The other replies are simplifying too much. Just adding or subtracting in a loop would be far too slow.
A multiplier will find the partial products by using AND gates, and then sum them, which is very similar to long multiplication as they teach you in school. This article explains it pretty well.
Division is more complicated. It’s sort of done like long division, but apparently that is slow and there’s some magic with two’s complements that can make it faster. Honestly I don’t fully understand it yet.
That article was really good. I feel like if someone explained it to me at a pub or party I could somewhat talk about it without sounding like a total ludite.
In formal definitions of arithmetics, division can be defined via multiplication: as a simplified example with real numbers, because a ÷ 2 is the same as a × 0.5, this means that if your axioms support multiplication you’ll get division out of them for free (and this’ll work for integers too, the definition is just a bit more involved.)
Mathematicians also subtract by adding, with the same logic as with division.
a/b is the unique solution x to a = bx, if a solution exists. This definition is used for integers, rationals, real and complex numbers.
Defining a/b as a * (1/b) makes sense if you’re learning arithmetic, but logically it’s more contrived as you then need to define 1/b as the unique solution x to bx = 1, if one exists, which is essentially the first definition.
Computers don’t subtract, and you can’t just add a negative, a computer can’t interpret a negative number, it can only store a flag that the number is negative. You need to use a couple addition tricks to subtract to numbers to ensure that the computer only has to add. It’s addition all the way down.
if your axioms support multiplication you’ll get division out of them for free
this is true… except when it isn’t.
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist
if your axioms support multiplication you’ll get division out of them for free^*^
^*^certain terms and conditions may apply. Limited availability in some structures, North Korea, and Iran. Known to the state of California to cause cancer or reproductive toxicity
Right. The cells are dividing in half, which would be represented in math form by 1/0.5 = 2. Dividing by one half is the same thing as multiplying by 2, and division in general is really just a visually simplified way to multiply by a fraction of 1.
Any time you divide by some fraction of 1, you will necessarily end up with a larger number because you’re doubling that division which reverses it back into multiplication, much in the same way as a negative x negative = positive. If that makes sense.
A mathematician would not be bothered by this. A high schooler taking algebra I might be though, if you phrased it the same way this post did.
Clever and I get the joke and it made me smile. If I recall my biology from 20 years ago I think the cell makes duplicates of its chromosomes then splits apart. So you have two cells inside one membrane that separates, 2 / 1 = 2. The way I first thought about it was one cell splitting in half, so half goes to one cell, the other half with the other, 1 / .5 = 2.
In short, I think the math works out fine, but the language you use to describe it can lead to comedy gold. You could say cells reproduce by division? I don't know, I'm not a biologist or mathematician. I'm a toilet poster.
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