Calculating Similarity: Comparing 0.00010 and 0.0037 Using Powers of Ten

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In summary, the conversation discusses the concept of numbers being within a power of ten of each other. This means the numbers have a ratio between 1/10 and 10. To determine if two numbers, 0.00010 and 0.0037, are within a power of ten of each other, their ratio is found. The result shows that these numbers are not within a power of ten of each other.
  • #1
eraemia
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"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?
 
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  • #2
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.
 
  • #3
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)
 
  • #4
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...
 
  • #5
eraemia said:
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.
 
  • #6
Thanks a lot for your help!
 

1. How do you calculate the similarity between two numbers using powers of ten?

To calculate similarity between two numbers using powers of ten, you can first write the numbers in scientific notation. Then, compare the exponents of the powers of ten. If the exponents are the same, you can compare the numbers in front of the powers of ten. If the exponents are different, you can move the decimal point of the number with the smaller exponent to match the exponent of the larger number. Then, compare the numbers in front of the powers of ten.

2. What is the significance of using powers of ten in comparing numbers?

Powers of ten are used to represent very large or very small numbers in a compact and easily comparable format. By using powers of ten, we can focus on the magnitude of the numbers rather than the specific values, making it easier to compare numbers with different scales.

3. Can you use powers of ten to compare numbers with different units?

Yes, powers of ten can be used to compare numbers with different units. However, it is important to note that the numbers being compared must be in the same unit system (e.g. both in metric units or both in imperial units) in order for the comparison to be meaningful.

4. How do you interpret the similarity between two numbers calculated using powers of ten?

The similarity between two numbers calculated using powers of ten can be interpreted as the ratio between the two numbers. For example, if the similarity is calculated to be 10^3, it means that one number is 1000 times larger than the other. This can also be expressed as a percentage, with a similarity of 10^3 representing a 100,000% difference between the two numbers.

5. Are there any limitations to using powers of ten in comparing numbers?

While powers of ten can be a useful tool for comparing numbers, there are some limitations to keep in mind. One limitation is that powers of ten do not take into account the precision or accuracy of the numbers being compared. Additionally, powers of ten may not be the best method for comparing numbers with very small differences in magnitude, as it may not accurately reflect the true difference between the numbers.

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