5.1 Decimals

Decimal Basics

Decimals let you represent parts of whole numbers using the base-10 system. You already use them constantly with money, measurements, and percentages. This section covers how to read, write, convert, compare, and round decimals.

Decimals in Standard and Word Form

Every decimal has two common ways to be written:

• Standard form uses digits and a decimal point: 3.14
• Word form spells it out using place value names: three and fourteen hundredths

Notice that the word "and" marks where the decimal point goes. Everything before "and" is the whole number part, and everything after uses the place value of the last digit.

Place values to the right of the decimal point each get ten times smaller:

• Tenths place (0.1)
• Hundredths place (0.01)
• Thousandths place (0.001)

So in 0.347, the 3 is in the tenths place, the 4 is in the hundredths place, and the 7 is in the thousandths place. The word form would be three hundred forty-seven thousandths (named after the smallest place value used).

Conversion Between Decimal Forms

Decimals to fractions: Write the digits after the decimal point over the place value of the last digit, then simplify.

$0.3 = \frac{3}{10}$

$0.25 = \frac{25}{100} = \frac{1}{4}$

Fractions to decimals: Divide the numerator by the denominator.

$\frac{3}{4} = 3 \div 4 = 0.75$

Mixed numbers to decimals: Convert just the fraction part to a decimal, then add it to the whole number.

$2\frac{3}{4} = 2 + 0.75 = 2.75$

Decimals to mixed numbers: Separate the whole number from the decimal part, convert the decimal part to a fraction, and simplify.

$2.75 = 2 + \frac{75}{100} = 2\frac{3}{4}$

Decimals on Number Lines

Number lines give you a visual way to place and compare decimals. Positive decimals sit to the right of 0, and negative decimals sit to the left. The further right a number is, the greater its value. So if you plot 0.3 and 0.5, you can see directly that 0.3 < 0.5 because 0.3 is to the left of 0.5.

When placing a decimal like 0.35 on a number line, find the interval between 0.3 and 0.4, then estimate halfway between them.

Ordering Decimals

To put decimals in order from least to greatest:

1. Line up the decimal points vertically.
2. Add trailing zeros so every number has the same number of decimal places (this doesn't change the value: 0.3 = 0.30).
3. Compare digits from left to right, one place value at a time.
4. The first place where the digits differ tells you which number is greater.

For example, ordering 0.5, 0.35, 0.4, and 0.3:

Rewrite as 0.50, 0.35, 0.40, 0.30. Comparing the tenths digit first gives you: 0.30 < 0.35 < 0.40 < 0.50.

Rounding Decimals

Rounding shortens a decimal to a specific place value. Here's the process:

1. Find the digit in the place value you're rounding to.
2. Look at the digit one place to the right of it.
3. If that digit is 5 or greater, increase the rounding digit by 1 (round up).
4. If that digit is less than 5, keep the rounding digit the same (round down).
5. Drop all digits to the right of the rounding place.

Example: Round 3.14159 to the nearest hundredth.

• The hundredths digit is 4. The digit to its right is 1, which is less than 5.
• Keep the 4 and drop everything after it: 3.14

Example: Round 0.867 to the nearest tenth.

• The tenths digit is 8. The digit to its right is 6, which is 5 or greater.
• Round up: 0.9

Real-World Applications of Decimals

• Money is written in decimal form. $3.50 means 3 dollars and 50 cents, where the cents are hundredths of a dollar.
• Measurements use decimals for precision: 1.5 meters, 0.25 liters.
• Percentages convert directly to decimals by dividing by 100: 75% = 0.75.
• Discounts and taxes rely on decimal multiplication. A $50 item with a 20% discount costs $50 \times (1 - 0.20) = 50 \times 0.80 = 40$ dollars.

Looking Ahead

As you continue with decimals, you'll work with adding, subtracting, multiplying, and dividing them. You'll also encounter scientific notation, which uses decimals and powers of 10 to write very large or very small numbers (like $3.2 \times 10^6$ for 3,200,000). For now, focus on getting comfortable reading, converting, comparing, and rounding decimals, since every operation you learn later builds on these skills.