1.1.7

Rational vs Irrational

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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.

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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 13\frac {1}{3} is .3\overline {.3}. The bar over the 3 indicates that the number 3 repeats infinitely.
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Rational numbers - 2

  • Every rational number can be written both as a ratio of integers pq \frac {p}{q}, 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.
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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.
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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.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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