Edward Frenkel
speaker
2,304 appearances
1 recordings
1 series
first heard Apr 2023
last heard Apr 2023
Edward Frenkel’s voice in public audio — every appearance, attributed to the second.
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That's just our past knowledge.
It's based on what our past knowledge is, and knowledge is limited.
What if we make the next step?
Today, for us mathematicians, complex numbers that we call them are not at all mysterious.
The idea is simply that
plot real numbers, that is to say all the whole numbers like 0, 1, and so on, 2, and so on, all fractions like 1 half or 3 halves or 4 over 3, but then also numbers like square root of 2 or pi, we plot them as points on the real line.
This is one of the perennial concepts even in our
very poor math curriculum at school.
But now imagine that instead of one line, you have one axis, you have a second axis, and so numbers now have two coordinates, x and y, and you associate to this point with coordinates x and y,
And the number x, which is a real number, plus y times square root of negative 1.
This is a graphical, geometrical representation of complex numbers, which is not mysterious at all.
Now, it took another 200 or 300 years for mathematicians to figure that out.
But initially, it looked like a completely crazy idea, you know.
The real part and the imaginary part.
The fact that you can add them up by adding together the real parts and the imaginary parts, that's easy.
But there is also a formula for the product, for the multiplication, which uses the fact that square root of minus one squared is minus one.
And the amazing thing is that that product, that multiplication satisfies the same rules, the same properties that are usual operation of multiplication for real numbers.
For instance, there is an inverse for every non-zero number that you can find.
Like number five has an inverse one over five.
But one plus I also has an inverse, for instance.
Showing 561–580 of 2,304 · page 29 of 116
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