(There is another "Euler"s Formula" about Geometry, this page is about the one used in Complex Numbers)
First, you may have seen the famous "Euler"s Identity":
eiπ + 1 = 0
It seems absolutely magical that such a neat equation combines:
And also has the basic operations of add, multiply, and an exponent too!
But if you want khổng lồ take an interesting trip through mathematics, you will discover how it comes about.
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Interested? Read on!
Discovery
It was around 1740, and mathematicians were interested in imaginary numbers.
An imaginary number, when squared gives a negative result
This is normally impossible (try squaring some numbers, remembering that multiplying negatives gives a positive, and see if you can get a negative result), but just imagine that you can bởi it!
And we can have this special number (called i for imaginary):
i2 = −1
Leonhard Euler was enjoying himself one day, playing with imaginary numbers (or so I imagine!), & he took this well known Taylor Series (read about those, they are fascinating):
ex = 1 + x + x22! + x33! + x44! + x55! + ...
And he put i into it:
eix = 1 + ix + (ix)22! + (ix)33! + (ix)44! + (ix)55! + ...
And because i2 = −1, it simplifies to:
eix = 1 + ix − x22! − ix33! + x44! + ix55! − ...
Now group all the i terms at the end:
eix = ( 1 − x22! + x44! − ... ) + i( x − x33! + x55! − ... )
And here is the miracle ... The two groups are actually the Taylor Series for cos và sin:
| cos x = 1 − x22! + x44! − ... |
| sin x = x − x33! + x55! − ... |
And so it simplifies to:
He must have been so happy when he discovered this!
And it is now called Euler"s Formula.
Let"s give it a try:
The answer is a combination of a Real và an Imaginary Number, which together is called a Complex Number.
We can plot such a number on the complex plane (the real numbers go left-right, and the imaginary numbers go up-down):
Let"s plot some more!
A Circle!
Yes, putting Euler"s Formula on that graph produces a circle:
And when we include a radius of r we can turn any point (such as 3 + 4i) into reix khung by finding the correct value of x và r:
Example: the number 3 + 4i
To turn 3 + 4i into reix size we bởi vì a Cartesian lớn Polar conversion:
r = √(32 + 42) = √(9+16) = √25 = 5x = tan-1 ( 4 / 3 ) = 0.927 (to 3 decimals)So 3 + 4i can also be 5e0.927 i
It is Another Form
It is basically another way of having a complex number.
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This turns out to lớn very useful, as there are many cases (such as multiplication) where it is easier khổng lồ use the reix form rather than the a+bi form.
Plotting eiπ
Lastly, when we calculate Euler"s Formula for x = π we get:
eiπ = cos π + i sin π
eiπ = −1 + i × 0 (because cos π = −1 & sin π = 0)
eiπ = −1
And here is the point created by eiπ (where our discussion began):