Why Does Squaring a Fraction Less Than 1 Result in a Smaller Number?

[AznBoi]

Why is that when a positive fraction with a value less than 1 is squared, the result is always smaller than the original fraction?? I find this rather weird, not mathematically, but brain teasingly. I guess I relate squares to times (2x) too much. because 1^2 is 1.. Please, can someone explain this to me using real life concepts!

 

[yenchin]

Erm? Half a half a cake is 1/4 of a cake, and 1/4 < 1/2.

 

[robert Ihnot]

Well, I remember from a Modern Algebra book:
If 0<r<1 then r^2<r by multiplication by r>0. And to go on with this, since r^3<r^2<r<1, etc., we can conclude that the limit of r^n as n goes to infinity is 0.

 

[Werg22]

Multiplication by a fraction with 1 as it numerator and another integrer as its denominator is defined as a division. Thus 1/2*1/2 is truly "one half devided by two" which is 1/4.

 

[matt grime]

Well, I remember from a Modern Algebra book:
If 0<r<1 then r^2<r by multiplication by r>0. And to go on with this, since r^3<r^2<r<1, etc., we can conclude that the limit of r^n as n goes to infinity is 0.

Not from that alone - it just shows that r^n is a decreasing sequence bounded below by zero, so it converges to something. It doesn't show it is zero. It is zero because of the extra piece of information that the limit L satisfies rL=L, thus r=1 (contradiction) or L=0.

 

[1016]

since fraction is means for division...
division means for separates something to small from big...
so once it is squared, which has boosted large way to splits and becomes tiny enough... therefore, get more smallest due to the exponentials.

 

[robert Ihnot]

matt grime: Not from that alone - it just shows that r^n is a decreasing sequence bounded below by zero, so it converges to something.

I thought about that, perhaps it requires a transfinite infinity to go to zero. Well, we can say if there is a positive value u>0 such that r^N>u for all N, then r>u^(1/N). But the limit of 1/N goes to zero. If it does not we have a epsilon>0, but we can set N>1/epsilon. Then by continuinity u^(1/N) goes to u^0 =1. This shows r>1, contradiction.

Or we have the Archimedean Principal, there is a N such that N*epsilon >1.

I believe Birkhoff and MacLane, "A Survey of Modern Algebra," solved that problem, but I don't have the book these days.

 

[AznBoi]

wah, you guys are using theorems and laws and such. The cake is good but that doesn't explain squaring much. =P If no one else can get a real life example of how this works, then I guess I'll just remember this as a rule..

 

[D H]

Why is this so counterintuitive? The product of some number and a number between 0 and 1 is smaller (closer to zero) than the original number.

What are the conditions under which x^2 &lt; |x|?
The inequality is obviously false for x=0. Thus we can assume x \ne 0.
Dividing both sides of the inequality by |x| yields the desired condition,

|x| &lt; 1

 

[robert Ihnot]

matt grime: Not from that alone - it just shows that r^n is a decreasing sequence bounded below by zero, so it converges to something.

Checking this out. In "Introduction to Real Analysis," Bartle and Sherbert do the exact problem on page 75 . They first prove that absolute value of
(x-x(0)) less than or equal to C*absolute value of a(n) where a(n) goes to zero. They here take it for granted that there is a sufficiently large N to achieve for the constant K, N>K of (e/C).

Having come that far, they make the substitution 0< b<1 =1/(1+a). They get: b^N=\frac{1}{(1+a)^N}&lt;\frac{1}{1+Na}&lt;\frac{1}{Na}. Since the final term is can be made less than a similar one for the series,1/m, which the previous theorem shows converges to 0, they are done. And again have made unspoken use of the Archimedean Principal.

So while they do recognize there is a difficulity here, your way involving the Least Upper Bound Axiom appears to be a lot simpler.

AxnBoi wah, you guys are using theorems and laws and such.

Well, it may be just as well to sidestep the whole mess, either forgetting about limit problems at this stage, or not bothering with axioms.

 

[matt grime]

I made use of the Archmidean principal. It is equivalent to the fact that an increasing sequence bounded above (or decreasing one bounded below) converges.

 

[yenchin]

wah, you guys are using theorems and laws and such. The cake is good but that doesn't explain squaring much. =P If no one else can get a real life example of how this works, then I guess I'll just remember this as a rule..

It's not counter-intuitive. There is nothing special with squaring. Squaring is just a special case. You can multiply any number between 0 and 1 with any number between 0 and 1, and the result will be less than either one of the numbers. E.g. 1/2 x 1/3 = 1/6. The "cake argument" still work here.

 

[robert Ihnot]

matt grime: It is equivalent to the fact that an increasing sequence bounded above (or decreasing one bounded below) converges.

I was kinda suspecting that.

 

[slider142]

A proper fraction (piece) of any (positive) amount is always less than the original amount. There's no reason to treat pieces that are less than a whole piece any differently.

 

[Gib Z]

Well explained slider142, and Welcome to Physicsforums.com.

 

[theperthvan]

If you times a number by x&lt;1, then it becomes smaller.

So if you square a fraction, you are multiplying it by something less than 1, so it gets smaller.

 

[DaveC426913]

wah, you guys are using theorems and laws and such. The cake is good but that doesn't explain squaring much. =P If no one else can get a real life example of how this works, then I guess I'll just remember this as a rule..

Two alternate ways of looking at this:

1] You're thinking that squaring always makes something larger.

The amount of water flow through a hose is dependent on its cross-sectional area.

A one inch diameter hose has an area of ~.8 sq inches.
A half inch diameter hose has an area of ~.2 sq inches.
The hose that is half the diameter has only one fourth the cross-section. Squaring a fractional number (such as 1/2) makes a much smaller number (1/4).



2] Forget about squaring for a moment. Or, at least, remember that squaring is merely a special case the multiplication of two separate numbers.

10 x 10 makes a large number, right?
10 x 1/2 makes a number smaller than 10, right?
1 x 1/2 makes a number smaller than 1, right?
9/10 x 1/2 makes a number smaller than 1. In fact, it makes a number smaller than 9/10. In fact, it makes a number smaller than EITHER of the two numbers.

Well 1/2 x 1/2 makes a number smaller than 1 too. In fact, it makes a number smaller than EITHER of the two numbers (which just happen to be the same).

So, don't think of squaring as anything more special than multiplying two fractions together.