Introduction to Mathematical Induction Alex Han

Chadwick International, 10th

“No human investigation can claim to be specific if it doesn’t pass the test of mathematical proof.”- Leonardo Da Vinci Observation is the basis of innovation. Numerous technological and mathematical theories and ideas first emerge from observing. Keen observation of the world around us is the key to constructing conjectures and hypotheses about the natural world; then, to prove them true or false would be the next step. Generally, proving a hypothesis true is much more difficult than proving something false, as one would only need to provide a counterexample to a statement to prove it false. It is impossible, however, to meticulously consider each and every case in the world in order to prove some statement true. The theory of natural selection is called a theory because although it is backed up by evidence, it is not completely proven as the definite law of nature. The statement “Everyone in the world wears glasses” would be proven false the moment you find someone who does not wear glasses. If you are in a room of 10 people, it would not be that hard to check each and every person. The moment you find someone without glasses, you have proven the statement to be false. However, what if all 10 people in the room do not wear glasses? You do not know if the statement is false, and you certainly do not know if it is true. To prove the statement true, you would have to check every person on earth if they wear glasses. In the worst case, you might have to check all 7.9 billion people. This may be fun, but realistically, it is impossible to go through every single possible case to prove the statement true. Regardless, there are numerous conjectures and theories that have been proven true, such as the Pythagorean Theorem, which states: in a right triangle, the sum of the squares of the two legs of the triangle equals to the length of the hypotenuse squared. There are various ways to prove conjectures; this article will introduce the basics of mathematical induction, which allows one to make sure that a certain statement is true in every case under

certain conditions. The specific ideas of mathematical induction will be elaborated on in later sections.

Mathematical induction is a technique of proof where you prove that a statement is true for every case. That is, say you have a statement P(x), for example. Through mathematical induction, you can prove that P(x) is true for every integer x1 by showing that P(1) is true, and for any positive integer k, if P(k) is true, P(k+1) is also true. Since you know that P(1) is true, you also know that P(2) is true, then P(3), then P(4), and so on. The starting point, or the Base Case of mathematical induction, is up to the person proving the statement. This may be hard to understand, so let’s take an analogy.

Suppose there is a penguin that wants to climb a ladder. We want to prove that the penguin can be on all steps of the ladder if we can successfully show that: 1.The penguin is on the ladder in the first place (for instance, on step 1). 2.If the penguin is on step k, the penguin can climb to the next step, step k+1. Step 1 proves a baseline where the claim holds true. Step 2 proves that one case being true can imply another, allowing the claim to hold true in infinitely many cases like a domino. Note that mathematical induction is a method of rigorously proving what has already been discovered, rather than a method of discovery. This means that mathematical induction is used to prove an already given statement, like “The penguin can be on all steps on the ladder” rather than discovering a new fact or theorem. Here is the abstract description of the steps of mathematical induction. Say you have to prove that the statement Sn is true for all positive integers n. Base Case: Prove that Sn is true when n=1. In the base case, which corresponds to step 1 in the penguin-ladder analogy, you provide a baseline for the ‘domino’ of induction to start. Where you begin depends up to you, or what the statement requires you to prove. You could set the base case at n=0, n=1, n=2, or anything as long as you can prove the statement holds.

Induction Hypothesis: Let k be a positive integer. Suppose Sn is true for n=k. That is, assume Sk is true. The induction hypothesis is the first part of the induction step, where you establish the assumption that Sn is true for some positive integer k. The condition for k can also be altered (e.g. some nonnegative integer k, some even number k, etc.). Induction Step: Assuming the induction hypothesis, prove that the claim is also true when n=k+1. That is, given Sk is true, prove that Sk+1 is also true. By the induction step, you have set up the domino of induction so that once you know the base case is true, you know all the next cases are true. The induction step can also be altered depending on the problem (e.g. prove that S2k is also true, prove that Sk+5

is also true, etc.). The theories will become much clearer when you actually try using it.

Claim: The number n3-n is divisible by 6, for all positive integers n.

Induction Hypothesis: Let k be a positive integer. Assume that the claim is true when n=k. That is, k3-k is divisible by 6. Base Case: Prove the statement is true when n=1. In this case, we can see that n3n=13-1=0, which is divisible by 6. Thus, the statement is true for the base case. Induction Step: Given the induction hypothesis, prove that the claim is true when n=k+1. That is, (k+1)3-(k+1) is divisible by k. Let’s expand the expression (k+1)3-(k+1). (k+1)3-(k+1) = k3+3k2+3k+1-k-1 = k3+3k2+2k = (k3-k)+3k2+3k By the induction hypothesis, we know that k3-k is divisible by 6. Thus, in order to show that the above is divisible by 6, we only need to show that 3k2+3k is divisible by 6. 3k2+3k=3k(k+1) k(k+1) is a multiple of 2 since one of two consecutive numbers is an even number, and 3k(k+1) is a multiple of 3 since 3 is multiplied. Thus, since 3k(k+1) is both a multiple of 2 and 3, it is a multiple of 6, (k+1)3-(k+1) is also a multiple of 6. Therefore, given that the claim is true for some positive integer k, we have proven that the claim is also true for a positive integer k+1. By the principles of mathematical induction, we have completed the proof.

Claim: Any integer n>=2 is either a prime or can be represented as a product of (not necessarily distinct) primes, i.e, in the form n=p1p2p3... pr, where pi(1ir) is a prime (Fundamental Theorem of Arithmetic, Existence part). For the sake of convenience, call this claim P(n).

Induction Hypothesis: Let k2 be an integer. Assume that the claim holds for n=k. That is, assume that P(k) is true: k can be expressed as k=p1p2p3... pr, where pi(1ir) is a prime. Base Case: Suppose that n=2. Then, the claim P(2) is true because 2 is a prime itself, thus satisfies the claim. Induction Step: Given the induction hypothesis, prove that the claim is true when n=k+1. That is, P(k+1) is true. We already know that k can be expressed as k=p1p2p3... pr, where pi(1ir) is a prime. Given this fact, try expressing k+1 in a form relevant to the induction hypothesis. k+1=p1p2p3... pr+1. How do we get from here to “k+1 is either a prime or can be represented as a product of primes”? We are stuck. In the practice problem given above, you may have noticed that it is indeed impossible to derive the statement in the induction step only given the induction hypothesis. In order to rigorously prove the claim above, we would have to utilize a different type, or a modified version of induction: strong induction.

Strong induction is another type of induction which has, as the name implies, a ‘stronger’ induction hypothesis [3]. The base case and induction step would be the same, only that the induction hypothesis makes a much bigger assumption. Suppose you need to prove that a statement Sn is true for all positive integers n. In strong induction, the base case, induction hypothesis, and induction step would be like the following.

Base Case: Prove that Sn is true when n=1. Induction Hypothesis: Let k be a positive integer. Suppose Sn is true for all positive integer nk. That is, assume that all S1, S2, S3, … up to Sk is true. In the domino of strong induction, the induction hypothesis assumes that all previous cases are true, given some k. In the penguin-ladder example, this would mean that we assume that the penguin can climb all steps up to step k. Induction Step: Assuming the induction hypothesis, prove that the claim is also true when n=k+1. That is, given S1, S2, S3, …, Sk is true, prove that Sk+1 is also true. In the penguin ladder example, we would have to prove that the penguin can climb step k+1, using that the penguin can climb steps 1, 2, 3, … k. Now, let’s go back to example 2 using strong induction. Induction Hypothesis: Let k2 be an integer. Assume that the claim holds for all nk. That is, assume that P(1), P(2), P(3), … P(k) is true. Base Case: Suppose that n=2. Then, the claim P(2) is true because 2 is a prime itself, thus satisfies the claim. Induction Step: Given the induction hypothesis, prove that the claim is true when n=k+1. That is, P(k+1) is true. We will do some casework. Case 1: If k+1 is a prime, then we are done because it satisfies the claim. Case 2: If k+1 is a composite number, then k+1 can be written as k+1=ab, for positive integers a and b which satisfy 2a, b<k+1. In other words, 2a, bk. By the induction hypothesis, we know that a and b can be a prime or can be represented as a product of primes. Multiplying a and b also results in a representation of k+1 as a product of primes. Therefore, the claim holds for n=k+1, and P(k+1) is true. By the principles of mathematical induction, we have completed the proof. It follows that P(n) is true for all n ≥ 2, i.e., every integer n ≥ 2 is either a prime or can be represented as a product of primes. Using strong induction, we could complete proofs that need an induction hypothesis that covers more cases. You may ask the question; “If strong induction gives us a stronger induction hypothesis, then why not use it in every case? Why use standard induction?” The answer to this is, it is not necessary in every case. If all you need to prove the case for n=k+1 is the assumption that the claim holds for n=k, then ordinary induction would be appropriate. Using strong induction would be using an unnecessarily large assumption when not needed.

Mathematical induction can be used to prove claims that are more practical, like say, the solution to the problem of the Tower of Hanoi [6]. The problem is as follows:

Let Tn be the minimum number of moves required for the monks to complete the task when there are initially n discs

There is a temple with three poles. One of the poles has a stack of 64 discs, where each one is smaller than the one beneath it. The monks of the temples are trying to transfer the discs from the pole to another pole. They can move only one disc at a time, and a larger disc can never be placed on top of a smaller one. How many individual moves are required to complete the task?

(nZ+). We can prove that Tn=2n-1 using mathematical induction.

Claim: Tn=2n-1 for all natural numbers n.

Induction Hypothesis: Suppose Tk=2k-1 for some natural number k. Base Case: You can simply move one disc from the original pole to another. The minimum number of moves required is 1, which is equal to 21-1=1. Therefore, the claim holds for n=1. Induction Step: We want to prove that Tk+1=2Tk+1, assuming the induction hypothesis is true. First, we can transfer the top k discs to another pole, which takes a minimum of Tk moves. Then, move the largest disc which was at the very bottom of the original pole to the leftover empty pole. This takes 1 move. Then, transfer the top k discs previously moved onto the largest disc. This takes Tk moves. Thus, it is possible to obtain that Tk=2k-1. By the induction hypothesis, we know that Tk=2k+1, and plugging this in, Tk+1 =2(2k+1)-1 = 2k+1+2-1 = 2k+1+1 Therefore, the claim is true for n=k+1, given that the claim is true for n=k. By the principles of mathematical induction, the claim is true.

Through mathematical induction, numerous theorems and statements can be proven true or false. Although the process requires a lot of assumptions thus can lead to various mistakes, when done carefully and accurately, the process of induction is a valuable method of proof.

Alternate proof for Practice Problem 1: The number n3-n is divisible by 6, for all positive integers n. Factoring the given term, we can represent n3-n as n(n+1)(n-1). This is a product of three consecutive integers. n(n+1)(n-1) is divisible by 2 because there has to be at least one even number in three consecutive integers. n(n+1)(n1) is divisible by 3 because there has to be at least one multiple of three in three consecutive integers. Since n3-n=n(n+1) (n-1) is a multiple of both 2 and 3, it is a multiple of 6. Therefore, the claim is true for all positive integers n.

[1] Andersson, P. (n.d.). PROOFS BY INDUCTION. Upenn. [2] Benedito, Staats, C., & Wibrow, M. (1962, August 1). How to make dominoes falling? TeX. https://tex. stackexchange.com/questions/149828/ how-to-make-dominoes-falling. [3] Davey, B. A. (n.d.). Proofs by Induction. Australian Mathematical Sciences Institute. [4] Depositphotos, I. (n.d.). 5,245 Crow cartoon Vectors, Royalty-free Vector Crow cartoon Images: Depositphotos®. Depositphotos. https://depositphotos. com/vector-images/crow-cartoon.html. [5] Hildebrand, A. J. (n.d.). Worksheet: Induction Proofs, II: Strong Induction. Application to Recurrences and Representation Problems. [6] Home. Report Inappropriate Project. (n.d.). https://sourceforge. net/projects/tower-of-hanoi/report_ inappropriate. [7] School Boy with Glasses Icon Vector Illustration Graphic Design. 123RF. (n.d.). www.123rf.com/photo_94205631_ stock-vector-school-boy-with-glassesicon-vector-illustration-graphic-design. html.

Peddie School, 9th

The fundamental definition of the exchange rate is ‘the value of one currency for the purpose of conversion to another’. Hence, the exchange rate means the price of one country’s currency in terms of the other. For example, 120 JPY is 1 USD (focuseconomics, 2014). The value changes when the money of one currency is expressed in another currency. The exchange rate is essential; it not only determines the value of various currencies but determines the trades of the countries. Therefore, the change in the exchange rate captures a lot of economic factors (CFI, 2021). The domestic and foreign currency value mainly impact the exchange rates and there are three central factors.

First, interest rate and inflation rate greatly impact the exchange rate because the higher interest rate in the domestic country will demand an increase in domestic currency as foreign investors seek to invest the domestic currency at the highest interest rate, but this is balanced by the inflation rate. The increase of the inflation rate in the domestic countries will lead to the decrease of the value of the domestic currency. Next, exports and imports impact the exchange rate. Exportation creates greater benefit than importing goods from foreign countries because exporting goods leads to an increase in the value of the domestic currency which increases the exchange rate. Lastly, economic decline will definitely affect the exchange rate. When a recession occurs, the interest rates will decrease which leads to less attraction from foreign investors and therefore the exchange rates will decrease. All these three factors can explain the importance of the political stability of the country, since the government actively influences the economy by producing goods and services, controlling taxes, and transferring the income. Also, political stability further attracts foreign investors due to less risk. Along with these impacts, there are constant changes in exchange rates (Labonte, 2010).

The exchange rate might seem like an unrelated issue for most people, but the indirect effects are deeply related to our daily lives. First, in this globalized era, imported products are common and sometimes even more common than domestic products. And the exchange rate has a significant effect on the prices of imported products. When the domestic country has a weaker currency, the price

of the imported goods for domestic people will increase. Therefore, when the domestic currency is strong, each individual would pay less for the imported products which is a great advantage. For example, if the euro declines 21.4% against the U.S. dollar and at the same time, the Canadian dollar tumbles 9.5% against the U.S. dollar. Then, Canadians, with a 15% appreciation against Europe, would be able to import European products at a lower price while the U.S. could import both Canadian and European products at a low price. If this happens, Europeans will have a terrible effect of importing Canadian and U.S. products at an expensive price while importing their products at a low cost. Thus, the price of imported products depends on the currency of exporting nations against the domestic currency. In reality, the U.S. dollar has topped most major countries leading to American consumers paying relatively lower price to import foreign products such as German automobiles or Japanese electronics. Next, the exchange rates can affect inflation and interest rates. If the domestic currency is weak, it can increase the inflation rate due to the higher price of foreign goods. This could lead the central bank to raise interest rates to counter inflation and to prevent currency plunging. Contrariwise, strong currency suppresses inflation and impedes the economy. Hence, the exchange rate has an indirect effect on the interest rates paid for car loans or interest received from the savings (Picardo, 2020).

Employments are also notably related to the exchange rate. Initially, the weak domestic currency boosts economic growth by increasing exports and selling the imports expensively. This coerces the customers to buy more domestic products. Rapid economic growth would lead to better employment prospects while strong domestic currency would be oppositely affected. Also, exchange rate fluctuations can have a significant impact on the investment portfolio even if holding only domestic investments. For instance, the strong dollar weakens global demand because commodity prices are generally priced in dollars. These low demands can affect the imports and values of domestic commodity producers, but some negative impacts will be mitigated by weak local currencies. Strong currencies can also affect sales and profits generated abroad. Investing in securities denominated in depreciation currency can reduce your total return while investing in securities denominated in high-value currencies can increase your total return. For example, while many European stock indices hit alltime highs in the first four months of 2015, when the dollar was very aggressively bullish, U.S. investors who invested in these stocks would actually have seen a significant decline in returns. Moreover, the exchange rates can impact real estate. Undervalued domestic currency can be like unlimited sales, all the reduction is goods, services, and the country’s assets. Only buyers capable of paying in a stronger foreign currency get the selling price because this attracts foreign tourists which is good for the domestic economy. However, not only foreign tourists but also foreign buyers seeking to spread cheap assets and outperform domestic buyers are also attracted. These foreign buyers have raised housing prices in countries with weak currencies. This leads to high housing prices and low supply that affects the rent price. Local demand for housing has been strong as central banks have kept interest rates low at all times to

revitalize the economy. This also had the effect of pushing the currency to its lowest level over the years and elevating fears of a global currency war. Lastly, the crucial impact of exchange rate fluctuation is like an iceberg that lies beneath the surface. The indirect effect exceeds the direct effect as the economy is influenced in both short and long terms. The indirect effect extends the prices in supermarkets, interest rates of loans and savings, returns on the investment portfolio, employment, and real estate (housing prices). This impact is significant on daily lives as people need goods, supplies, jobs, houses and to use banks. A seemingly unrelated exchange rate has a substantial effect on the daily lives and economic status of the general people.

As there are rapid changes in the economy, trading in any investment market is challenging as most beginners lose money. However, by education, practice, and experience people can earn money. In fact, forex (FX), the currency market, is the largest investment market worldwide and it annually continues to expand. The average daily turnover of the foreign exchange market in 2019 was $6.6 trillion, which increased by 100% since 2007. Currency exchange is available 24 hours except for the weekends. But the idea of 24 hours of trading is deceptive because there are three sessions of European, Asian, and United States trading. Despite the overlapping sessions, the main currencies of each market are traded during specific hours. This implies that certain currency pairs would get better chances of trading volumes during certain sessions. For example, traders who stay with dollars will only find the market during the United States trading session. Additionally, currencies are traded in various sized lots. The micro-lot is 1,000 units of currency. For instance, if the account funds are in U.S. dollars, a microlot would be $1,000 of the base currency. Furthermore, the mini-lot is 10,000 units and the standard-lot is 100,000 units

(Parker, 2021).

Moreover, in currency markets, every currency trading is done in pairs. It only works in buy and sell; in order to buy one currency an individual should sell one. And all currencies are usually priced in the fourth decimal place. A pip, or percentage in point, is the minimum increment of the trade. Hence, one pip means 1/100 or 1%. Trade beginners or retailed currencies are often traded in micro-lots because one pip in a microlot is only a 10-cent change in the price. Therefore, it does not make massive losses when the trade does not produce the intended result. On the other hand, one pip in a mini-lot is worth $10. In single trading sessions, some currencies are moved as 100 or more pips which results in easier management of potential losses for small investors by trading in microlots or mini-lots. Unexpectedly, there are only 18 currency pairs available for trade. Although there are other currencies than the 18, the currencies that are traded most often are the U.S. dollar (USD), Canadian dollar (CAD), euro (EUR), British pound (GBP), Swiss franc (CHF), New Zealand dollar (NZD), Australian dollar (AUD) and the Japanese yen (JPY). Despite that currency trading is not easy, the fewer options make the trades and the portfolio management easier. Besides, many stock traders often take interest in currency markets due to their relations. The main relationship is supply and demand. When there are more demands for the currency the price increases and when there are excessive amounts of the currency the price decreases. Learning about the currency trades is easy, however, the actual experiences are extremely hard. Countries such as Russia, Cuba, North Korea, India and China still impose foreign exchange controls. The governments of the listed countries restrict and control trading currencies. Majority of countries have removed the foreign exchange controls but still it is necessary for poor countries. However, some countries with vast economies choose to impose restrictive controls. Imposing exchange controls creates huge limitations as people are restricted to own or use foreign currencies. People are only allowed to exchange currencies under government’s approval and even under approval there are limits on the amount they can carry. Moreover, the exchange rates are all regulated by the government. All currency exchanges are recorded by the government using personal identifications. Fortunately, most UN member countries are not allowed to control the currencies since the Bretton Woods Conference of 1944. Also, only certain economies are allowed to impose currency controls according to the IMF’s articles and promotions.

Imposing currency control is not always a disadvantage though. It is a huge benefit for weaker economies as there are various ways to control the excessive fluctuations. There are some countries that stabilized their economy after imposing exchange controls. However, there are some risks of the government being capable of controlling currencies as weapons to wage currency wars. Currencies are supposed to naturally rise and fall but government control removes the idea of free trade. International firms directly suffer from these systems as it becomes much more complicated to monitor the exchanges under government control. Small firms from Russia are having a hard time due to these limiting systems. Therefore, “In the interest of

achieving the UN’s strategic development goals, countries which still impose currency controls should abandon these and move actively toward free markets” (Biggs, 2020).

To conclude, the exchange rate is the change in price from one to another currency. However, there are various reasons for its change such as interest rates, inflation rates, exports, imports, economy, and government stability. It greatly affects people’s daily lives like employment and prices of foreign products. Thus, the public should be aware of it. Further, there are governments imposing currency control which is not recommended but only for countries with extremely weak economies. Additionally, despite the challenging trades of the popular currency market, plenty of experiences can help making great profits.

Biggs, J. (2021, March 8). Foreign exchange controls – And why countries still impose foreign exchange controls. Meldium. https://www.meldium. com/foreign-exchange-controls-andwhy-countries-still-impose-foreignexchange-controls/#:~:text=What's%20 common%20between%20 Russia%2C%20Cuba,and%20selling%20 of%20foreign%20currency.

Exchange rate - oveview, how it works, importance. (2021, February 18). Retrieved March 11, 2021, from https:// corporatefinanceinstitute.com/resources/ knowledge/economics/exchange-rate/

FocusEconomics. (2014, March 29). What is an exchange rate? - definition, data and forecasts. Retrieved March 11, 2021, from https://www.focus-economics. com/economic-indicator/exchange-rate

Labonte, M. (n.d.). The Size and Role of Government: Economic Issues. Retrieved June 14, 2010, from https://fas. org/sgp/crs/misc/RL32162.pdf

Parker, T. (2021, March 04). The basics of currency trading. Retrieved March 11, 2021, from https://www. investopedia.com/financial-edge/0412/ the-basics-of-currency-trading.aspx

Picardo, E. (2020, September 16). Understand the indirect effects of exchange rates. Retrieved March 11, 2021, from https://www.investopedia. com/articles/forex/053115/understandindirect-effects-exchange-rates. asp#:~:text=Exchange%20rates%20 have%20a%20significant,foreign%20 goods%20to%20some%20extent.