Rational Numbers 8 Maths — Revision Notes
Rational numbers are numbers that can be expressed as the ratio of two integers. They form an important part of the number system and have properties simil
TL;DR: Rational numbers are numbers that can be expressed as the ratio of two integers. They form an important part of the number system and have properties…
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Rational numbers are numbers that can be expressed as the ratio of two integers. They form an important part of the number system and have properties simil
Definition and Representation
- Rational number is a number that can be expressed as p/q where p and q are integers and q is not zero
- Examples include 2/3, -5/7, 4/1, and 0/5
- Every integer is a rational number since it can be written with denominator 1
- Rational numbers can be positive, negative, or zero
- Equivalent rational numbers represent the same value; 2/3 equals 4/6
Standard Form of Rational Numbers
- Standard form is when numerator and denominator have no common factor other than 1
- Denominator should be positive in standard form
- To reduce to standard form, divide both by their greatest common divisor
- Example: 8/12 in standard form is 2/3
- -2/3 is in standard form but -2/-3 should be written as 2/3
Operations on Rational Numbers
- Addition: a/b + c/d = (ad + bc)/bd
- Subtraction: a/b - c/d = (ad - bc)/bd
- Multiplication: a/b x c/d = ac/bd
- Division: a/b ÷ c/d = a/b x d/c = ad/bc
- Commutative and associative properties hold for addition and multiplication
Properties of Rational Numbers
- Closure property: sum, difference, product of rationals are rational
- Additive identity is 0; a + 0 = a for any rational a
- Multiplicative identity is 1; a x 1 = a for any rational a
- Additive inverse of a/b is -a/b; their sum is zero
- Multiplicative inverse of a/b is b/a; their product is one
Representation on Number Line
- Rational numbers can be plotted on a number line between integers
- Positive rationals are to the right of zero; negatives to the left
- Between any two rational numbers, infinitely many rational numbers exist
- Division method: divide the segment into equal parts to mark fractional positions
- 1/2 lies midway between 0 and 1
Key Terms
- Rational Number: A number that can be expressed as p/q where p and q are integers and q is not zero
- Standard Form: Rational number in form where numerator and denominator have no common factors
- Multiplicative Inverse: For a/b, the multiplicative inverse is b/a; their product equals 1
- Additive Inverse: For a number x, the additive inverse is -x; their sum equals 0
Frequently Asked Questions
Is every integer a rational number?
Yes, every integer can be expressed as a rational number with denominator 1. For example, 5 equals 5/1.
How many rational numbers exist between 0 and 1?
Infinitely many. Between any two rational numbers, infinitely more rational numbers can be found.
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