Thread: Why do we square numbers?

I understand how to square a number but I'm a little lost in my understanding as to why we would do it in certain circumstances?

For example.

If I take a number, say 20% and take another number 10% and divide 10 into 20 I get the figure 2.

You could liken this to an investment where the investment returns 20% and you would still keep that investment until it was returning 10% but not below. This would make 10% your minimum return. The resulting 2 would represent how many times the original value of the whole investment you would pay in order to achieve that 10% return. This is just hypothetical.

What would be achieved by squaring the 2 (making it 4). Is there some assumption to be made by squaring a number. If my question isn't clear then would anyone know if there is anything out there which explains the theory behind the reasons for squaring a figure and what is achieved. I often see this in maths but aren't sure what the reasons are for it.

Thanks.

Originally Posted by abdomtom

I understand how to square a number but I'm a little lost in my understanding as to why we would do it in certain circumstances?

For example.

If I take a number, say 20% and take another number 10% and divide 10 into 20 I get the figure 2.

You could liken this to an investment where the investment returns 20% and you would still keep that investment until it was returning 10% but not below. This would make 10% your minimum return. The resulting 2 would represent how many times the original value of the whole investment you would pay in order to achieve that 10% return. This is just hypothetical.

What would be achieved by squaring the 2 (making it 4). Is there some assumption to be made by squaring a number. If my question isn't clear then would anyone know if there is anything out there which explains the theory behind the reasons for squaring a figure and what is achieved. I often see this in maths but aren't sure what the reasons are for it.

Thanks.

Well, for starters I would have thought the idea was pretty useful for finding the area of things like squares, for example.

Hi, thanks for the response.
It is indeed useful for that but I'm trying to work out if squaring a number (say, in the example I gave), would say something about the result of 4. For example, if I originally would pay 2 times as much for an investment because my minimum return was only half of what it is generating now then what would squaring the number achieve. I'm thinking of a specific example but it's far to extensive for me to put on a forum. The general concept is there.

I've tried to think of reasons for say, Squaring a Standard Deviation to get the Variance. It not only changes a possibly negative number to a positive figure but it more than doubles that figure as well. What meaning does squaring give the result?

Thanks

Originally Posted by abdomtom

Hi, thanks for the response.
It is indeed useful for that but I'm trying to work out if squaring a number (say, in the example I gave), would say something about the result of 4. For example, if I originally would pay 2 times as much for an investment because my minimum return was only half of what it is generating now then what would squaring the number achieve. I'm thinking of a specific example but it's far to extensive for me to put on a forum. The general concept is there.

I've tried to think of reasons for say, Squaring a Standard Deviation to get the Variance. It not only changes a possibly negative number to a positive figure but it more than doubles that figure as well. What meaning does squaring give the result?

Thanks

It is the otherway around squaring per se has no significance, you square because it does something you need done.

Aside: we do not square the standard deviation to get the variance. It is the variance which is of fundamental, we square root it to get the SD because that gives us something in the same units as the data and because a lot of things of statistical interest scale as the square root of the variance. Also, by definition SD is always non-negative.

CB

Thanks again but I'll put it another way this time.

If I have an investment which grows at 20% and I only require 10%. If I pay twice as much for that investment then I'll receive the 10% on the whole amount I invested. (never mind why I would do that). I say twice as much because 20% divided by 10% = 2.

If I take the original 20% return (the actual money I received) and re-invest it into the investment returning 20% (before I reinvested the returns) then that investment would grow. Either the numerator or the denominator would increase or both? I'm not sure.

What I do know is, if I was to square the number 2 (originally calculated) to give me 4, that number 4 now would mean that I would be paying 4 times the value for the original investment as opposed to 2. I squared the 2 because this is apparently the way to increase the original investment (by 4 in this case) when I reinvest the original 20% into that original investment earning 20%.

This is a financial fact but I don't understand why squaring works in this case. I was wondering if there's a way of analyzing this that I'm not seeing.


Thanks again.

Part of the problem is that you are being very loose with operations and the meanings behind them.
Let's just try labeling some of these values.
20% = .2 is a grow rate of an investment.
10% = .1 is another growth rate (which you call "required", but that only distracts us from the issue).
.2÷.1 = 2 ...... but this is essentially meaningless, unless you are trying to compare growth rates.
We have to label the investment. Call it P (for "principal").
What do you mean by "If I pay twice as much for that investment"? That English is so ambiguous...
How about "I invest twice as much"? That way, you can just compare the original amount, P, with twice that... which is 2P.
Now what do you mean by "the original 20% return"? Is it the principal plus interest, or just the interest? Then "the original 20% return" is either 1.2P (120% of P) or just .2P
Can you see why this is confusing?
To make things worse, you then go on to say that you are re-investing the "20% return" BEFORE you "reinvested the returns".

So you're jamming a whole bunch of numbers and computations together, but you haven't been very strict with what you are talking about.

Also, when you ask about the numerator or denominator increasing, that is misguided.