Lesson 9: Understand 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

To convert from decimal notation to scientific notation:

1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, $n$, that the decimal point was moved.
3. Write the number as a product with a power of 10. If the original number is greater than 1, the power is $10^n$. If the original number is between 0 and 1, the power is $10^{-n}$.

Examples

• To convert 37,000, move the decimal 4 places to the left to get 3.7. Since the original number was greater than 1, the result is $3.7 \times 10^4$.

• To convert 0.0052, move the decimal 3 places to the right to get 5.2. Since the original number was between 0 and 1, the result is $5.2 \times 10^{-3}$.