Calculating Similarity: Comparing 0.00010 and 0.0037 Using Powers of Ten

[eraemia]

"Power of Ten"?

Homework Statement

How do I calculate whether 0.00010 and 0.0037 are "within a power of ten" of each other?

Homework Equations

10^1?

The Attempt at a Solution

I simply don't understand the language within a power of ten. What is a power of ten? 10^x?

 

[Doc Al]

Find their ratio.

For example: A number between 2.2 and 220 is within a power of 10 (10 times smaller or 10 times bigger) of the number 22.

 

[mgb_phys]

To specify large or small numbers we use powers of ten.
A power means how many times a number is multiplied by itself, so 4 is 2 to the power 2, 100 is 10 to the power 2 ie 10*10, a 1000 is 10 to the 3, ie 10*10*10.
It is much easier to read a billion as 10^9 rather than 1,000,000,000.
A negative number means divide by 10, so 0.1 = 10^-1 ie 1/10, 0.0001 is 10^-3, 1/(10*10*10)

To quote a number in powers of ten, also know as standard form or scientific notation, move the decimal point to after the first digit and count the number of places (number of x10) you have moved it.

so 123456.678 = 1.2345678 * 10000 = 1.2345678*10*10*10*10*10 =1.2345678^10^5
with numbers smaller than one just use a negative sign.
0.00001234 = 1.234 /10000 = 1.234 / 10*10*10*10 = 1.234^10-4

This is also important to show accuracy, if I say something is 1000kg do I mean exactly 1000.0 or do I mean roughly 1000?
With this notation I can say either 1*10^3 (between 500 and 1500) or 1.000*10^3 (between 999.5 and 1000.5)

 

[eraemia]

Okay, so

0.00010 * 10 = 0.0010
0.00010 / 10 = 0.000010

0.0037 * 10 = 0.037
0.0037 / 10 = 0.00037

Now, 0.00010 is not within a power of ten of 0.0037, because 0.00010 < 0.00037.

Neither is 0.0037 within a power of ten of 0.00010, because 0.0037 is > 0.0010.

I don't know if that's correct or not...

I still don't understand, I'm afraid... You showed that 2.2 and 220 are within a power of ten of ANOTHER number. But how do I show that the above two numbers are a power of ten WITHIN EACH OTHER? Thanks so much for the help. I know I'm stupid...

 

[Doc Al]

Okay, so

0.00010 * 10 = 0.0010
0.00010 / 10 = 0.000010

0.0037 * 10 = 0.037
0.0037 / 10 = 0.00037

Now, 0.00010 is not within a power of ten of 0.0037, because 0.00010 < 0.00037.

Good!

Neither is 0.0037 within a power of ten of 0.00010, because 0.0037 is > 0.0010.

Good. Realize that these checks are equivalent, so you only need to check once. (If A is 10x B, then B is 1/10x A.)

I don't know if that's correct or not...

I still don't understand, I'm afraid... You showed that 2.2 and 220 are within a power of ten of ANOTHER number. But how do I show that the above two numbers are a power of ten WITHIN EACH OTHER?

Just the way you did! To be within a power of 10 of 0.0037, a number must be within 0.00037 and 0.037.

The easy way: Find the ratio of the numbers. If the ratio is between 1/10 and 10, then the numbers are within a power of 10 of each other.

 

[eraemia]

Thanks a lot for your help!