# which number is larger?

• July 28th 2010, 05:41 AM
sfspitfire23
which number is larger?
Fellas, I have a question regarding quantitative comparisons.

Say we have, in column A, (78)(243) and in column B, (77)(244).

I'm asked to determine which is larger.

Is it a good strategy to pick out the first two numbers in each variable? So, in column A we would take (78)(24) and B would be (77)(24). Thus A is larger because 78>77.

What about A= (90,021)(100,210) and B=(90,210)(100,021)? Using my method, we have (900)(100) in A and (902)(100) in B. Thus, B is larger.

Is it possible to do this? Is there some law?
• July 28th 2010, 05:47 AM
Ackbeet
What does the notation (78)(243) mean? I mean, I normally think usual arithmetic product of the two numbers. But in number theory, you might have a different context.
• July 28th 2010, 05:48 AM
sfspitfire23
Sorry, it is the product
• July 28th 2010, 05:53 AM
Ackbeet
Ok, so I would say that, given two numbers whose sum is a constant, their product is maximized when the two numbers are closer together (global max is when they are equal). So I would say instantly that the product in Column A is greater. You can prove this using calculus: assume $x+y=\text{constant}=c.$ The goal is to maximize the product $xy=x(c-x).$ So, let

$f(x)=x(c-x).$ Then

$f'(x)=c-2x.$ Setting this equal to zero implies

$c-2x=0,$ or $x=c/2=y.$ Done.
• July 28th 2010, 05:54 AM
undefined
Quote:

Originally Posted by sfspitfire23
Fellas, I have a question regarding quantitative comparisons.

Say we have, in column A, (78)(243) and in column B, (77)(244).

I'm asked to determine which is larger.

Is it a good strategy to pick out the first two numbers in each variable? So, in column A we would take (78)(24) and B would be (77)(24). Thus A is larger because 78>77.

What about A= (90,021)(100,210) and B=(90,210)(100,021)? Using my method, we have (900)(100) in A and (902)(100) in B. Thus, B is larger.

Is it possible to do this? Is there some law?

(78)(243) = (77 + 1)(243) = 77 * 243 + 243

(77)(244) = (77)(243 + 1) = 77 * 243 + 77
• July 28th 2010, 05:57 AM
sfspitfire23
Very nice I get it, thanks!
• July 28th 2010, 06:00 AM
Also sprach Zarathustra
Try to prove the following for a, b integers greater than 1.

a*b>(a-1)*(b+1)
• July 28th 2010, 06:06 AM
Ackbeet
You're welcome. undefined and AsZ's methods are also both entirely valid. AsZ's method might be generalized by proving that

ab > (a-m)(b+m).
• July 28th 2010, 06:22 AM
undefined
Quote:

Originally Posted by Also sprach Zarathustra
Try to prove the following for a, b integers greater than 1.

a*b>(a-1)*(b+1)

You forgot to mention the restriction $\displaystyle a \le b$. (Edit: Possibly it was an intentional omission though.)