Euler's Identity (Complex Numbers)

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 formerly 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 leonardo dillo leonardo doilo was born on the 15th of april 1707 in basel in 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 who convinced paul that his son may have a greater future in the field of mathematics [Music] 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 instant petersburg euler would eventually replace bernoulli as the head of the mathematics department but concerned about the continuing turmoil in russia euler left some 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 led 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 one dollar at the beginning of the year and awarded himself a total of a hundred percent 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 five three 2. 71828182845904523536 oh two eight seven four seven one it was euler who named this number e after his own name and that has been the symbol that has stuck but how is e actually calculated the humble calculator knows only four 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.

to this we add 1 divided by one factorial this gives us two we then add one divided by one times two otherwise known as two factorial this gives us two point five we then add one divided by one times two times three or three factorial this gives us two point six six six we then add one 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 to the precise value of euler's number the beautiful thing about this equation is that 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 one 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 three factorial plus x to the four over four factorial and so on now it just so happens that e to the 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 divided by 1 factorial minus x cubed over 3 factorial plus x to the 5 over 5 factorial minus x to the seven over seven factorial and so on again it's easy to predict the continuation of the series as each time the denominator is increasing by two 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 one minus x squared over two 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 two each time only this time instead of starting from one they begin from two the infinite series for cosine sine and e look so similar to each other that you might be forgiven 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's 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 out this series 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 minus 1.

the same would be true when x was cubed again when x was raised to the power of 6 and seven 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 minus one the problem is such a number doesn't exist the brilliant thing about mathematicians is that when they're 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 just jolly will go out and invent one as it happens euler's identity wasn't the only place where it would have been useful to have a squared number equally minus one 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 us a negative result namely minus one by now you might have tried plugging the number minus one into your calculator and hit the square root button your calculator unless it's a very clever one will have gunnery refuse 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 the power of i the square root of minus one times x this will give us one plus i x plus i x squared over two factorial plus i x cubed over three factorial plus i x to the four over four factorial and so on but we just said that i is the square root of minus one therefore if we square it we'll just get minus one just look at that term over there there we have i x squared so we could rewrite those brackets as minus x squared now i cubed is the same as i squared times i so that term can be written as i squared which is equal to minus 1 times i times x cubed i to the power of four is the same as writing i squared times i squared so minus one times minus one equals one so that term there is just x to the power of four this is beginning to look like the series we worked out before for cosine of x plus sine of x now we have some terms that are real i.