Rational vs Irrational
Maths: Edexcel A Level Pure Maths, topic 1.1.7
Rational & Irrational Numbers
A rational number is a number that can be written as a ratio of two integers. An irrational number is a number that cannot be written as the ratio of two integers.
Rational numbers
- A rational number is a number that can be written as a ratio of two integers.
- In general, any decimal that ends after a number of digits (such as 7.3 or −1.2684) is a rational number.
- We can use the place value of the last digit as the denominator when writing the decimal as a fraction.
- The decimal for is . The bar over the 3 indicates that the number 3 repeats infinitely.
Rational numbers - 2
- Every rational number can be written both as a ratio of integers , where p and q are integers and q≠0, and as a decimal that stops or repeats.
- Since all integers can be written as a fraction whose denominator is 1, the integers (and so also the counting and whole numbers) are rational numbers.
Irrational numbers
- Are there any decimals that do not stop or repeat? Yes!
- The number π (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.
- We use three dots (…) to indicate the decimal does not stop or repeat.
- E.g. π=3.141592654...
- The square root of a number that is not a perfect square is a decimal that does not stop or repeat.
- These are all irrational numbers.
Summary
- If the decimal form of a number:
- Repeats or stops, the number is a rational number.
- Does not repeat and does not stop, the number is an irrational number.
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