Powers of Ten and Scientific Notation

Scientific notation is a method of saving time and space by counting zeros instead of writing them out. For small numbers it doesn't save space so much, but it can be used to specify the number of significant figures in numbers that end with one or more zeros, numbers such as 5000 or 1500.

It is not really 'scientific' at all, but it is often used to describe things that are very large, such as the mass of the earth (6 x 10²² kilograms), or very small, such as the mass of an electron (9 x 10-28 kilograms).

It is also useful in resolving ambiguous zeros in determining the number of significant figures. Since our number system is based on '10' it is called a decimal system ("deci" = "ten" in Latin). Numbers are in columns with each column representing a power of ten.

On the far right is the 'ones' column, then to the left is the 'tens' column, the 'hundreds' column, and so on.

A decimal point represents the beginning of a fraction so that the first column to its right is the 'tenths' column, then to its right is the 'hundreths' column, and so on.

In this system zeros are placeholders; they represent an empty column.

Each power of ten moves the decimal point one place to the right or left.

Multiplying by ten adds a zero (2 x 10 = 20) or moves the decimal point to the right (2.72 x 10 = 27.2)

Dividing by ten subtracts a zero (20/10 = 2) or moves the decimal point to the left (27.2/10 = 2.72)

HOW TO DO IT: Zeros		Count how many zeros in the number. Make that number a power of ten. Multiply the first digit by that power of ten.
2 zeros --> 10 x 10 --> 102	100 = 1 x 102	200 = 2 x 102	300 = 3 x 102	400 = 4 x 102
3 zeros --> 10 x 10 x 10 --> 103	1000 = 1 x 103	2000 = 2 x 103	3000 = 3 x 103	4000 = 4 x 103
4 zeros --> 10 x 10 x 10 x 10 --> 104	10000 = 1 x 104	20,000 = 2 x 104	30,000 = 3 x 104	40,000 = 4 x 104

HOW TO DO IT: Decimals		Place the decimal point after the first significant digit. Count how many digits, including zeros to the right or left of the decimal point. Make that number a power of ten. Multiply the resulting number by that power of ten.
2 decimals --> 10 x 10 --> 102	176 = 1.76 x 102	241 = 2.41 x 102	327 = 3.27 x 102	496 = 4.96 x 102
3 decimals --> 10 x 10 x 10 --> 103	1762 = 1.762 x 103	2417 = 2.417 x 103	3274 = 3.274x 103	4963 = 4.963 x 103
4 decimals --> 10 x 10 x 10 x 10 --> 104	17,625 = 1.7625 x 104	24,178 = 2.41 x 102	32,741 = 3.2741 x 104	49,632 = 4.9632 x 104

SCIENTIFIC NOTATION AND ROUNDING

Rounding numbers in scientific notation is done the same as in decimal notation.

• Example: 249.8 is rounded to 3 significant figures to become 2.50 x 10²

• Example: the average distance from the Sun to the Earth is 93,000,000 mi.

Using powers-of-ten notation, this can be written as

93,000,000 miles = 9.3 x 10^7 mi or 9.3 x10⁷ mi

(The carat ^ is sometimes used to mean "raised to the power". So 10^2 is the same as 10² or "ten raised to the second power".)

Likewise, a very small number can be expressed using the powers-of-ten notation. For example, the thickness of a piece of paper is about 0.0001 m. This can be written as

0.0001 m = 1 x 10^-4 m or 1x10^-4 m

Note that the exponent counts the number of spaces right or left of the first significant figure.