MathematicsChapter 1: Real Numbers
Laws of the Four Operations
On the real numbers, addition and multiplication stay inside R: that is closure.
Learning objectives
- Explain Laws Of The Four Operations in simple mathematical language.
- Apply closure, commutative, associative correctly.
- Apply identities, inverses, distributive correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
Statement
On the real numbers, addition and multiplication stay inside R: that is closure. Order of two numbers does not matter: that is commutative. Grouping of three does not matter: that is associative. The additive identity is 0 (a + 0 = a) and the multiplicative identity is 1 (a x 1 = a). The additive inverse of a is -a. The multiplicative inverse of a (a != 0) is 1a; zero has none. Distributive law links the two operations: a(b + c) = ab + ac.
Where / symbols
- closure
- On the real numbers, addition and multiplication stay inside R: that is closure.
- commutative
- Order of two numbers does not matter: that is commutative.
- associative
- Grouping of three does not matter: that is associative.
- additive identity
- The additive identity is 0 (a + 0 = a) and the multiplicative identity is 1 (a x 1 = a).
- 0
- The additive identity is 0 (a + 0 = a) and the multiplicative identity is 1 (a x 1 = a).
- multiplicative identity
- The additive identity is 0 (a + 0 = a) and the multiplicative identity is 1 (a x 1 = a).
- 1
- The additive identity is 0 (a + 0 = a) and the multiplicative identity is 1 (a x 1 = a).
- additive inverse
- The additive inverse of a is -a.
- -a
- The additive inverse of a is -a.
- multiplicative inverse
- The multiplicative inverse of a (a !
Relationship
- a(b + c) = ab + ac
Overview
Group the laws so you can quote them in one line each:
Closure, Commutative, Associative
Closure: a + b and a x b are real if a and b are real. Commutative: a + b = b + a and a x b = b x a. Associative: (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c). Subtraction and division are not commutative (5 - 3 is not 3 - 5) and not associative in general.
Examples 2 + 5 = 5 + 2 · (1 + 2) + 3 = 1 + (2 + 3) · 5 - 3 != 3 - 5 (not commutative)
Identities, Inverses, Distributive
Additive identity 0 does nothing under addition. Multiplicative identity 1 does nothing under multiplication. Additive inverse -a undoes a by addition: a + (-a) = 0. Multiplicative inverse 1a undoes a by multiplication: a x (1a) = 1, but 0 has none. Distributive: a(b + c) = ab + ac, the bridge from brackets to expansion.
Examples 7 + 0 = 7 · 7 x 1 = 7 · 4 x (1/4) = 1, but 1/0 undefined
WHAT — Core idea
On the real numbers, addition and multiplication stay inside R: that is closure. Order of two numbers does not matter: that is commutative. Grouping of three does not matter: that is associative. The additive identity is 0 (a + 0 = a) and the multiplicative identity is 1 (a x 1 = a). The additive inverse of a is -a. The multiplicative inverse of a (a != 0) is 1/a; zero has none. Distributive law links the two operations: a(b + c) = ab + ac.
WHY — Why it matters
These laws let you rearrange algebra without changing value. Closure tells you the answer is still a real number. Identities name the 'do nothing' numbers. Inverses name the 'undo' numbers. Distributive is the reason you may expand or factor.
HOW — Method and examples
1. CLOSURE, COMMUTATIVE, ASSOCIATIVE: Closure: a + b and a x b are real if a and b are real. 2. IDENTITIES, INVERSES, DISTRIBUTIVE: Additive identity 0 does nothing under addition.
HOW — Answer in an exam
Zero has an additive inverse (namely 0) but no multiplicative inverse. Never write 10. Subtraction is adding the inverse; division is multiplying by the inverse. Do not claim subtraction is commutative.
Explanation
These laws let you rearrange algebra without changing value. Closure tells you the answer is still a real number. Identities name the 'do nothing' numbers. Inverses name the 'undo' numbers. Distributive is the reason you may expand or factor.
In simple words
Zero has an additive inverse (namely 0) but no multiplicative inverse. Never write 10. Subtraction is adding the inverse; division is multiplying by the inverse. Do not claim subtraction is commutative.
Key points
- closure
- commutative
- associative
- additive identity
- Closure
- Commutative
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Laws of the Four Operations
Sticky Notes · Quick Revision
WHAT
- On the real numbers, addition and multiplication stay inside R: that is closure.
Key Points
- On the real numbers, addition and multiplication stay inside R: that is closure.
- Closure: a + b and a x b are real if a and b are real.
- Additive identity 0 does nothing under addition.
- These laws let you rearrange algebra without changing value.
HOW
- 1. CLOSURE, COMMUTATIVE, ASSOCIATIVE: Closure: a + b and a x b are real if a and b are real.
- 2. IDENTITIES, INVERSES, DISTRIBUTIVE: Additive identity 0 does nothing under addition.
Examples
- 2 + 5 = 5 + 2
- (1 + 2) + 3 = 1 + (2 + 3)
- 5 - 3 != 3 - 5 (not commutative)
- 7 + 0 = 7
- 7 x 1 = 7
Common Mistake
- Zero has an additive inverse (namely 0) but no multiplicative inverse.
- Correction: Never write 10.
Exam Tip
- Zero has an additive inverse (namely 0) but no multiplicative inverse. Never write 10. Subtraction is adding the inverse; division is multiplying by the inverse. Do not claim subtraction is commutative.
60-second revision
- On the real numbers, addition and multiplication stay inside R: that is closure.
- Closure: a + b and a x b are real if a and b are real.
- Additive identity 0 does nothing under addition.
- These laws let you rearrange algebra without changing value.
- Zero has an additive inverse (namely 0) but no multiplicative inverse.
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