concentration of 
FeSCN2+
and absorbency had a positive correlation as seen by the near-linear plot on graph 1. 
Part 2 Discussion
Again, a similar process as part 1 was done instead with different volumes of solutions. Prior to placing the solutions into the spectrophotometer, they were diluted as well for consistency. The concentration versus absorbency had a positive correlation as well as indicated by the near-linear graph 2. In addition, the equilibrium needed to be found. In general, it had a negative trend from test tubes 1 to 5. There was a slight discrepancy in the equilibrium of test tube 2 being 87.479 whereas the equilibrium of test tube 1 was 166.07 and the equilibrium of test tube 3 was 101.96. This may have been an error in calculation; otherwise, they display a downward trend in equilibrium.
Sources of Error
The nitric acid in week 1 were not prepared for the group. It may explain why we needed to dilute our solution in the first place. Perhaps it was contaminated. Additionally, some of the pipettes did not function well and continued to leak. This could explain discrepancies in volume. Although the goal was to fill the test tubes each with 10 mL of solution comprising of their specific dosages, that may not be accurate due to the pipettes.
Changes to the Experiment
A suggested change to the experiment is to allow each group to create their own necessary component i.e. the nitric acid. There are too many possible sources of error that cannot be accounted for when relying on others. Also, working equipment should be available to eliminate the need to compensate for volume. The pipettes should have been the most accurate measure of volume during the experiment, and if they are not functioning correctly, the accuracy of the entire experiment is questionable.
Conclusion
The purpose of this experiment was to find the equilibrium constant of five different concentrations of
FeSCN2+.
As mentioned before, a high equilibrium constant correlates with a high bonding affinity. In the experiment it was seen that low thiocyanate levels have high bonding affinity. In test tube 1, there was only 1 mL KSCN added which had an equilibrium constant of 166.07. In test tube 2, there was only 2 mL KSCN added which had an equilibrium constant of 87.479. In test tube 3, there was only 3 mL KSCN added which had an equilibrium constant of 101.96. In test tube 4, there was only 4 mL KSCN added which had an equilibrium constant of 58.261. In test tube 5, there was only 5 mL KSCN added which had an equilibrium constant of 52.7. As one can see, low thiocyanate levels result in a stronger bond to Fe+3.
Although there was a slight discrepancy on test tube 2 having a lower equilibrium constant that test tube 3, the rest of the data indicate that low thiocyanate levels have high affinity bonds. This discrepancy could be attributed to the fact that another group created a key ingredient of the experiment or that the pipettes were not functioning correctly and did not deliver accurate volumes of each of the solutions.
A major concern with this data was that it was only performed once. There should have been at least 3 trials of it. Perhaps if the equilibrium constant of part 1 was found, it could confirm the results of part 2 as well.
Research Connection
An experiment by Silvia et al. aims to find the thiocyanate concentration in human saliva. ³ As mentioned before, thiocyanate is naturally occurring¹ and human saliva can be used in lieu of the potassium thiocyanate that was used in the lab. Saliva samples were collected from both smokers and non-smokers then it was diluted with DI water. The goal of their experiment was to see which method is most efficient in detecting high thiocyanate levels indicative of smoking status. They found that the micropumping multicommutation flow system was the best option because it had high success rates of distinguishing between smokers and non-smokers.
References

¹ Butts, W. C.; Kuehneman, M.; Widdowson, G. M.Automated method for determining serum thiocyanate, to distinguish smokers from nonsmokers. Clin. Chem.1974, 20, 1344–1348.
² Tsuge, K.; Kataoka, M.; Seto, Y.Cyanide and thiocyanate levels in blood and saliva of healthy adult volunteers. J. Health Sci.2000, 46, 343–350.
³ Silva Junior, J., Farias, M., Silva, V., Montenegro, M., Araujo, A., Lavorante, A., & Paim, A. P. Spectrophotometric Determination of Thiocyanate in Human Saliva Employing Micropumping Multicommutation Flow System. Spectroscopy Letters. 2010, (3), 213.

 

Finding ‘G’ Using Simple Pendulum Experiment

Abstract
This report shows how to find an approximate of ‘g’ using the simple pendulum experiment. There are many variables we could see into, some of them are displacement, angle, damping, mass of the bob and more. However the most interesting variable is, the length of the swinging pendulum.
The relationship between the length and the time for one swing (the period) has been researched for many years, and has allowed the famous physicists like Isaac Newton and Galileo Galilei to get an accurate value for the gravitational acceleration ‘g’. In this report, we will replicate their experiment, and will find an accurate value for ‘g’. Finally it will be compared with the commonly accepted value of 9.806 m/s2 .
Introduction

A simple pendulum performs simple harmonic motion, i.e its periodic motion is defined by an acceleration that is proportional to its displacement and directed towards the Centre of motion. Equation 1 shows that the period T of the swinging pendulum is proportional to the square root of the length l of the pendulum:

With T the period in seconds, l the length in metres and g the gravitational acceleration in m/s2. Our raw data should give us a square-root relationship between the period and the length. Furthermore, to find an accurate value for ‘g’, we will also graph T2 versus the length (l) of the pendulum. This way, we will be able to obtain a straight-line graph, with a gradient equal to 4π2g-1 .
Equipment and Method
For this investigation, limited resources like, clamps, stands, a metre ruler, a stopwatch, a metal ball (bob), and some string were used. The experimental set-up was equal to the diagram, shown in figure 1.
In this investigation, the length of the pendulum was varied (our independent variable) to observe a change in the period (our dependent variable). In order to reduce possible random errors in the time measurements, we repeated the measurement of the period three times for each of the ten lengths. We also measured the time for ten successive swings to further reduce the errors. The length of our original pendulum was set at 100 cm and for each of the following measurements, we reduced the length by 10 cm.
 

Figure 1
As stated earlier, it was decided to measure the time for ten complete swings, in order to reduce the random errors.
These measurements would be repeated two more times, and in total ten successive lengths were used, starting from one metre, and decreasing by 10 cm for each following measurement.
A metre ruler was used to determine the length of the string. One added difficulty in determining the length of the pendulum was the relative big uncertainty in finding the exact length, since the metal bob added less than a centimeter to our string length, measured from the bob’s centre. This resulted in an uncertainty in length that was higher than one would normally expect. The table clamp was used to secure the position of the tripod stand, while the pendulum was swinging.
After the required measurements, one experiment was carried out to find the degree of damping in our set-up. Damping always occurs when there is friction, but exactly how significant the degree of damping in our experimental set-up was, remained uncertain.
Depending on the degree of damping, it may or may not have a significant effect on our measurements.
All measurements were taken under the same conditions, using the same metal bob, the same ruler, in the same room, and at approximately 26 degrees Celsius.

Data Collected
Table 1
In table 1 the ±o.46 sec uncertainty in time was obtained by comparing the spread for the different measurements. The time measurement for the 0.50 metre length, had the largest spread (±0.4 seconds), and was therefore used as the uncertainty in the time measurement.
In table 1 the theoretical uncertainty in the length measurement would be 0.05 cm (a metre ruler was used). However, in the experimental set-up, the two end points (the one tied to the clamp, and the one tied to the metal bob) gave rise to a bigger uncertainty, as the exact end-points could not be precisely determined. We estimated the uncertainty in length to be 0.5 cm, or 0.005 metres.
These data in table 1 need to be processed, before we can continue our analysis. First of all, the average of the three trials need to be found, which will reduce our error. Secondly, the time for one swing (or one period) must be found, which will reduce our absolute error, but not our percentage error.
It should also be noted, that for all the measurements, a constant, and small, angle of maximum displacement (amplitude) was used. The angle was kept between 5° to 7°, small enough to ignore the friction present in our experimental set-up.
Apart from these measurements, one more experiment was done to see how much damping was present in our set-up. It took, on average, between 100 and 150 swings, before the motion had seemed to stop. This showed that there was damping present, but this did not significantly affect the measurement of just ten swings.

Table 2 shows the processed data and the uncertainties.

While drawing the graph for the data in table 2, the relationship between the variables used is clearly not a linear one. The suggested square-root relationship shows it, and to linearise this curve, it must be interchanged and the axis must be modified. (the graph is shown in Graph 1)

Table 3
Based on the theory of Simple Harmonic Motion and equation 1, it should be a linear relationship between T2 and Length. When graphing these two modified variables, the regression line must be linear, passing through point (0,0) and with a gradient equal, or close to 4π2g-1 .

Graph 2
Conclusion and Evaluation
Graphing the length against T2 clearly shows a linear relationship, in agreement with the theory. The actual line of best fit does not go through (0,0) which suggests a systematic error in our experiment. But when graphing a line of best fit, with the condition it should pass through (0,0), we find a line with a gradient of 4.128 and a correlation coefficient of 0.993, which further suggests a very strong linear correlation between our chosen variables.

The value for ‘g’ can be calculated by dividing 4π2 with the gradient of the line of best fit;

The uncertainty in this value was found, by taking half the difference of the lowest possible value for ‘g’ and the highest possible value for ‘g’:
Comparing our calculated value for the gravitational acceleration ‘g’ with the accepted theoretical value gives us an error of 2.5%, well within the error margins that we calculated. This is a reasonable result, given the equipment and the time constraints that we faced.
Looking at our graph, we cannot identify any outliers. However, our data values suggest a line of best fit that does not pass through (0,0). When we do fit a linear regression onto our data values, that passes (0,0), we see that the line does not ‘hit’ all the horizontal error bars (the uncertainty in the length). This may suggest a systematic error in the measurement of the length of our pendulum.

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Further Improvements
To reduce the systematic error in the length measurements, one should take accurate measurements of the diameter of the metal bob used. In this experiment, it looks as if we systematically used a length for the pendulum that was too short. If 1 cm was added to our data, we would get a value for ‘g’ that is equal to the theoretical value of 9.806 m/s2 .
The theoretical value used, is the average value for ‘g’ on Earth, and may be slightly different from the one that was measured.
Instead of three measurements, taking five measurements would be better, as it would not take too much extra time, and this would further reduce our uncertainty in the measurement of the period of swing.
Alternatively, measuring the time for 20 swings, instead of 10 swings, would also reduce the uncertainty in time.
Lastly, a photogate could be used in the future, to measure the period with higher precision. A nice extension to this experiment would be the use of different metal bobs, of different diameter and/or mass. This would allow us to calculate the effect of air resistance on this experiment.
References

http://physics.nist.gov/cgi-bin/cuu/Value?gn
http://www.practicalphysics.org/go/Experiment_480.html
http://en.wikipedia.org/wiki/Simple_harmonic_motion#Mass_on_a_spring
http://www.phys.utk.edu/labs/simplependulum.pdf

 

Finding Difference Between the Squares of any Two Natural Numbers

One of the basic arithmetic operations is finding squares and difference between squares of two natural numbers. Though there are various methods to find the difference between squares of two natural numbers, still there are scopes to find simplified and easy approaches. As the sequence formed using the difference between squares of two natural numbers follow a number patterns, using number patterns may facilitate more easy approach. Also, this sequence has some general properties which are already discussed by many mathematicians in different notations. Apart from these, the sequence has some special properties like sequence – difference property, difference – sum property, which helps to find the value easily. The sequence also has some relations that assist to form a number pattern.

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This paper tries to identify the general properties, special properties of finding difference between the squares of any two natural numbers using number patterns. A rhombus rule relationship between the sequences of numbers formed by considering the difference between squares of the two natural numbers has been defined. A new method to find a2 – b2 also has been introduced in some simple cases. This approach will help the secondary education lower grade students in identifying and recognizing number patterns and squares of natural numbers.
Mathematical Subject Classifications: (2010) 11A25, 11A51, 40C99, 03F50
DIFFERENCE BETWEEN SQUARES OF TWO NATURAL NUMBERS RELATIONS, PROPERTIES AND NEW APPROACH
Introduction
Mathematics, a subject of problem solving skills and applications, has wide usage in all the fields. Basic skills of mathematical applications in number systems used even in day – to – day life. Though calculators and computers have greater influences in calculations, still there is a need to find new easy methods of calculations to improve personal intellectual skills.
As there has been growing interest, in mathematics education, in teaching and learning, many mathematicians build simple and different methods, rules and relationships in various mathematical field. Though various investigations have made important contributions to mathematics development and education (2), there still room for new research to clarify the mutual relationship between the numbers and number patterns.
In natural numbers, various subsets have been recognized by ancient mathematicians. Some are odd numbers, prime numbers, oblong numbers, triangular numbers and squares. These numbers shall be identified by number patterns. Recognizing number patterns is also an important problem-solving skill. Working with number patterns leads directly to the concept of functions in mathematics: a formal description of the relationships among different quantities.
One of the basic arithmetic operations is finding squares and difference between squares of two natural numbers. Already many proofs and relationships were identified and proved in finding difference between squares of two natural numbers. We use different methods to find the difference between squares of two natural numbers. That is, to find a2 – b2. Though, this area of research may be discussed by early mathematicians and researchers in various aspects, still there are many interesting ways to discuss the same in teaching.
Teaching number patterns in secondary level education is most important issue as the students develop their analytical and cognitive skills in this stage. Different arithmetic operations and calculations need to be introduced in such way that they help the students in lifelong learning. Easy and simplified approaches will support the students in logical reasoning.
This paper tries to identify the general properties, special properties of finding difference between the squares of any two natural numbers using number patterns. Also, this paper tries to define the rhombus rule relationship between the sequences of numbers formed by the differences of squares of two natural numbers. A new method to find a2 – b2 also has been introduced in some simple cases.
These may be introduced in secondary school early grades, before introducing algebraic techniques of finding a2 – b2 to develop the knowledge and understanding of number patterns. This will help to recognize and apply number patterns in further level.
Literature Review
To find the difference between the squares of any two natural numbers, we use different methods. Also, we use various rules to find the square of a natural number. Some properties were also been identified by the researchers and mathematicians.
Methods used to find the difference between squares of two natural numbers
Direct Method
The difference between the squares of two natural numbers shall be found out by finding the squares of the numbers directly.
Example: 252 – 52 = 625 – 25 = 600
Using algebraic rule
The algebraic rule a2 – b2 = (a – b)(a + b) shall be applied to find the difference between the squares of two natural numbers.
Example: 252 – 52 = (25 – 5)(25 + 5) = 20 x 30 = 600
Method when a – b = 1(2)
“The difference between the squares of every two consecutive natural numbers is always an odd number, and that it is equal to the sum of these numbers.”
Example: 252 – 242 = 25 + 24 = 49
Methods used to find the square of a natural number
Using Algebraic Method
The algebraic rules shall be used to find the square of natural number other than the direct multiplication. In general, (a + b)2, (a – b)2 are used to find the squares of a natural number from nearest whole number.
Example: 992 = (100 – 1)2
= 1002 – 2(100)(1) + 12 = 10000 – 200 + 1
= 9801
Square of a number using previous number(8)
The following rule may be applied to find the square of a number using previous number.
(n + 1)2 = n2 + n + (n+1)
Example: 312 = 302 + 30 + 31 = 900 + 30 + 31 = 961
The Gilbreth Method of finding square(9)
The Gilbreth method uses binomial theorem to find the square of a natural number. The rule is
n2 = 100(n – 25) + (50 – n)2
Example: 992 = 100(99 – 25) + (50 – 99)2
= 7400 + 2401 = 9801
Other than the above mentioned methods various methods are used based on the knowledge and requirements.
Properties of differences between squares of the natural numbers
2.3.1. The difference between squares of any two consecutive natural numbers is always odd.
To prove this property, let us consider two consecutive natural numbers, say 25 and 26
Now let us find 262 – 252
262 – 252 = (26 + 25)(26 – 25) [Using algebraic rule]
= 51 x 1 = 51, an odd number
2.3.2. The difference between squares of any two alternative natural numbers is always even.
To prove this property, let us consider two alternative natural numbers, say 125 and 127
Now let us find 1272 – 1252
1272 – 1252 = (127 + 125)(127 – 125) [Using algebraic rule]
= 252 x 2 = 504, an even number
Some other properties were also identified and discussed by various mathematicians and researchers.
Number Patterns and Difference Between the Squares of Two Natural Numbers – Discussions and Findings
Some of the properties stated above shall be proved by using number pattern. Number patterns are interesting area of arithmetic that stimulates the logical reasoning. They shall be applied in various notations to identify the sequences and relations between the numbers.
3.1. Sample Table for the difference between squares of two natural numbers
To find the properties and relations that are satisfied by the sequences formed by the differences between the squares of two natural numbers, let us form a number pattern. For discussion purposes, let us consider first 10 natural numbers 1, 2, 3 … 10.
Now, let us find the difference between two consecutive natural numbers.
That is, 22 – 12 = 3; 32 – 22 = 5; and so on.
Then the sequence will be as follows: 3, 5, 7, 9, 11, 13, 15, 17 and 19.
The sequence is a set of odd numbers starting from 3.
i.e., Difference 1: {x| x is an odd number greater than or equal to 3, x Î N}
In the same way, let us form the sequence for the difference between squares of two alternative natural numbers.
That is, 32 – 12 = 8, 42 – 22 = 12, and so on.
Then the sequence will be: 8, 12, 16, 20, 24, 28, 32 and 36
Thus the sequence is a set of even numbers and multiples of 4 starting from 8.
i.e., Difference 2: {x| x is an multiple of 4 greater than or equal to 8, x Î N}
By proceeding this way, the sequences for other differences shall be formed.
Let us represent the sequences in a table for discussion purposes.
In Table 1, N is the natural number.
S is the square of the corresponding natural number.
D1 represents the difference between the squares of two consecutive natural numbers. That is, the difference between the numbers is 1.
D2 represents the difference between the squares of two alternate natural numbers. That is, the difference between the numbers is 2.
D3 represents the difference between the squares of 4th and 1st number. That is, the difference between the numbers is 3, and so on.
3.2. Relationship between the row elements of each column
Now, let us discuss the relationship between the elements of rows and columns of the table.
From the above table,
Column D1 shows that the difference between squares of two consecutive numbers is odd.
Column D2 shows that the difference between squares of two alternate numbers is even.
The other columns show that the difference between the squares of two numbers is either odd or even.
From the above findings, the following properties shall be defined for the difference between squares of any two natural numbers.
3.3. General Properties of the difference between squares of two natural numbers:
The difference between squares of any two consecutive natural numbers is always odd.
Proof: Column D1 proves this property.
This may also be tested randomly for big numbers.
Let us consider two digit consecutive natural numbers, say 96 and 97.
Now, 972 – 962 = 9409 – 9216
= 493, an odd number
Let us consider three digit consecutive natural numbers, say 757 and 758.
Thus, 7582 – 7572 = 574564 – 573049
= 1515, an odd number
This property may also be further tested for big numbers and proved. For example, let us consider five digit two consecutive natural numbers, say 15887 and 15888.
Then, 158882 – 158872 = 252428544 – 252396769
= 31775, an odd number
Apart from these, the property shall also be easily derived by the natural numbers properties. As the difference between two consecutive numbers is 1, the natural number property “The sum of odd and even natural numbers is always odd”, shall be applied to prove this property.
The difference between squares of any two alternative natural numbers is always even.
Proof: Column D2 proves this property.
This may also be verified for big numbers by considering different digit natural numbers as discussed above.
Apart from this, as the difference between two alternate natural numbers is 2, the natural numbers property “A natural number said to be even if it is a multiple of two” shall also be used for proving the stated property.
The difference between squares of any two natural numbers is either odd or even, depending upon the difference between the numbers.
Proof: The other columns of Table 1 prove this property.
In Table 1, as D3 represents the sequence formed by the difference between two natural numbers whose difference is 3, an odd number, the sequence is also odd. Thus, the property may be proved by testing the other Columns D4, D5, …
Also, the addition, subtraction and multiplication properties of natural numbers prove this property.
Example:
112 – 62
Here the difference (11 – 6 = 5) is odd.
So, the result will be odd.
i.e. 112 – 62 = 121 – 36 = 85, an odd number
122 – 82
Here the difference (12 – 8 = 4) is even.
So, the result will be even.
i.e. 122 – 82 = 144 – 64 = 80, an even number
3.4. Special Properties of the difference between squares of the two natural numbers
Table 1 also facilitates to find some special properties stated below.
Sequence Difference Property
Table 1 shows that the sequences formed are following a number pattern with a common property between them. Let us consider the number sequences of each column.
Let us consider the first column D1 elements. D1: 3, 5, 7, 9, 11 … …
As D1 represents the difference between the squares of two consecutive natural numbers, let us say, a and b with a > b, the difference between them will be 1.
That is a – b = 1
Let us consider the difference between the elements in the sequence.
The difference between the numbers in the sequence is 2.
Thus the difference between the elements of the sequence shall be expressed as, 2 x 1. Thus, Difference = 2(a – b)
Now, let us consider the second column D2 elements. D2: 8, 12, 16, 26, … … …
As D2 represents the difference between the squares of two alternative natural numbers, the difference between the natural numbers, say a and b is always 2. That is a – b = 2
If we consider the difference between the elements in the sequence, the difference is 4.
Thus, the difference between the elements in the sequence shall be expressed as 2 x 2.
That is, difference = 2 (a – b)
In the same way, D3: 15, 21, 27, 33, … … …
D3 represents the difference between squares of the 4th and 1st numbers, difference is 3. That is a – b = 3
The difference between the numbers in the sequence is 6.
Thus, difference = 2 x 3 = 2(a – b)
All other columns also show that the difference between the numbers in the corresponding sequence is 2 (a – b)
Thus, this may be generalized as following property:
“The difference between elements of the number sequence, formed by the difference between any two natural numbers, is equal to two times of the difference between those corresponding natural numbers.”
Difference – Sum Property:
From Table 1, we shall also identify another relationship between the elements of the sequence formed.
Let us consider the columns from table 1 other than D1.
Consider D2: 8, 12, 16, 20
This sequence shall be formed by adding two numbers of Column D1.
i.e. 8 = 3 + 5
12 = 5 + 7
16 = 7 + 9
20 = 9 + 11
And so on.
Thus, if the difference between the natural numbers taken is 2, then the number sequence of the difference between the two natural numbers shall be formed by adding 2 natural numbers.
Consider D3: 15, 21, 27
This sequence shall be formed by adding three numbers from Column D1.
i.e. 15 = 3 + 5 + 7
21 = 5 + 7 + 9
27 = 7 + 9 + 11
And so on.
Thus, if the difference between the natural numbers taken is 3, then the number sequence of the difference between the two natural numbers shall be formed by adding 3 natural numbers.
This may also be verified with respect to the other columns.
Table 2 shows the above relationship between the differences of the squares of the natural numbers.
Now the above relation shall be generalized as
“If a – b = k > 1, then a2 – b2 shall be written as the sum of ‘k’ natural numbers”
As Column D1 elements are odd natural numbers, this property may be defined as
“If a – b = k > 1, then a2 – b2 shall be written as the sum of ‘k’ odd natural numbers”
As these odd numbers are consecutive, the property may be further precisely defined as:
“If a – b = k > 1, then a2 – b2 shall be written as the sum of ‘k’ consecutive odd natural numbers”
3.5. New Method to find the difference between squares of two natural numbers
Using the above difference – sum property, the difference between squares of two natural numbers shall be found as follows.
The property shows that, a2 – b2 is equal to sum of ‘k’ consecutive odd numbers. Now, the principal idea is to find those ‘k’ consecutive odd numbers.
Let us consider two natural numbers, say 7 and 10.
The difference between them 10 – 7 = 3
Thus, 102 – 72 = sum of three consecutive odd numbers.
102 – 72 = 100 – 49 = 51
Now, 51 = Sum of 3 consecutive odd numbers
i.e., 51 = 15 + 17 + 19
Let we try to find these 3 numbers with respect to either the first number, let us say, ‘a’ or the second number, say, ‘b’.
Assume, for ‘b’
As general form for odd numbers is either (2n + 1) or (2n – 1), as b 15 = 2(7) + 1 = 2b + 1
17 = 2(7) + 3 = 2b + 3
19 = 2(7) + 5 = 2b + 5
Thus, 102 – 72 shall be written as the sum of 3 consecutive odd numbers starting from 15.
i.e. starting from 2b + 1
This idea may also be applied for higher digit numbers. Let us consider two 3 digit numbers, 101 and 105. Let us find 1052 – 1012
Here the difference is 4. Thus 1052 – 1012 shall be written as the sum of 4 consecutive odd numbers.
The numbers shall be found as follows:
Here b = 101
The first odd number = 2b + 1 = 2(101) + 1 = 203
Thus, the 4 consecutive odd numbers are: 203, 205, 207, 209
So,
1052 – 1012 = 203 + 205 + 207 + 209 = 824
This shall be verified for any number of digits. Let us consider two 6 digit numbers 100519, 100521. Let us find 1005212 – 1005192
Here the difference is 2. Thus 1005212 – 1005192 shall be written as the sum of two odd numbers.
Applying the same idea,
The first odd number = 2(100519) + 1 = 201039
Thus the 2 consecutive odd numbers are: 201039, 201041
1005212 – 1005192 = 201039 + 201041 = 402080
The above result shall be verified by using other methods.
For example: 1052 – 1012
1052 – 1012 = 11025 – 10201 = 824 (Using Direct Method)
1052 – 1012 = (105 + 101) (105 – 101) = 206 x 4 = 824 (Using Algebraic Rule)
Thus, this idea shall be generalized as follows:
“a2 – b2shall be found by adding the (a – b) consecutive odd numbers starting from 2b + 1”
This shall also be found using the first term ‘a’. As a > b, let us consider (2n – 1) form of odd numbers.
From Table 1, 102 – 62 = 13 + 15 + 17 + 19 = 64
Here, 2a – 1 = 2(10) – 1 = 19
2a – 3 = 2(10) – 3 = 17
2a – 5 = 2(10) – 5 = 15
2a – 7 = 2(10) – 7 = 13
Thus, as the difference between the numbers is 4, 102 – 62 shall be written as the sum of four consecutive odd numbers in reverse order starting from 2a – 1.
Thus proceeding, this may be generalized as,
“a2 – b2shall be found by adding the (a – b) consecutive odd numbers starting from 2a – 1 in reverse order”
Finding the first number of each column
Let us check the number pattern followed by the first numbers of each column. From Table 1, the first numbers of each column are: 3, 8, 15, 24 …
Let us find the difference between elements of this sequence.
The difference between two consecutive terms of this sequence is 5, 7, 9 …
i.e. D2 – D1 = 8 – 3 = 5; D3 – D2 = 7; D4 – D3 = 9 and so on.
As D2 represents the difference between two alternate natural numbers, (say a and b) which implies that the difference between a and b is 2.
Now, 5 = 2 (2) +1
i.e. 2 times of the difference between the numbers + 1
In the same idea, D3 – D2 = 15 – 8 = 7
As D3 represents the difference between squares of the 4th and 1st natural numbers, (say a and b) which implies that the difference between a and b is 3.
Thus, 7 = 2(3) + 1
This also shows that the difference shall be found by
= 2 times of the difference between the numbers + 1
Thus,
“The first term of the each column shall be found by adding the previous column first term with 2 times of the difference between the numbers + 1”
Finding the elements row – wise
The elements of the table shall also be formed in row wise.
If we check the elements of each row, we can find that they follow a number pattern sequence with some property.
Let us consider the elements of row when N = 5: 20, 40, 60, 80
20 = 2 x 5 x 2
Here, 5 represent the row natural number.
2 represent the difference between the elements using which the column is formed.
Thus Row element = 2 x N x difference
In the same way, 40 = 2 x 5 x 4
= 2 x N x difference
Thus, the elements shall be formed by the rule:
“Row Element = 2 x N x difference”
This shall be applied for middle rows also.
For example, let us consider the row between 5 & 6:
The elements in this intermediate row are: 11, 33, 55, 77, 99
Here N is the mid value of 5 & 6. i.e. N = 5.5
Let us consider the elements and apply the above stated rule.
11 = 2 x N x difference
= 2 x 5.5 x 1
In the same way other elements shall also be formed.
Thus the elements of the table shall be formed in row wise using the stated rule.
Rhombus Rule Relation
Let us consider the elements in D2, D3 and D4.
Consider the elements in the rhombus drawn, 24, 33, 39 and 48
24 + 48 = 72
33 + 39 = 72
Thus the sums of the elements in the opposite corners are equal.
The other column elements also prove the same.
Thus, Rhombus Rule Relation:
“Sum of the elements the same row of the sequence of alternative columns is equal to the sum of the two elements in the intermediate column”
Application of the Properties in Finding the Square of a number
The square of a natural number shall be found by various methods. Here is one of the suggested methods.
This method uses nearest 10’s and 100’s to find the square of a number.
This method is also based on the algebraic formula a2 – b2 = (a – b)(a + b)
If a > b, b2 = a2 – (a2 – b2)
If b > a, b2 = a2 + (b2 – a2)
Example: Square of 32
As we need to find 322, let us assume b = 32.
The nearest multiple of 10 is 30. Let a = 30
Here b > a. b2 = a2 + (b2 – a2)
322 = 302 + (322 – 302)
Using the Difference – Sum Property,
322 = 900 + 61 + 63 = 1024
Example 2: Square of 9972
Let b = 997
Nearest multiple 10 is 1000. Let a = 1000
Here a > b, so b2 = a2 – (a2 – b2)
9972 = 10002 – (10002 – 9972)
Using Difference – Sum Property,
9972 = 1000000 – (1995 + 1997 + 1999)
= 994009
Conclusion
Though this method shall be applied to find the difference between squares of any two natural numbers, if the difference is big, it will be cumbersome. Thus, this method shall be used for finding the difference between squares of any two natural numbers where the difference is manageable. The properties shall be used for easy calculation.
This properties and approach shall be introduced in secondary school lower grade levels, to make the students to identify the number patterns. This approach will surely help the students to understand the properties of squares, difference and natural numbers. The new approach will surely help the students in developing their reasoning skills.
Limitations
As number systems, number patterns and arithmetic operations have wide applications in various fields, the above properties, rules and relations shall be further studied intensively based on the requirements. Thus, new properties and relations shall be identified and discussed with respect to other nations.