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Remedial Algebra I25 chapters | 248 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will learn that the mathematical constant e has a name. You will also learn how it is used in mathematics along with what characteristics it has.

The number ** e** is sometimes called 'Euler's number' because Leonhard Euler, the Swiss mathematician, used this particular letter to represent this mathematical constant. I think it's easy to remember the name of this number. The letter 'e' is also the first letter in Leonhard Euler's last name!

As a mathematical constant, the number *e* remains the same value no matter what math you do to it. It will always be approximately 2.71828. I say approximately because the number *e*, like the number *pi*, is an irrational number.

Being an **irrational number**, the number *e* is a real number that goes on forever and cannot be written as the fraction of two numbers. Just like how you can't write a simple fraction to represent *pi*, you can't write a fraction to represent *e*. You have to use the symbols to accurately represent these numbers.

But we can approximate irrational numbers. Just like we approximate the number *pi* to 3.14, we approximate *e* to 2.71828. We do, however, keep in mind that the number does go on forever, and by using these approximations, our answers will also be approximations and won't be 100% accurate.

There are two ways to calculate *e*, but both involve infinity.

The first is when we calculate (1 + 1 / *n*)^*n* when *n* is infinity. Sure we can't calculate infinity; we make calculations that get progressively higher. Can we ever reach infinity? No, we can't, but we can get closer and closer to it if we keep making higher and higher calculations. As we do this, we see that we get closer and closer to the number *e*. For example, (1 + 1 / 100,000)^100,000 = 2.71827, which is correct to four decimal places. If we keep making higher calculations, our answer will be closer and closer to the value of *e*.

The second way of calculating *e* is to keep adding onto this series 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5!. . . The exclamation mark is the symbol for factorial, which means we take the number and multiply it by all the numbers it takes to count to that number starting with 1. So 5! means we multiply 1, 2, 3, 4 and 5 together. 4! means we multiply 1, 2, 3 and 4 together. The more we add, the closer we get to the value of *e*. Try it for yourself.

The number *e* has a very important use in math as the base of the natural logarithm. You might have seen equations such as *y* = ln (*x*). What this means is that if we take the number *e* to the *y* power, we will get *x* as our answer. So, ln (5) will equal 1.609, because if we take *e* to the power of 1.609 we will get 5 as our answer. On calculators, you will see a button for ln, which is this natural logarithm function.

Another famous use of the number *e* is used by bankers when they make continuous compounding calculations. What is this? Well, when you have a savings account, your interest, the money you earn, is usually calculated once a month or once a year. This is called periodic compounding. When you increase the frequency of these calculations to the point where it is continuously being calculated and you are paid continuously, it is called continuous compounding.

The formula for how much you would get after a period of time of continuous compounding is FV = PV* *e*^(Rt) where FV is the final value, PV is the amount you currently have in the bank, R is the annual percentage rate and t is your period of time. For example, if you had $10,000 in the bank, and you wanted to see how much money you would have after five years of continuous compounding with an annual percentage rate of two percent, you would use the formula and plug in your numbers to get FV = 10,000 * *e*^(0.02 * 5). Evaluating this, you get 10,000 * *e*^(0.02 * 5), which is equal to 10,000 * *e*^(0.1), which is equal to 10,000 * 1.1 = 11,051.71. So, after five years, you would have $11,051.71 in the bank. Not bad for just letting your money sit for five years.

Let's review what we've learned about the number *e*. We've learned that the number ** e** is sometimes called Euler's number and is approximately 2.71828. Like the number

After watching this lesson, you should be able to:

- Recall the source of the number
*e*and its approximate value - Calculate the value of
*e*using infinity and factorials - Apply
*e*in mathematical equations through the natural logarithm and in compounding calculations.

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Remedial Algebra I25 chapters | 248 lessons | 1 flashcard set

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