Math & History: Scientific Notation

I. Intro

Scientific notation (also called Standard form in Britain) is used by scientists, mathematicians, and engineers when they are working with very large or very small numbers. Numbers are written in this format:

a x 10^b

The letter a is decimal number where 1 ≤ a < 10, and b is an integer.

When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1.

II. Converting A Number To Scientific Notation

1. Move the decimal point so that the number is rewritten and contains only ONE non-zero digit to the left of the decimal point.

2. Count how many places you moved the decimal point in Step 1. This number becomes the exponent.

3. If you moved the decimal to the left, the exponent is positive. If you moved the decimal to the right, the exponent is negative.

Example:

Write 450,000 in scientific notation

The unwritten decimal point in this number follows the last zero. Move the decimal point from this position to the left and stop just before the last digit, giving you 4.5.
The decimal point was moved five places to the left, so the exponent is a positive 5. So, in scientific notation:

450,000 = 4.5 x 10⁵.

Example:

Write 0.0000000102 in scientific notation

We moved the decimal point to the right eight places so the exponent is 8. Since we moved to the right the exponent is a negative.

1.02 x 10^-8

III. Converting From Scientific Notation To A Decimal Number

To convert from scientific notation to a decimal number, simply reverse the process.

Example:

Write 4.5 x 10⁵ as a decimal number

The decimal is a positive number so we move the decimal five places to the right.

4.5 x 10⁵ = 450,000

III. Conceptual Understanding