7.1: Scientific notation

Scientific notation is used to write very large numbers or very small numbers (close to zero) or in a compact way.

Writing very large numbers using scientific notation

For example, rather than writing 2,400,000,000, we can write 2.4 x 10⁹. What does this mean? We read “10⁹” as “10 to the power 9” or simply, “10 to the 9.” The “9” in “10⁹” is an exponent that means “multiply 10 by itself 9 times,” i.e., 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10. So, 2.4 x 10⁹ = 2.4 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 2,400,000,000. Sometimes a calculator cannot display exponents, so 2.4 x 10⁹ is written as 2.4 E 9.

The number in front of the “power of 10” part is a decimal number with 1 nonzero digit to the left of the decimal point (i.e., in the “ones” column). Thus, 2.413 x 10⁹ is a number written in scientific notation, but these numbers are not: 0.2413 x 10⁹, 24.13 x 10⁹, and 241.3 x 10⁹.

Here are some more examples of numbers written in scientific notation:

• 2.413 x 10² = 2.413 x 100 = 241.3
• 2.413 x 10³ = 2.413 x 1,000 = 2,413
• 2.413 x 10⁴ = 2.413 x 10,000 = 24,130
• 2.413 x 10⁵ = 2.413 x 100,000 = 241,300
• 2.413 x 10⁶ = 2.413 x 1,000,000 = 2,413,000

To write the decimal equivalent of the number 2.413 x 10ⁿ, move the decimal point n places to the right in 2.413, filling in zeros as needed. So, 2.4 x 10⁹ is 2,413,000,000 (since we have to fill in 9 – 3 = 6 zeros to the right of the “3”).

To write a large number in scientific notation, count the number of digits to the right of the first digit and before the decimal point. So, 2,413,000,000 is 2.413 x 10⁹ (since we can think of 2,413,000,000 as 2,413,000,000.0 and there are 9 digits to the right of 2 and before the decimal point).

Writing very small numbers using scientific notation

What happens if we continue the pattern of the five examples above in the reverse direction, i.e., with smaller and smaller numbers?

• 2.413 x 10¹ = 2.413 x 10 = 24.13
• 2.413 x 10⁰ = 2.413 x 1 = 2.413
• 2.413 x 10^–1 = 2.413 x (1/10¹) = 0.2413
• 2.413 x 10^–2 = 2.413 x (1/10²) = 0.02413
• 2.413 x 10^–3 = 2.413 x (1/10³) = 0.002413

Thus, 2.413 x 10^–9 is 2.413 x (1/10⁹) = 0.000000002413. It is easy to lose track of all those zeros in 0.000000002413, so 2.413 x 10^–9 is both a more compact way to write the number, as well as being less prone to error when used in calculations.

To write the decimal equivalent of the number 2.413 x 10^–n, move the decimal point n places to the left in 2.413, filling in zeros as needed. So, 2.413 x 10^–9 is 0.000000002413 (since we have to fill in 9 – 1 = 8 zeros to the left of the “2”).

To write a small number in scientific notation, count the number of zeros to the right of the decimal point and to the left of the first nonzero digit and add one. So, 0.000000002413 is 2.413 x 10^–9 (since there are 8 zeros to the right of the decimal point and to the left of the “2”).

The video below works through some examples of writing numbers using scientific notation.