# Multiplying decimals

Multiplying decimals is similar to dividing decimals in that we want to use our understanding of decimals as a base 10 numeration system to allow us to simplify the multiplication problem while still accounting for the decimal.

To multiply decimals:

- Count the number of significant digits after the decimal point in all of the numbers being multiplied. Call this number n. By significant, we mean that the digit is either non-zero, or is a zero that affects the value of the number
- Now multiply all the numbers as you would normally, ignoring the decimal points
- Count n digits from the right of the product, then write a decimal point. This is the solution

Examples

1. Solve 25 × 0.5:

- There is 1 digit after the decimal point in 0.5, and 0 after 25, so n=1. Technically we could add infinite 0s after the decimal point, but they would not be significant 0s since 25 is the same as 25.0, which is the same as 25.000, and so on. For a zero to be significant, there needs to be at least one non-zero digit that follows it.
- 25 × 0.5 → 25 × 5 = 125
- Since n=1, we count 1 decimal place from the right of the product, and place the decimal point:

2. Solve 10.31 × 5.723:

- There are 2 digits after the decimal point in 10.31, and 3 digits after the decimal point in 5.723, so n=5.
- 10.31 × 5.723 → 1,031 × 5,723 = 5,900,413
- Since n=5, we count 5 decimal places from the right of the product, and place the decimal point:

### Why does this work?

Each digit after the decimal point (and those before as well) represents a power of 10. When we count the number of digits after the decimal point in each number, we are essentially counting the powers of 10 involved in the multiplication problem; each digit represents 10^{1}. The steps for multiplying decimals are a simplified version of what is shown below. Using example 2 above, the following is what is actually happening, and why the steps above for multiplying decimals work:

Above, we found that n=5 is the sum of the number of significant digits after the decimal place of each number. Referencing the figure, we had to multiply 10.31 by 100 and 5.723 by 1000 to shift the decimal place before multiplying.

100 × 1000 = 100,000 = 10^{5}

So, since we changed the problem by multiplying the numbers by the equivalent of 10^{5}, or 100,000, we then have to divide the solution we found by 100,000, which is the equivalent of shifting the decimal point 5 times from the rightmost digit.

Essentially, we are changing the values in the multiplication problem by factors of 10 to make the problem simpler to compute, then changing the solution back by the appropriate factor of 10.