Properties of Real Numbers

Commutative Properties

  • Commutative Property of Addition

    • Changing the order of addends does not change the sum.

    • Formula: a+b=b+aa + b = b + a

    • Example: 3+5=5+3=83 + 5 = 5 + 3 = 8

  • Commutative Property of Multiplication

    • Changing the order of factors does not change the product.

    • Formula: a×b=b×aa \times b = b \times a

    • Example: 4×7=7×4=284 \times 7 = 7 \times 4 = 28

Associative Properties

  • Associative Property of Addition

    • Changing the grouping of addends does not change the sum.

    • Formula: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

    • Example: (1+2)+3=1+(2+3)=6(1 + 2) + 3 = 1 + (2 + 3) = 6

  • Associative Property of Multiplication

    • Changing the grouping of factors does not change the product.

    • Formula: (a×b)×c=a×(b×c)(a \times b) \times c = a \times (b \times c)

    • Example: (2×3)×4=2×(3×4)=24(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24

Distributive Property

  • Multiplication distributes over addition and subtraction.

  • Formula for addition: a×(b+c)=a×b+a×ca \times (b + c) = a \times b + a \times c

  • Formula for subtraction: a×(b−c)=a×b−a×ca \times (b - c) = a \times b - a \times c

  • Example: 3×(4+5)=3×4+3×5=273 \times (4 + 5) = 3 \times 4 + 3 \times 5 = 27

Identity Properties

  • Additive Identity Property

    • Adding zero to any real number leaves the number unchanged.

    • Formula: a+0=aa + 0 = a

    • Example: 9+0=99 + 0 = 9

  • Multiplicative Identity Property

    • Multiplying any real number by one leaves the number unchanged.

    • Formula: a×1=aa \times 1 = a

    • Example: 7×1=77 \times 1 = 7

Inverse Properties

  • Additive Inverse Property

    • The sum of a number and its negative (opposite) equals zero.

    • Formula: a+(−a)=0a + (-a) = 0

    • Example: 6+(−6)=06 + (-6) = 0

  • Multiplicative Inverse Property

    • The product of a non-zero real number and its reciprocal equals one.

    • Formula: a×1a=1a \times \frac{1}{a} = 1 where a≠0a \neq 0

    • Example: 5×15=15 \times \frac{1}{5} = 1

Zero Property of Multiplication

  • The product of any real number and zero is zero.

  • Formula: a×0=0a \times 0 = 0

  • Example: 8×0=08 \times 0 = 0

Closure Properties

  • Closure Property of Addition

    • The sum of any two real numbers is a real number.

    • Formula: a+b∈Ra + b \in \mathbb{R} for all a,b∈Ra, b \in \mathbb{R}

  • Closure Property of Multiplication

    • The product of any two real numbers is a real number.

    • Formula: a×b∈Ra \times b \in \mathbb{R} for all a,b∈Ra, b \in \mathbb{R}