How maths can help you get a good deal at the bank
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<p>Few people love mathematics.</p>
<p>A common refrain among students is, “Why do I have to learn this stuff? When will I need it?” But having a working knowledge of the basic concepts is essential in daily life as an adult.</p>
<p>We use them when counting cash, calculating mortgage payments and filling out tax returns.</p>
<p>In fact, it was financial matters such as loans, interest payments and gambling that spurred the development of a lot of early mathematics.</p>
<p>Negative numbers, for example, were needed to represent debt, and the mathematical rules for their use were worked out in India and the Islamic world by the 7th century.</p>
<p>One money problem that was carefully analysed in the 17th century concerned compound interest – a familiar enough concept today.</p>
<p>Just like modern banks, the money lenders of the day competed for customers using interest rates as incentives.</p>
<p>But when making comparisons the customer always has to be careful of the small print.</p>
<p>Interest rates are normally expressed on an annual basis.</p>
<p>For example, a simple 5% annual interest means that $100 investment becomes $105 at the end of one year.</p>
<p>But if interest is credited, say, every six months, the customer gets a higher overall annual return.</p>
<p>To keep the arithmetic simple, imagine a bank that paid 100% annual interest (that would be nice!).</p>
<p>If credited annually, that rate of interest would turn $100 into $200 at the end of the year.</p>
<p>But if credited every six months, then $50 gets credited to the account after six months, so at the end of the year the original capital has earned $100, but the $50 credited after six months will itself earn $25 interest over the second half of the year.</p>
<p>So by offering biannual compound interest, the bank would pay the customer $125 interest at the end of one year rather than $100.</p>
<p>A customer who started with $100 would now have $225 in the account.</p>
<p>If the interest is paid quarterly, the deal is even better, amounting to a little over $244 at the end of the year.</p>
<p>The more often the interest is credited, the higher the final total.</p>
<p>But it is a process of diminishing returns: the total goes up by a smaller and smaller amount the more frequently you credit the interest.</p>
<p>Crediting every day would yield a bit over $271. That is to say, the original capital will have been boosted 2.71 times.</p>
<p>All of which raises the question: what would be the upper limit to this compounding process?</p>
<p>Mathematicians were pondering this even back in the 17th century.</p>
<p>In 1683, the mathematician Jacob Bernoulli found the answer: 2.7182818… (the ellipsis indicates that this number is an unending decimal).</p>
<p>It is an <a href="https://cosmosmagazine.com/mathematics/the-square-root-of-2">irrational number</a> and, like π<span style="font-family: inherit;">, proved to be a fundamental mathematical constant that turns up in fields as diverse as accounting, physics, engineering, statistics and probability theory. </span></p>
<p><span style="font-family: inherit;">It is such an important number it is given a letter all its own: e. </span></p>
<p><span style="font-family: inherit;">Peruse any textbook on science, engineering or economics, and you will see the symbol e scattered throughout. </span></p>
<p><span style="font-family: inherit;">It is most often used in connection with “exponential growth” – a term that has entered the popular lexicon, though it is often misused. </span></p>
<p><span style="font-family: inherit;">The correct meaning refers to a specific type of rapid, runaway growth in which a quantity doubles in a fixed time, and then doubles again, and again, ad infinitum. </span></p>
<p><span style="font-family: inherit;">The population of bacteria in a dish, for example, will increase exponentially if their growth is unrestrained. </span></p>
<p><span style="font-family: inherit;"> One familiar example of exponential growth is Moore’s Law, named after Gordon Moore, co-founder of Intel. </span></p>
<p><span style="font-family: inherit;">After noticing in 1965 that the size of transistors was rapidly shrinking, which meant more of them could fit onto a computer chip, he predicted that processing power would double roughly every two years (and the price would drop by half). </span></p>
<p><span style="font-family: inherit;">Remarkably, this exponential growth has remained more or less consistent for several decades, though nobody expects it to go on forever. </span></p>
<p><span style="font-family: inherit;">And e makes a surprise appearance in less obvious places, too. </span></p>
<p><span style="font-family: inherit;">My favourite example is e’s application to the secretary problem. </span></p>
<p><span style="font-family: inherit;">Imagine there are 100 applicants </span><span style="font-family: inherit;">to be randomly interviewed </span><span style="font-family: inherit;">for a secretarial job. </span></p>
<p><span style="font-family: inherit;">At the end of each interview, the interviewer must give the applicant an irrevocable decision as to whether they’ve got the job. </span></p>
<p><span style="font-family: inherit;">It would be risky to see them all, dismissing the first 99, because the 100th interviewee would have to be given the job regardless of quality.</span></p>
<p><span style="font-family: inherit;">The conundrum is this: to maximise the probability of getting the best candidate, how many should be interviewed before selecting the first remaining candidate who trumps the ones already seen? </span></p>
<p><span style="font-family: inherit;">It turns out the answer is 100/e, or about 37. This result is worth remembering by people who like to play the dating game methodically. </span></p>
<p><span style="font-family: inherit;">So mathematical knowledge isn’t just useful at tax time. </span></p>
<p><span style="font-family: inherit;">Perhaps if more people knew maths could help them find love, more would be willing to embrace it.</span></p>
<p><em><span style="font-family: inherit;">Image credit: Shutterstock</span><span style="font-family: inherit;"></span></em></p>
<p><em><span style="font-family: inherit;">This article was originally published on <a rel="noopener" href="https://cosmosmagazine.com/science/mathematics/explore-the-potential-of-exponential-growth/" target="_blank">cosmosmagazine.com</a> by Paul Davies. </span></em></p>
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