What Is
Complex
Analysis?
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Hear the words "imaginary number", and plenty of people assume it's a fake number. In fact, the name was an insult. In 1637, Descartes called it "imaginary". He meant it as a put-down. The numbers we use every day all live on one straight line. Positive to the right, negative to the left.
Multiplying by minus one is an about-turn: from the right side over to the left. So is there a number that turns you only halfway, so that doing it twice is a full about-turn? There is. Multiply by it once, and you turn a right angle to point straight up. Once more, another right angle, and you land on minus one.
That right-angle turn is called i. And i times i is minus one. Now numbers aren't squeezed onto a line. They fill a whole flat sheet. Every point is a number. These are complex numbers. A function is just a machine: put a number in, get a number out.
Feed it complex numbers, and it's a complex function. In one go, it moves every point on the whole sheet to somewhere new. Let's try a little experiment: cover the sheet in tiny squares and hand it to the machine. Most machines stretch and crumple the squares, and every right angle is gone.
But one kind of smooth machine is strange. However hard it bends the net, every tiny square stays a square, and every right angle stays a right angle. Mathematicians call these analytic functions. And complex analysis is mostly the study of them. They have an even stranger habit: know how one goes in one tiny patch, and you know how it goes across the entire sheet.
Like seeing a short arc and drawing the whole circle from it. That stubbornness is exactly what makes them useful. In 1910, Joukowsky used one of these machines to turn a circle into the cross-section of an aircraft wing.
Air flowing around a circle is easy to work out. Map it across, and you know how air flows around the wing. The alternating current in your walls swings back and forth. Since 1893, engineers have pictured it as a spinning arrow and worked it out with complex numbers, where multiplying means turning. So complex analysis studies machines that move the whole sheet without breaking a single right angle.
It also hides the most famous puzzle in mathematics. In 1859, Riemann used one of these machines to look at how the prime numbers are spread out. He guessed that all of this machine's non-trivial zeros stand on one single vertical line. The first ten trillion have been checked, and they're all on the line. But to this day, nobody can prove that not a single one steps off it.
Whoever proves it wins a million dollars. What's your guess: do they all stay in line? Which math word makes your head spin? Tell me in the comments.
"Imaginary" numbers
aren't imaginary
Hear "imaginary number"
and you'd think it's fake.
The name was actually an insult. In 1545, a book by Cardano was the first to print the square root of a negative number. In 1637, Descartes called such numbers "imaginary" — and he meant it as a put-down. The name stuck.
Multiply once,
turn a right angle
Everyday numbers live on one straight line: positive to the right, negative to the left. Multiplying by minus one is an about-turn — from the right side over to the left.
Is there a number that turns you only halfway, so doing it twice is a full about-turn? Yes: multiply by it once and you turn a right angle, pointing straight up. That right-angle turn is called i.
i times i is two right angles —
an about-turn. It equals minus one.
Numbers fill
a whole sheet
Now numbers aren't squeezed onto a line — they fill a whole flat sheet. Every point on it is a number: so many steps right, so many steps up. These are complex numbers.
A machine that
moves the whole sheet
A function is just a machine: put a number in, get a number out. Feed it complex numbers and it's a complex function — in one go, it moves every point on the sheet somewhere new.
Try a little experiment: cover the sheet in tiny squares, hand it to the machine, and see what the squares turn into.
Most machines stretch and crumple the squares, and every right angle is gone.
But one kind of smooth machine is strange: however hard it bends the net, every tiny square stays a square, every right angle stays a right angle.
These are called
"analytic functions".
Complex analysis is mostly
the study of them.
See a little,
know it all
They have an even stranger habit: know how one behaves in one tiny patch, and its behaviour across the whole sheet is settled — like seeing a short arc and drawing the whole circle from it.
That stubbornness is exactly what makes them useful.
A circle becomes
an aircraft wing
In 1910, the Russian scientist Joukowsky used one of these machines to turn a circle into the cross-section of a wing. Air flowing around a circle is easy to work out; map it across, and you know how air flows around the wing.
The power in your walls
is spinning
The alternating current in your walls swings back and forth. Since 1893, engineers have pictured it as a spinning arrow and worked it out with complex numbers, where multiplying means turning.
In one line
Complex analysis studies machines that move the whole sheet without breaking a single right angle.
A million dollars
on one straight line
In 1859, Riemann used one of these machines to look at how the prime numbers are spread out. He guessed that all of its "non-trivial zeros" stand on one vertical line.
The first ten trillion have been checked, and they're all on the line. But to this day, nobody can prove not a single one steps off it. In 2000, the Clay Mathematics Institute made it one of seven "Millennium Prize Problems": prove it and win a million dollars.
and nobody knows the answer yet.
