Numerical Structure Math Example 1

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Example 1

medium
Identify which properties (commutative, associative, distributive, identity, inverse) are illustrated by each equation: (a) 3+5=5+33 + 5 = 5 + 3, (b) (2×3)×4=2×(3×4)(2 \times 3) \times 4 = 2 \times (3 \times 4), (c) 7(x+2)=7x+147(x + 2) = 7x + 14, (d) −8+8=0-8 + 8 = 0.

Solution

  1. 1
    (a) 3+5=5+33 + 5 = 5 + 3: order of addition changed, result unchanged. Commutative property of addition.
  2. 2
    (b) (2×3)×4=2×(3×4)(2 \times 3) \times 4 = 2 \times (3 \times 4): grouping of multiplication changed. Associative property of multiplication.
  3. 3
    (c) 7(x+2)=7x+147(x+2) = 7x + 14: multiplication distributed over addition. Distributive property.
  4. 4
    (d) −8+8=0-8 + 8 = 0: a number plus its opposite equals the additive identity 00. Additive inverse property.

Answer

(a) Commutative; (b) Associative; (c) Distributive; (d) Additive inverse.
The structural properties of arithmetic (commutative, associative, distributive, identity, inverse) define the rules that numbers obey regardless of their specific values. Recognising these properties underlies algebra, as they allow expression manipulation with confidence.

About Numerical Structure

The underlying patterns, relationships, and algebraic properties—like commutativity and distributivity—that organize numbers into coherent systems.

Learn more about Numerical Structure →

More Numerical Structure Examples

Example 2 hard

Use the distributive property to compute [formula] mentally, and explain the structural reasoning.

Example 3 easy

Simplify using the appropriate property: (a) [formula], (b) [formula].

Example 4 medium

Show that [formula] using the structure of fraction arithmetic, and verify with [formula], [formula]

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