1.1.7
Rational vs Irrational
Rational & Irrational Numbers
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
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
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
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...
- We use three dots (…) to indicate the decimal does not stop or repeat.
- 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
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.
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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