Comparing Repeating Decimals
Comparing repeating decimals requires writing out the repeating decimal digits fully so that a direct place-by-place comparison can be made. The repeating decimal 0.3333... compared to 0.34 can only be correctly determined by expanding 0.3333 to several decimal places — revealing it is less than 0.34. This Grade 7 math skill from Saxon Math, Course 2 develops precise decimal number sense and teaches students that terminating and repeating decimals require careful place value analysis for accurate comparison.
Key Concepts
Property To accurately compare a repeating decimal with another number, you must expand the repeating decimal by writing out its digits. This allows for a direct, place by place comparison to see which number is greater.
Examples Compare $0.3$ and $0.\overline{3}$: $0.30 < 0.33...$, so $0.3 < 0.\overline{3}$ Arrange $0.\overline{6}, 0.6, 0.65$: This is $0.66..., 0.60, 0.65$. The order is $0.6, 0.65, 0.\overline{6}$ Compare $0.12$ and $0.1\overline{2}$: $0.1200 < 0.1222...$, so $0.12 < 0.1\overline{2}$.
Explanation Don't let the little bar over the numbers fool you! To find the true champion in a comparison, write out the first few decimal places for each number and put them head to head. The bigger one will be obvious!
Common Questions
How do I compare a repeating decimal to another number?
Write out several decimal places of the repeating decimal to make a direct comparison. For example, 0.3 repeating = 0.3333..., which is less than 0.34 because 3 < 4 in the hundredths place.
What is a repeating decimal?
A repeating decimal has one or more digits that repeat infinitely. For example, 1/3 = 0.3333... and 1/7 = 0.142857142857... A bar is written over the repeating block.
Is 0.3 repeating equal to 1/3?
Yes. 0.3333... = 1/3. To verify, multiply 0.3333... by 3: you get 0.9999... = 1, confirming 3 times (1/3) = 1.
How do I order a list of decimals that includes repeating decimals?
Convert repeating decimals to as many decimal places as needed to compare. Align them at the decimal point and compare digit by digit from left to right.
When do students learn about comparing repeating decimals?
Repeating decimal comparison is a Grade 7 skill. Saxon Math, Course 2 covers it in Chapter 3 alongside rational number classification.
What are common mistakes when comparing repeating decimals?
A common mistake is treating the repeating digit as if it stops at one or two decimal places. Always write several digits of the repeating pattern to ensure an accurate comparison.
How do repeating decimals connect to fractions?
Every fraction that does not simplify to a denominator with only 2s and 5s as factors produces a repeating decimal when divided. Understanding this connection builds rational number fluency.