Section 10.5: Reading Scientific Notation

Property

A number is expressed in scientific notation when it is of the form:
a x 10^n
where "a" is greater than or equal to 1 and less than 10, and "n" is an integer. Scientific notation is a useful way of writing very large or very small numbers.

Examples

• For a large number like 4,000, we write it as 4 x 1000, which becomes 4 x 10^3 in scientific notation.
• For a small number like 0.004, we write it as 4 x (1/1000), which becomes 4 x 10^-3 in scientific notation.
• The population of the world, over 6,850,000,000, can be written more simply as 6.85 x 10^9.

Explanation

Think of scientific notation as a compact, secret code for huge or tiny numbers. The first number (the coefficient) holds the most important, significant digits, while the power of 10 acts as an instruction manual, telling you exactly how many places to move the decimal point to see the number's true size.

Property

A number is in scientific notation if it is expressed as the product of a number between 1 and 10 and a power of 10.

To Write a Number in Scientific Notation.

1. Locate the decimal point so that there is exactly one nonzero digit to its left.
2. Count the number of places you moved the decimal point: this determines the power of 10.

a. If the original number is greater than 10, the exponent is positive.
b. If the original number is less than 1, the exponent is negative.

Examples

• To write a large number in scientific notation: $475,000,000 = 4.75 \times 10^8$.
• To write a small number in scientific notation: $0.000082 = 8.2 \times 10^{-5}$.
• To perform a calculation: $\frac{9.6 \times 10^8}{3 \times 10^3} = (\frac{9.6}{3}) \times 10^{8-3} = 3.2 \times 10^5$.