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Math Education: What Is The Number E In Math?

By Yann, published on 27/02/2019 We Love Prof - AU > Academia > Maths > Definition And Use Of The Number E In Maths!

When you think about maths probably what comes to mind is Addition and subtraction, Multiplication and division, Negative numbers, algebra, the distinctive number i,  differential equations, Geometry, Trigonometry (trig) or fractions etc. But while you are learning complex or even basic math, it is unlikely that you think about the definition or history of parts of your calculations. Because you will be busy calculating, solving and checking the math problems so that you have the right answer.

It can, however, be useful to understand the story behind the symbols, numbers and mathematical principles that you use in your everyday math class. If you find yourself with a mental block why not take a closer look into some of the histories of your equations.

Today we will take a look at the number e and understand why it is made up some of the most important numbers in mathematics.

rational numbers which can be converted into a fraction. An irrational number is a number which can not be made into a fraction and whose decimals go on infinitely. Photo Source: Unsplash

What Is An Irrational Number?

An irrational number is a number which can not be made into a fraction and whose decimals go on infinitely.

Rational numbers can be written as a fraction as you can see 1.5 can be written as 3/2 and 7 can be written as 7/1. These are rational numbers which can be converted into a fraction and whose decimals do not continue. Thus as the name suggests rational numbers are opposite to irrational numbers. Rational numbers also have a decimal development which is called periodic.

For example,

The ratio 2/7 = 0.285714285714285714 …

The digits after the decimal point are a logical and recurring sequence of decimals.

The other common irrational numbers are:

  1. Pythagorean theorem (pi and ) (3,14159265358979323846264338327950288419716939937510582 …), which has been the object of research by scholars since ancient times
  2. Golden Ratio () (1.6180339887498948482…) the digits keep going without forming a pattern.

What Is The Number E When Learning math?

The number e is a well known irrational number along with square roots and the golden ratio among others. The number e is an essential number in maths, and it is opposite to rational numbers. It has an infinite written decimal which does not repeat in any kind of pattern.

The numerical value of e truncated to 50 decimal places is:

  • 71828182845904523536028747135266249775724709369995… (Wikipedia)

The numerical value of e truncated to 100 decimal places is:

  • 71828182845904523536028747135266249775724709369995957
    49669676277240766303535475945713821785251664274…

How Useful Is The Number E Outside Of Mathematics?

e is not just a mathematical irrational number saved for fractions, mathematical models or statistics. It is also quite useful for a variety of other standards. A few of these are

  • Economics: to calculate Compound interest ( the growth of interest paid continuously.
  • Solving issues with electrical circuits and dynamic voltages
  • Decay and growth issues in biology: To measure the multiplication of living cells.
  • Newtons law of heating and cooling
  • Plane waves
  • In physics and also computer science

Basic History Of The Number E 

The number e is called Euler’s number after Leonhard Euler, who worked with the number. It was also worked on by Scottish mathematician John Napier. However, the person who discovered e seems to be up for debate in mathematical circles despite it being called by the name of Mr Euler. The number e appeared in the 17th century with the development of logarithms, with Napier’s research.

In Napier’s reference book dating from 1614, Napier presents a tool to simplify mathematical calculations: the logarithm. Of course in the 17th century, calculators and computers did not exist. But these geniuses still invented the mathematics that forms the foundation of the mathematics that we use today.

In the 3rd century BC, Archimedes who is known for many things including contributing to the creation of maths and calculus. He had found to multiply numbers written as exponents, that the exponents must be added together. Doing this allowed them to create more significant numbers than they had ever thought possible before.

Napier’s method was to extend Archimedes’ work by developing a plan for making additions instead of multiplications, subtractions instead of divisions, and division by 2 in place of extractions from square roots. The first tables of decimal logarithms with 8 decimals were invented.

The number e appeared in the 17th century The digits after the decimal point are a logical and recurring sequence of decimals. Photo Source: Unsplash

Mathematicians History Of The Study Number E 

  1. John Napier (1550-1617)
    1. Napier was a Scottish mathematician who invented logarithms. Napier wanted to reduce the multiplication of the large numbers required in astronomy and navigation. Although Napier didn’t name the number e specifically his work created the foundation by which is it known. His published books on the topic of logarithms detail this clearly.
  2. Jacob Bernoulli (1654-1705),
    1. Bernoulli was a mathematician works to find the maximum value of interest on loans by using the compound interest technique. Adding the accumulated interest to the initial amount deposited and then re-calculating to maximise the gains on the original deposit.
    2. With £1 given at a 100% interest rate with an annual capital interest calculation, the debt will be £2 by the end of the year. But, if we change just one thing which is to calculate the capital interest monthly, we will have £2.61 by the end of the year. If the interest is calculated daily, it will be even higher at £2.71.
    3. Bernoulli also recognised that compound interest stagnates as you increase the frequency of calculation. For example, the interest from calculating the interest ever second is the same as if you calculate it daily (£2.77). There are no additional gains by increasing the frequency after a certain point. This was how Bernoulli was introduced to the number
  3. Leonhard Euler (1707-1783),
    1. Euler was a Swiss mathematician who became interested in the number e with his demonstration of the irrationality of the number on the basis of continuous fraction development. The first letter ‘e’ taken from the word exponential was then given to this process. Euler determines the developmental series of e using factorization.

How To Calculate The Number E?

As we know now the number e is an irrational exponential number. Mathematically, this quality makes it a notoriously difficult figure to calculate accurately. But there are a few ways that get close to estimating the value, although these answers are never exact.

Calculation One

Calculating The Value Of e As A Limit
c=lim  n →∞(1+1/n)n

As the value of n gets bigger, you get closer to the real value of e.

Calculation Two

Calculating the value of e number using an infinite series

C = 1+1/1i+1/2i+…

! means it is factorial, for example 3! equals 3x2x1 and this is the factorial function. The more of the calculations created the closer to the real value of Euler’s number that you will get. However, you would have to continue infinitely to get the number to show accurately.

Euler's number after Leonhard Euler The number e is called Euler’s number. Photo Source: Unsplash

How To Remember The Number E?

To remember the first ten places of the number e memorizes the saying below. If you count the letters of each of the words, you will get the first ten decimals of the number e. Let’s take a look at some learning fun in this fun math quote to learn complex numbers like the number e.

Remembering the first 9-12 digits with a pattern

You can break the number up to remember the first 9 decimals.

  • 7
  • 1828
  • 1828

This is easy to remember as the 2.7 is simple and 1828 repeats twice.

Add to these 9 digits and move up to 12 digits easily by remembering the angles of an Isosceles Right Triangle.

  • 45
  • 90
  • 45

This gives you = 2.7 1828 1828 45 90 45

Remembering the first 12 digits with a phrase

  • = 2.
  • Express = 7
  • e = 1
  • remember = 8
  • to = 2
  • memorise = 8
  • a = 1
  • sentence = 8
  • to = 2
  • memorise = 8
  • this = 4
  • first = 5

This gives you = 2.71828182845

Resources To Learn Math Using The Number E For The Mathematics Student

Let’s face it for the maths student and learner there are many things to learn before gaining mastery in the subject. You must know the arithmetic, graphing equations, differential equations, equivalent fractions, scientific notation, probability, area and volume, number patterns, an exponential function, logarithm, complex numbers, etc. It is a long list.

But thankfully today we have computers, calculators, math tutors, peer reviewed textbooks, highly qualified math teachers, lesson plans, math worksheets and free online resources to support our learning and understanding. Math lessons in the past gave the student access to almost none of those resources as they didn’t exist. But even without this support some of the best and most highly skilled mathematicians came out of the past era.

The number e is a logarithmic calculation, which can be difficult to understand for a student studying in the high school math classroom, the college level maths curriculum or other grade level mathematics courses. When this kind of Mathematical concepts are introduced, it is always possible to get extra help to support your continued development.

Tutoring for math help, is one of the best options that you can have while struggling with complex math concepts. For teachers who teach you already, it could be useful to let them know that you are struggling and ask their advice on how you can progress. Because a math teacher may offer private tutoring or maybe they can help you put together a peer study group where students can support each other. So you don’t have to do it alone; Computational mathematics, other people, calculators and other math equipment, can all support your learning.

While a math tutor does cost money, it is the best investment you can make to get yourself past any difficulties. If private math tutorials or tutoring is too expensive an investment for your math curriculum however you can use free math exercises, math games like Sudoku, jigsaw puzzles, quizzes, math videos worksheets or math courses that can be found online to further your mathematics education. This interactive math, Subtract boring study routines and start counting cool math and fun math into your study routine. This is an effortless way to work at multiplying your understanding and math practice.

If you like to learn about special numbers read more about pi and its history, the curious prime numbers or the secretive perfect numbers.

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