Scientific Notation

Purpose of Scientific Notation

Often the magnitudes of numbers become so large or small that it becomes tedious to write out all of the zeros. In addition, when a number contains many zeros it becomes difficult to quickly judge the size of the number. For example, try to quickly tell which number in each set is the largest:

Set 1: 8567000000000000              35460000000000000

Set 2: 0.0000000009321                  0.00000000001834

Things could be made easier in the first set by adding commas after every three digits, but little can be done with the second set. Scientific notation helps overcome these problems with standard notation.

Using Scientific Notation

To write a number using scientific notation, record the number as a decimal and multiply it by an appropriate power of ten. For example, the following numbers can be written in scientific notation as shown:

435000 = 4.35 x 10⁵ 0.000693 = 6.93 x 10^-4

Keep the following things in mind when using scientific notation:

1 One and only one digit should be to the left of the decimal point. Furthermore, this digit must be 1-9 (i.e. not zero).

2 Every positive power of ten indicates moving the decimal to the right one position. Add zeros as necessary.

3 Every negative power of ten indicates moving the decimal to the left one position. Add zeros as necessary.

4 The greater the power of ten, the larger the number. If the powers of ten are the same, the greater the decimal, the larger the number.

With this in mind, the above sets can be written in scientific notation and compared quickly.

Set 1: 8.567 x 10¹⁵ < 3.546 x 10¹⁶ (15 < 16)

Set 2: 9.321 x 10^-10 > 1.834 x 10^-11 (-10 > -11)

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