Convert between ordinary and standard index form representations
Here’s an annoyingly large number:
5,974,200,000,000,000,000,000,000,000
It's the mass of the sun in grams
Here’s an annoyingly tiny number:
0.000000000000000000000000000000910938188
It’s the mass of an proton in grams
From the examples above, you can see that sometimes people can’t avoid having to work with really really big numbers, or really really tiny numbers. Our usual number system just isn’t equipped to deal with them. It’s awkward, messy, and it would be very easy to make a mistake writing out all those zeroes.
To counter this problem, we can write numbers in something known as ‘Standard Index Form.’
A standard index form number is ALWAYS written as: A x 10 ^{n}
So the mass of the earth in grams 5,974,200,000,000,000,000,000,000,000 becomes the following in standard index form:
5.9742 x 10^{28}
When turning a ridiculously large number in to a standard index form we must remember that the number infront of the decimal point must always be between 0 and 10. Secondly we must include all the significant figures in for A and finally there is always a 10 when expressing a number in standard form.
The positive power of 28 depicts how many times the decimal place was moved to the left to get to the number 5.9742.
The mass of an electron: 0.000000000000000000000000000000910938188, is written as the following in standard index form: 9.1098188 x 10^{32}
As we discussed previously when writing a number in it's standard index form we must always include all of the significant figures in the original number, make sure that the number infront of the decimal point is between the numbers 1 and 10 and finally there is also a 10 in the expression.
The negative power of 32 shows how many times decimal point was moved to the right to get to our new number of 9.1098188.
Now that we understand how to write ridiculousy large and small numbers in to standard form let's try it one more time for each.
Mass of uranus: 641,693,000,000,000,000,000,000 = 6.41693 x 10^{23}
Mass of a neutron: 0.00000000000000000000000000167262158 = 1.67262158 x 10^{27}
Now try it out by yourself, pick a really small number and a really largenumber and see if you can switch them from the ordinary and standard forms!
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