Lesson 3: Compare Decimals

Property

Decimals are another way of writing fractions whose denominators are powers of 10.

The “th” at the end of the name tells you that the number is smaller than one. To name a decimal, name the number to the left of the decimal, write “and” for the decimal point, name the number to the right, and state the place value of the last digit. To write a decimal from words, use “and” to place the decimal point, then write the number parts accordingly.

Examples

• To name the decimal 8.9, you say "eight and nine tenths".
• To name the decimal -12.345, you say "negative twelve and three hundred forty-five thousandths".
• To write "thirty-one and seven hundredths" as a decimal, you write 31.07.

Explanation

Decimals are just special fractions where the bottom number is 10, 100, or another power of ten. The word “and” separates the whole part from the fractional part, telling you exactly where the decimal point goes.

Property

To compare two decimals, start from the leftmost digit and compare the digits in each place value. The first place where the digits differ determines which number is greater. Use the symbols $>$ (greater than), $<$ (less than), or $=$ (equal to).

Examples

• To compare $0.528$ and $0.541$, we see the tenths are the same ($5$). Comparing the hundredths, we find that $2 < 4$, so $0.528 < 0.541$.
• To compare $0.7$ and $0.689$, we start with the tenths place. Since $7 > 6$, we know that $0.7 > 0.689$.
• To compare $3.45$ and $3.450$, we can add a trailing zero to $3.45$ to get $3.450$. Since all digits are identical, $3.45 = 3.450$.

Explanation

When comparing decimals, always begin with the largest place value on the left and move to the right. Align the decimal points to ensure you are comparing corresponding place values (ones to ones, tenths to tenths, etc.). The first pair of digits that are not equal will tell you which decimal is larger. If needed, you can add zeros to the end of a decimal without changing its value to make comparisons easier.

Property

On a horizontal number line, decimal values increase from left to right. If a decimal $a$ is to the right of a decimal $b$, then $a > b$. If $a$ is to the left of $b$, then $a < b$.

Examples

• To compare $0.547$ and $0.542$, we can look at a number line segment from $0.540$ to $0.550$. Since $0.547$ is to the right of $0.542$, we know that $0.547 > 0.542$.
• To compare $0.19$ and $0.198$, we can write $0.19$ as $0.190$. On a number line, $0.198$ is to the right of $0.190$, so $0.198 > 0.19$.

Explanation

A number line is a visual tool for ordering and comparing decimals. To compare decimals to the thousandths, it helps to "zoom in" on a small segment of the number line. By plotting the points, you can visually determine which number is greater based on its position. The number further to the right is always the larger value.