There have been a number of Scientific discoveries that seemed to be purely scientific curiosities that later turned out to be incredibly useful. Hertz famously commented about the discovery of radio waves: “I do not think that the wireless waves I have discovered will have any practical application.”
Are there examples like this in math as well? What is the most interesting “pure math” discovery that proved to be useful in solving a real-world problem?
Complex numbers. Also known as imaginary numbers. The imaginary number
iis the solution to√-1. And it is really used in quantum mechanics and I think general relativity as well.A complex number is just two real numbers stitched together. It’s used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.
In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn’t, then no one would use complex numbers, because computers can’t process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.
That’s not quite accurate because the two numbers have a relationship with each other. i^2 = - 1, so any time you square a complex number or multiply two complex numbers, some of the value jumps from one dimension to the other.
It’s like a vector, where sure, certain operations can be treated as if the dimensions of the vector are distinct, like a translation or scale. But other operations can have one dimension affecting the other, like rotation.
I don’t think this is really an accurate way of thinking about them. Yes, they can be mapped to a 2d plane, so you can represent them with their two real-numbered coordinates along the real and imaginary axes, but certain operations with them (eg. multiplication) can be done easily with complex numbers but are not obvious how to carry out with just grid points. (3,4) * (5,6) isn’t well-defined, but (3+4i) * (5+6i) is.
It’s used extensively in electronic circuit design (where it’s called “j”, as "i’ already meant electronic current).
Also signal processing has i or j all over it.
I’m the akshually guy here, but complex numbers are the combination of a real number and an imaginary number. Agree with you, just being pedantic.
Sure, but 1 is a real number. 😜
Yes, and 1 is also a complex number.
Of course, but 1 is the loneliest number.
2 is as bad as 1: it’s the loneliest number since the number 1.
Electromagnetics as well.