Lesson 46: Writing Decimal Numbers in Expanded Notation

New Concept

We can express decimal numbers in expanded notation and mentally multiply them by powers of ten by shifting the decimal point.

This is how we write a decimal in expanded notation:

When multiplying by 10 or 100, we get the same effect by shifting the decimal point one or two places to the right.

What’s next

This is just the start. Next, you'll apply these concepts with worked examples and practice problems to build your skill and speed.

Property

We write a number like $4.025$ in expanded notation this way: $(4 \times 1) + \left(2 \times \frac{1}{100}\right) + \left(5 \times \frac{1}{1000}\right)$. The zero is usually not included.

Examples

$2.05 = (2 \times 1) + \left(5 \times \frac{1}{100}\right)$
$(7 \times 10) + \left(8 \times \frac{1}{10}\right) = 70.8$
$0.205 = \left(2 \times \frac{1}{10}\right) + \left(5 \times \frac{1}{1000}\right)$

Explanation

Think of expanded notation as deconstructing a decimal! You're breaking it down into a sum of its parts. Each digit gets multiplied by its specific place value, like ones, tenths, or hundredths. It’s a super clear way to see the true value behind each digit. Zeros are just placeholders, so we often leave them out.

Property

To multiply a decimal by $10$, shift the decimal point one place to the right. To multiply by $100$, shift the decimal point two places to the right.

Examples

$0.35 \times 10 = 3.5$
$2.5 \times 100 = 250$
$0.125 \times 100 = 12.5$

Explanation

Forget complex calculations! Multiplying a decimal by $10$ or $100$ is just a simple shift. While the digits are actually moving one or two places to the left to get bigger, the easiest way to see it is by moving the decimal point to the right. It’s the same result, but way faster for your brain!

Property

You can simplify a fraction with decimals by multiplying the numerator and denominator by the same power of 10. For example: $\frac{1.2}{0.4} \times \frac{10}{10} = \frac{12}{4}$.

Examples

$\frac{1.5}{0.5} = \frac{1.5}{0.5} \times \frac{10}{10} = \frac{15}{5} = 3$
$\frac{2.5}{0.05} = \frac{2.5}{0.05} \times \frac{100}{100} = \frac{250}{5} = 50$
$\frac{6}{0.3} = \frac{6}{0.3} \times \frac{10}{10} = \frac{60}{3} = 20$

Explanation

Who wants to divide by a decimal? Not me! Here’s a brilliant shortcut. By multiplying both the top and bottom of the fraction by the same power of 10, you can slide those pesky decimal points away. Since you’re essentially multiplying by $1$ (like $\frac{10}{10}$), the value stays the same, but the problem becomes much easier!