2 squared is 4, which is more than 2. However, 0.2 squared is 0.04, which is less than 0.2. Can someone explain this apparent contradiction where a number between 0 and 1 squared becomes less than the original number but everything else (more than 1) becomes more?

Question

And I understand for example if I’m talking about meters, it’ll become square meters so the comparison is not apples to apples anymore. But in situations where there is no unit (for example, a math equation where you need to find x, whatever X is), why is this not contradictory?

In: 0

Answer 1

Consider a financial setting, where multiplying a price by 75% is replacing the price p by .75p, or (3/4)p. We call that a 25% discount, since you removed 25% of the price: .75p = p – .25p. In any case, multiplication by .75 makes the price *smaller*, not larger. And what if you applied two 25% discounts in a row? First the price changes from p to .75p, and then the new price .75p changes to .75(.75p) = (.75)^(2)p. Shouldn’t the latest price be even smaller than after the first 25% discount?

Hopefully this explains why .75p < p and .75(.75p) < .75p < p, so (.75)^(2)p < p. Now take the price p to be one dollar (or one peso, etc.), so p = 1. Then we’re left with (.75)^(2) < .75 < 1.

Answer 2

What’s the contradiction? It makes some numbers smaller, and others – bigger. There is no rule, that an operation must treat all numbers equally. Some operations may do that, but squaring is just not one of them.

Answer 3

When you square a number smaller than 1 you are multiplying a number by a number smaller than 1 so you get a number that is smaller than the original number.

When you square a number that’s larger than 1 you are multiplying a number by a number that’s larger than 1 so you get a number that is larger than the original number.