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I construct a rectangle of 16 dots by 24 dots in my mind, then count 1, 2, 3, 4, 5, etc, till reaching 384. https://twitter.com/asusarla/status/1332450052749537281
Historically, it was a major breakthrough in mathematics, discovered in the 1960s, to recognize that this could be tackled like so:
1 * 2 = Two
6 * 4 = Twenty-four
Two + Twenty-four - (6 - 1) * (4 - 2) = Sixteen
So the answer is Two hundred + Sixteen tens + Twenty-Four = 384.
1 * 2 = Two
6 * 4 = Twenty-four
Two + Twenty-four - (6 - 1) * (4 - 2) = Sixteen
So the answer is Two hundred + Sixteen tens + Twenty-Four = 384.
This is called Karatsuba multiplication, and I'm not joking when I say it was a major breakthrough. Ordinarily, to multiply a pair of two-digit numbers, you have to do four single-digit multiplications. Here, we got away with three single-digit multiplications.
In case it wasn't clear exactly what we did, we turned (ax + b) * (cx + d) into Ex^2 + Fx + G, where E = a * c, G = b * d, and F = E + G - (b - a) * (d - c). The x is our base ten, and the a, b, c, and d are the digits of the numbers we start with.
Even for numbers with more than two digits, we can use this same Karatsuba idea to turn multiplying a pair of long numbers into three multiplications of numbers with half as many digits. And then turn each of those into three multiplications of quarter-length numbers, and so on.
When multiplying two N-digit numbers, the familiar elementary school method requires N^2 many single-digit multiplications, while this iterated Karatsuba method gets away with essentially N^(log(3)/log(2))) ≈ N^1.58 many single-digit multiplication steps.
(Plus some addition and subtraction steps, but not much more of those in either method, on top of the single-digit multiplication steps. It's often reasonable to gauge the overall time taken by just looking at the number of single-digit multiplication steps.)
Andrey Kolmogorov, a major mathematician, conjectured in 1960 that the N^2 was as good as it gets, but then his student Anatoly Karatsuba discovered this, the first time in history anyone realized how to multiply long numbers more efficiently than the elementary school way.
That's not the end of the story, though. Shortly after that, yet another Russian, Andrei Toom, realized how to generalize Karatsuba's idea to become even more efficient, bringing the exponent in N^(log(3)/log(2)) down to log(2k - 1)/log(k), which gets as close to 1 as you like.
And then people realized how to connect this to the Fast Fourier Transform to make it even more efficient, on the order of N log(N) many single-digit multiplication steps.
Like everyone one-upping each other's speed records right after Bannister proved the sub-four minute mile possible at all.
These later ideas are not so much harder than Karatsuba's idea to explain either, but I know no one cares after the first few posts of a thread like this. They're also only really a win for very long numbers, because they increase the overhead of the addition etc steps a bit.
