Scientific Notation

Scientific notation is a way to write very large or very small numbers so that they are easier to read and work with.  You express a number as the product of a number greater than or equal to 1 but less than 10 and an integral power of 10.  When a positive number greater than or equal to 10 is written in scientific notation, the power of 10 used is positive.  When the number is less than 1, the power of 10 used is negative.

Which is greater: 391000000000000000000 or 86400000000000000000?

To tell, you have to count all those zeros. Unless you have really good eyes, it will probably give you a headache.

Scientific notation was developed by scientists who use really really big numbers and really really small numbers all the time, and were sick of getting headaches from counting lots of zeros.

The first thing to remember is your powers of 10. (If you're confused by this table, see the pages on exponents and the properties of exponents.)

Powers of 10
10-5 = 0.00001
101 = 10
10-4 = 0.0001
102= 100
10-3 = 0.001
103 = 1,000
10-2 = 0.01
104 = 10,000
10-1 = 0.1
105 = 100,000
100 = 1
106 = 1,000,000

For positive powers of 10 , the exponent is the same as the number of zeros after the 1. The negative powers of 10 show how many places there are to the right of the decimal point.

If you counted the big numbers at the beginning of the problem, you found that 391000000000000000000 has 18 zeros. So you can write it as

391 × 1018

Much easier to read! But, to make scientific notation standard, there is a convention that the first number in the product should be greater than or equal to 1, and less than 10. So, we divide 391 by 100 (or 102) to get 3.91. Then we make up for it by multiplying the second number by 102. So, we end up with the number in scientific notation:

3.91 × 1020

The other big number at the top of the page was 8.64 × 1019. When the numbers are written in scientific notation it's much easier to compare them and do calculations.

The same thing works for small numbers, like 0.000076. First move the decimal point five points to the right to get 7.6 (which is between 1 and 10). To compensate, multiply by 10-5:

0.000076 = 7.6 × 10-5