Repeating Decimals
Decimals with a repeating block are rational and convertible to fractions.
Key idea
Decimals with a repeating block are rational and convertible to fractions.
In depth
When long division produces a remainder it has seen before, the digits cycle forever. Denominators with prime factors other than 2 and 5 give repeating decimals. A bar over the repeating block shows the cycle.
Worked example
1/7 = 0.142857 142857..., a cycle of length 6. To convert 0.333...: let x = 0.333..., 10x − x = 3, x = 1/3.
Formula
Real-world uses
- Understanding why 1/3 = 0.333… in digital calculations
- Recognising rounding limits in computing
Common mistake
Believing 0.999... is less than 1. They are equal: x = 0.999..., 10x − x = 9, x = 1.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Repeating Decimals?
Decimals with a repeating block are rational and convertible to fractions. It is a topic at the advanced level in Arithmetic & Number Sense.
What is the formula for Repeating Decimals?
The key formula is 0.333… = 1/3.
Where is Repeating Decimals used in real life?
Real-world examples: Understanding why 1/3 = 0.333… in digital calculations; Recognising rounding limits in computing.
What should I learn before Repeating Decimals?
Make sure you are comfortable with Fractions to Decimals, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Repeating Decimals?
Believing 0.999... is less than 1. They are equal: x = 0.999..., 10x − x = 9, x = 1.
How can I practise Repeating Decimals?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Repeating Decimals?
Move on to Percentages next.
Related advanced topics in Arithmetic & Number Sense
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.
- Square Roots of Non-squares – Approximate or simplify surds like √50 = 5√2.