Inequality with modulus question

[vodkasoup]

Homework Statement

What is the maximum value of 'n' such that the modulus of pi-22/7 < 10^-n?

Homework Equations

The Attempt at a Solution

I have worked out that pi > -10^-n + 22/7 , and pi < 10ⁿ +22/7. I also know that 10^-n is equivalent to 1/10ⁿ. I do not know where to go after this however.

 

[tiny-tim]

hi vodkasoup!

What is the maximum value of 'n' such that the modulus of pi-22/7 < 10^-n?

π = 3.14159265…

22/7 = 3.14285714…​

what is π - 22/7 ?

what is it less than?

 

[vodkasoup]

Hi tiny-tim, thanks for the reply :)

Haha, I didn't even notice the 22/7 relating to pi!

So π - 22/7 is, according to my calculator, -0.001 ... which is less than 0.

I still can't quite see how to relate this to 10^-n. I'm a bit slow today, haha.

It seems that no matter how much I increase the value of n, -0.001 will always be lower.

Am I on the right track with the 1/10ⁿ thing?

 

[tiny-tim]

hi vodkasoup!

So π - 22/7 is, according to my calculator, -0.001 ... which is less than 0.

that's ok, the question asks you about the modulus, doesn't it?

I still can't quite see how to relate this to 10^-n.

what's 10^-1? what's 10^-2? what's 10^-3? …

 

[vodkasoup]

Hmm. I think I may be a bit unclear on the meaning of 'modulus'. Is it the 'distance' to 0 of a given point?

If that is so, would the maximum value of n be 2, because 1 / 10 to the third power is 0.001, which does not satisfy the inequality?

 

[tiny-tim]

Hmm. I think I may be a bit unclear on the meaning of 'modulus'. Is it the 'distance' to 0 of a given point?

yes, modulus is the magnitude

so, for a number x, the modulus, |x|, is always positive (or zero)

If that is so, would the maximum value of n be 2, because 1 / 10 to the third power is 0.001, which does not satisfy the inequality?

yes

 

[vodkasoup]

Thanks so much for your help tiny-tim! Sorry I took so long to reply here; I've been so busy over the last couple of days. Very much appreciate your help.