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Mathematics · Chapter 2: Real Numbers
Properties of Real Numbers
Closure, commutative, associative, distributive, identity and inverse properties, and the properties of equality and order.
Key termRememberExampleWatch out
01Concept
When we add or multiply real numbers, they follow some simple rules. You have used them since primary class without naming them. Knowing their names helps you explain each step of your working.
Properties of addition and multiplication
| Property | Addition | Multiplication |
|---|---|---|
| Closure | a + b is a real number | a × b is a real number |
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
| Identity | a + 0 = a (0 is the additive identity) | a × 1 = a (1 is the multiplicative identity) |
| Inverse | a + (−a) = 0 (−a is the additive inverse) | a × (1/a) = 1, a ≠ 0 (1/a is the multiplicative inverse) |
Distributive property: a(b + c) = ab + ac. 7 × 102 = 7 × 100 + 7 × 2 = 714. That is how shopkeepers multiply in their heads.
Properties of equality
- Reflexive: a = a
- Symmetric: if a = b, then b = a
- Transitive: if a = b and b = c, then a = c
- Additive: if a = b, then a + c = b + c
- Multiplicative: if a = b, then ac = bc
These are the rules that let you do the same thing to both sides of an equation.
Properties of order (inequalities)
- Trichotomy: for any a and b, exactly one is true: a < b, a = b or a > b.
- Transitive: if a < b and b < c, then a < c.
- Adding: if a < b, then a + c < b + c.
- Multiplying by a positive number: if a < b and c > 0, then ac < bc.
- Multiplying by a negative number: if a < b and c < 0, then ac > bc (the sign turns round).
Closure in smaller sets
Some sets are not closed. Natural numbers are not closed under subtraction: 3 − 5 = −2 is not natural. Irrational numbers are not closed under multiplication: √2 × √2 = 2 is rational.
02Real World Examples
Mental maths at the shop
A shopkeeper working out 6 packets at Rs 99 thinks 6 × 100 − 6 = Rs 594: the distributive property.
Order of shopping
Adding prices in any order gives the same bill total: the commutative and associative properties of addition.
Balancing scales
A sabzi-wala's balance stays level if you add the same weight to both pans: the additive property of equality.
03Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Name the property used: (a) 5 + 9 = 9 + 5 (b) 3(x + 4) = 3x + 12 (c) 7 + (−7) = 0.Show answer
Answer
- (a) Commutative property of addition
- (b) Distributive property
- (c) Additive inverse
Q2.What are the additive and multiplicative inverses of −2/3?Show answer
Answer
Additive inverse: 2/3. Multiplicative inverse: −3/2.
Q3.Use the distributive property to work out 25 × 98 mentally.Show answer
Answer
25 × 98 = 25 × (100 − 2) = 2500 − 50 = 2450.
Q4.Is the set of integers closed under division? Give a reason.Show answer
Answer
No. 3 ÷ 2 = 1.5 is not an integer.
Q5.If x < 4, what can you say about −3x?Show answer
Answer
Multiplying by −3 reverses the inequality: −3x > −12.
Q6.Why does 0 have no multiplicative inverse?Show answer
Answer
Because 0 × anything = 0, never 1.
Long questions
Q7.Explain the properties of equality and show how they are used to solve 3x − 5 = 10.Show answer
Answer
If a = b we may add, subtract, multiply or divide both sides by the same number (not dividing by 0), and the two sides stay equal.
- 3x − 5 = 10
- Add 5 to both sides (additive property): 3x = 15
- Multiply both sides by 1/3 (multiplicative property): x = 5