# Thread: which number is larger?

1. ## 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?

2. 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.

3. Sorry, it is the product

4. 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 $\displaystyle x+y=\text{constant}=c.$ The goal is to maximize the product $\displaystyle xy=x(c-x).$ So, let

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

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

$\displaystyle c-2x=0,$ or $\displaystyle x=c/2=y.$ Done.

5. 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

6. Very nice I get it, thanks!

7. Try to prove the following for a, b integers greater than 1.

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

8. 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).

9. 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 $\displaystyle \displaystyle a \le b$. (Edit: Possibly it was an intentional omission though.)