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
Squaring is multiplying a number by itself. Think of it in more everyday terms. What’s half of a half? A quarter. That’s 0.5^2. Likewise, you can raise 1 to any power and it’ll always be 1 because it’s multiplied by 1.
Try it with fractions instead of decimals. Squaring anything less than a whole number you are multiplying a fraction by a fraction, which will result in a smaller number.
For example:
1/4 * 1/4
You are finding a quarter of a quarter, which is 1/16. The denominator is getting larger but the numerator remains the same.
Squaring a number is essentially multiplying that number by itself. So, this not a squaring issue, but more of a multiplication issue. Whenever you multiply by between 0 and 1, you get a smaller number.
When you multiply a whole number by a whole number, you get a bigger whole number. When you multiply a fraction by a fraction, you get a bigger fraction. However, a bigger fraction is technically a smaller number. 0.5 x 0.5 is 0.25. 0.25 is more fractured of a number than 0.5, but that makes it numerically less. Or in fraction form, 1/2 x 1/2 = 1/4. In this case, then numerator of the fraction become larger. However, larger numerator create a number of lower value.
It makes sense in the real life too. Let’s say you have half a pie. John then gets half of that half, so what does John get? That’s a quarter. By multiplying a half by a half, you have less than half.
When you are multiplying any number by a number less than 1, the result will necessarily be less than the original number.
You might find it easier to think of squaring as multiplying, and multiplying by a fraction (or a decimal) as dividing (by its reciprocal). So squaring a decimal is the same as taking a number and dividing it by a number, which naturally results in a smaller number.
When you square something, it’s shorthand for saying “MULTIPLY IT BY ITSELF”. So think that way.
2 squared is 2 multiplied by 2, that’s “two twos”. 3 squared is 3 multiplied by 3, that’s “three threes”.
***When you multiply any positive number by something bigger than 1, you get a larger number.*** So squaring any number bigger than 1 results in a larger number.
Now let’s look at 0.2. It’s actually one-fifth. When you square it, you’re saying 0.2 * 0.2. That’s the same as saying “one fifth of one fifth”.
A fifth of *anything* is SMALLER than that thing. And that’s what you’re doing.
How about 0.5? That’s actually one half. Why you square it, you’re saying 0.5 * 0.5 or “one half of one half”. Once again, half of anything is SMALLER than that thing.
***When you multiply any positive number by something between 0 and 1, you get a smaller number.*** So squaring any number between 0 and 1 results in a smaller number.
well, we can try to describe how is that there is no cobtradiction:
we ask when does a number squared is less than itself:
x²<x
x² – x < 0
x(x-1)<0
and restricting only over positive values of x we see this
x-1 < 0 (dividing by x)
x < 1
so we see that when x is less than 1, x² is less than x
What do you think the contradiction here is, exactly?
Well think of it as a square.
+—-+—-+
| X | | A
+—-+—-+
| | | A
+—-+—-+
A A
if you square 1/2 x 1/2 of a square (A x A), you end up with 1/4 of the original square.
vs squaring 2A x 2A = 4A^2. If A = 1/2 then 4A^2 = 1
Multiplying anything by a number between 0 and 1 will result in a smaller number. Squaring a number (or any other exponent) is just another way to write multiplication.
Perhaps it will also help to look at a graph: https://www.wolframalpha.com/input?i=plot+y+%3D+x%5E2+and+y+%3D+x+for+x+%3D+0+to+2
When x > 1, x^2 is larger than x. When x < 1, x^2 is smaller than x. When x is exactly 1, they’re equal.
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