Multiplying and Dividing Real Numbers; Properties of Real Numbers 1.4 Multiplying and Dividing Real Numbers; Properties of Real Numbers 1. Multiply integers. 2. Multiply more than two numbers. 3. Multiply rational numbers. 4. Find the multiplicative inverse of a number. 5. Divide rational numbers.

Multiply integers. Ways to Show Multiplication 2 ∙ 3 2(3) (2)(3) 2 3 Factors Product Ways to Show Multiplication 2 ∙ 3 2(3) (2)(3) 2 3 2x

Properties of Multiplication Property Example Word Form Symbolic Form Multiplicative Property of 0 0(4) = 0 5(0) = 0 The product of a number and 0 is 0. 0 ∙ a = 0 Multiplicative Identity The product of a number and 1 is that number. 4(1) = 4 1 ∙a = a Commutative Property of Multiplication Changing the order of factors does not affect the product. 3(5) = 5(3) ab = ba Associative Property of Multiplication Changing the grouping of three or more factors does not affect the product. (2 ∙ 3) ∙ 4 = 24 2 ∙ (3 ∙ 4) = 24 a(bc) = (ab)c Distributive Property of Multiplication over Addition A sum multiplied by a factor is equal to the sum of that factor multiplied by each addend. 2(3 + 4) = 2(3) + 2(4) 3(x + 2) = 3x + 3(2) a(b + c) = ab + ac

Multiplying Numbers with Different Signs 3(4) = 12 3(3) = 9 When multiplying two numbers that have different signs, the product is negative. 3(2) = 6 3(1) = 3 3(0) = 0 3(–1) = –3 3(–2) = –6 3(–3) = –9

Multiplying Numbers with the Same Sign –3(4) = –12 When multiplying two numbers that have the same sign, the product is positive. –3(3) = –9 –3(2) = –6 –3(1) = –3 –3(0) = 0 –3(–1) = 3 –3(–2) = 6 –3(–3) = 9

Multiplying Integers = –14 –2(7) 3(–5) = –15 –6(–2) = 12 5(–2) = –10 Different = answer negative 3(–5) Different = answer negative = –15 –6(–2) Same = answer positive = 12 5(–2) = –10 Different = answer negative 3(6) Same = answer positive = 18 – 3(– 8) Same = answer positive = 24

Multiplying Integers –2(–2) = +4 –2(–2) (–2) = –8 –2(–2) (–2) (–2) = +16 –2(–2) (–2) (–2) (–2) = –32 The product of an even number of negative factors is positive, whereas the product of an odd number of negative factors is negative.

Multiplying Integers –2(3)(5) = –30 3(–5)(–2) = 30 –6(–2)(–1)(–3) = 36 5(–2)(–1) = 10 2(6)(1)(2) = 24 – 7(–3)(–2) = –42

Multiply (–6)(–3)(7). a) 126 b) 126 c) –63 d) 63 1.4 Copyright © 2011 Pearson Education, Inc. 1.4

Multiply (–6)(–3)(7). a) 126 b) 126 c) –63 d) 63 1.4 Copyright © 2011 Pearson Education, Inc. 1.4

Divide integers. Same sign rules as multiplication! Dividend Quotient Divisor Dividend Divisor Quotient Quotient Divisor Dividend Division is defined in terms of multiplication. because 4 ∙ 2 = 8. Same sign rules as multiplication!

Dividing Signed Numbers When dividing two numbers that have the same sign, the quotient is positive. When dividing two numbers that have different signs, the quotient is negative.

Dividing Integers = –3 –21 ÷ 7 15 ÷ (–3) = –5 –16 ÷ (–2) = 8 = –5 = 3 Different = answer negative 15 ÷ (–3) Different = answer negative = –5 –16 ÷ (–2) Same = answer positive = 8 = –5 Different = answer negative Same = answer positive = 3 Same = answer positive = 4

Multiplying Rational Numbers 2 1 Multiplying Rational Numbers Reduce common factors. Multiply numerators. Multiply denominators.

Multiplying Rational Numbers 2 1 4 3 2 1 1 2

Multiplying Rational Numbers 1 1 3 2 3 8 5 5 2 1

Dividing Rational Numbers 2 1 Dividing Rational Numbers Change divide to times and write the reciprocal or multiplicative inverse of the divisor (second fraction) Reduce common factors. Multiply numerators. Multiply denominators.

Dividing Rational Numbers 2 1 1 3

Multiplying Decimal Numbers 1 place + 2 places 7.6 0.24 304 + 1520 1824 3 places 1.824 –1.824 Different signs = negative.

Multiplying Decimal Numbers 1 place + 1 places 6.6 0.5 330 2 places 3.30 Same signs = positive. 3.30

Dividing Decimal Numbers Move the decimal point in the divisor so it is an integer. Move the decimal point in the dividend the same number of places. Place the decimal in the quotient directly above the new decimal point in the dividend.

Dividing Decimal Numbers 44.64 ÷ (3.6) Same signs = positive. 12.4

Divide a) b) c) d) 14.7 ÷ (0.03) 49 49 490 490 1.4 Copyright © 2011 Pearson Education, Inc. 1.4

Divide a) b) c) d) 14.7 ÷ (0.03) 49 49 490 490 1.4 Copyright © 2011 Pearson Education, Inc. 1.4

Division Involving 0 Undefined. Indeterminate. What times 0 equals 3?

Division Involving 0 Undefined Undefined Indeterminate