Euler’s Identity

Euler’s Identity

Could mathematics ever be described as beautiful?

If you are a religious person then perhaps the answer might be yes because mathematics is the language that allows us to describe, with utter precision, the intricate way in which God created his universe.

If you are a scientist, then perhaps you might think so too because there are those formulae that link, most beautifully, two areas of knowledge that may have seemed hitherto unrelated.

The famous American physicist Richard Feynman seemed to think so about one formula in particular which he called “our jewel” and “the most remarkable formula in mathematics”. He was referring to Euler’s identity first expressed in 1748 by Leonhard Euler.

Leonhard Euler was born on the 15^th of April 1707, in Basel, Switzerland. His father, Paul was a Church minister so religion was an important part of his formative years. His father was friendly with the Bernoulli family and it was Johann Bernoulli, one of the foremost mathematicians of his time that convinced Paul that his son may have a greater future in the field of mathematics.

The Bernoulli family, a dynasty of famous mathematicians, has had a huge effect on the way we live our lives today as it was one of their number, Daniel, who would go on to describe what is now known as the Bernoulli Effect, the principle that enables aircraft to fly.

It was Daniel Bernoulli who, in 1727, secured Euler a post at the Imperial Russian Academy of Sciences in Saint Petersburg. Euler would eventually replace Bernoulli as the head of the mathematics department, but concerned about the continuing turmoil in Russia, Euler left St. Petersburg in 1741 to take up a post at the Berlin Academy.

It was while Euler was in Berlin that he published his “Introduction to Analysis of the Infinite” in which he expressed what is now known as Euler’s formula, a special case of which lead to Euler’s identity.

Strangely enough, the special number “e”, known as Euler’s number and used in Euler’s formula was not actually discovered by Euler himself. It was discovered, instead by another of the Bernoulli clan, Jacob Bernoulli who came across it whilst working on the principle of compound interest.

Jacob Bernoulli found that if he deposited $1 at the beginning of the year and awarded himself a total of 100% interest over that year, no matter how many times he divided the interest payments throughout the year, he couldn’t get past a grand total of $2.72 by the end of the year.

The exact number is actually: 2.718281828459045235360287471352662497757…

It was Euler who named this number “e” after his own name and it has been that symbol that has stuck.

But how is “e” actually calculated. The humble calculator knows only 4 mathematical operations: addition, subtraction, multiplication and division.

The number “e” can actually be calculated by an infinite series of additions, multiplications and divisions like this:

• We start with 1.
• We then add 1 divided by 1 factorial. This gives us 2.
• We then add 1 divided by 1 x 2, otherwise known as 2 factorial. This  gives us 2.5.
• We then add 1 divided by 1 x 2 x 3 (or 3 factorial). This  gives us 2. 666
• We then add 1 divided by 4 factorial. This  gives us 2.708
• We then add 1 divided by 5 factorial. This  gives us 2.716

As we add more and more terms into the equation, we get closer and closer the precise value of Euler’s number. The beautiful thing about this equation is it is very easy to predict what the next term will be as each time, the denominator factorial is increasing by one.

Now e, as we have written it here, is like writing e to the power of 1. If we wanted to make this more general and raise e to the power of x, then the infinite series changes to:

• 1
• plus x over 1 factorial
• plus x-squared over 2 factorial
• plus x-cubed over 3 factorial
• plus x to the 4  over 4 factorial
• … and so on.

Now it just so happens that e^x is not the only thing that can be calculated by such an infinite series. The trigonometric functions Sine and Cosine can also be calculated by infinite series.

• Sine of X is equal to:
• X over 1 factorial
• Minus x-cubed over 3 factorial
• Plus x-to the 5 over 5 factorial
• Minus x-to the 7 over 7 factorial
• … and so on.

Again it is easy to predict the continuation of this series as each time the denominator is increasing by 2 as is the power to which we are raising x. The sign of each term keeps alternating between a plus and a minus.

Something similar happens with the Cosine of x too.

• X
• Minus x-squared over 2 factorial
• Plus x-to the 4 over 4 factorial
• Minus x-to the 6 over 6 factorial
• Plus x-to the 8 over 8 factorial
• … and so on.

Again, like the Sine series, the denominator and power terms increase by 2 each time only this time, instead of starting from 1, they begin from 2.

The infinite series for Cosine, Sine and e look so similar to each other, that you might be mistaken for thinking that there is a relationship between them. If we were to add the Cosine and Sine series together, would that give us Euler’s number?

Well, unfortunately not. There is a little problem. The problem is that the signs between the terms in the Sine and Cosine series keep changing whereas in the series which calculates Euler’s  number, they don’t. If only there was some way we could sort this out, some number we could multiply the x term by to make the two equations equal each other.

In order to make the equations equal, the something we would need to multiply x by when it was squared, would have to equal -1. The same would be true when x was cubed, again when x was raised to the power of 6 and 7 and so on. The problem is, when we square a number, the result is ALWAYS positive, not negative. What we need is a number, that when squared, is equal to -1. The problem is, such a number doesn’t exist.

The brilliant thing about mathematicians is that when they are on their way to some wonderful mathematical discovery, they don’t let a little thing like “numbers not existing” stop them. After all, mathematics is the science of numbers, so if a number, rather inconveniently, doesn’t exist… well, they jolly well go out and invent one.

As it happens, Euler’s identity wasn’t the only place where it would be useful to have a squared number equaling -1. The same problem was happening all over mathematics.

And so was born the imaginary number “i”. “i”, is the only number that, when squared, gives a negative result, namely: -1.

By now, you might have tried plugging the number -1 into your calculator and hit the square root button. Your calculator (unless it’s a very clever one) will have gallantly refused to give you an answer. Maybe the word “Error” is currently showing on its readout. That’s because your calculator is a rational piece of electronics and cannot deal with numbers which don’t exist.

However, in the eyes of a mathematician, just because a number doesn’t exist, doesn’t mean that it cannot be useful. What happens if we take Euler’s number and raise it, not to the power of x, but to i (the square root of -1) times x?

This will give us:

• 1 plus ix
• Plus ix-squared over 2 factorial
• Plus ix-cubed over 3 factorial
• Plus ix-to the 4 over 4 factorial
• …and so on.

…but we just said that i is the square root of -1. Therefore, if we square it we will get -1. Just look at that term over there: We have just THAT!  i-squared times x-squared so we could rewrite these brackets as minus x-squared.

Now i-cubed is the same as writing i-squared times i, so that term could be written as i² (which is equal to minus 1) times i times x-cubed.

i-to the power 4 is the same as writing i-squared times i-squared, so -1 times -1 equals 1. So that term here is just x-to the power 4.

This is beginning to look like the series we worked out for cos(x) + sin(x).

Now we have some terms that are real (i.e. are not multiplied by i) and some terms that are imaginary (the terms that are multiplied by i). Let’s highlight the real terms in blue and the imaginary terms in red.

Now let’s move everything around so that we group the real and imaginary terms together.

All the imaginary terms are multiplied by i so we can factor i out.

Now the real terms are the terms we used to calculate cosine of x

And the imaginary terms are the terms we used to calculate sine of x. It’s just we’ve multiplied them all by “i”.

So there is a relationship between Euler’s number and cosine and sine, it’s just we have to raise e to the power of ix and multiply the sine term by i in order to make the maths work.

This is Euler’s formula which we mentioned at the beginning of this video.

I said that a special case of Euler’s formula lead to Euler’s identity which the physicist Richard Feynman found so beautiful. That special case occurs when x is equal to pi.

Cosine of pi is equal to -1 and sine of pi is equal to zero, so e to the i pi is equal to -1.

Rearranging this slightly gives us Euler’s identity:

e to the i pi plus 1 equals zero.

What is so amazing about this little identity is that it links together, in one simple relationship, four completely different concepts in mathematics. Euler’s number, Pi, The Cosine and Sine functions and the imaginary number i.

And THAT, in mathematical terms, is beautiful!