70 Multiplication and Division

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Identity, Grouping, and Signs

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Let s review how signs work in multiplication and division Then we ll proceed to the three major laws that govern the interplay between multiplication, division, addition, and subtraction

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Notation for multiplication When you want to multiply two numbers, you can use the familiar times sign and put the numerals for the factors on either side This symbol ( ) looks like a tilted cross or a letter x Another symbol you ll often see is the small, elevated dot ( ) When a number is multiplied by a variable, or when a variable is multiplied by another variable, you ll see their symbols run together without any space between Parentheses are placed around complicated expressions when they are multiplied by each other

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When you see 3 7, it means 3 times 7 When you see 3 7, it means 3 times 7 When you see 4a, it means 4 times a When you see ab, it means a times b When you see abc, it means a times b times c When you see a(b c), it means a times (b c) When you see (a + b)(c + d ), it means (a + b) times (c + d )

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Notation for division In this book, we ll use the forward slash (/) to indicate division In arithmetic, you sometimes see the dash with two dots ( ), but that s rarely used in algebra When expressions are complicated, the dividend (the number you want to divide) can be placed on top of a long horizontal line, and the divisor (the number you divide by) is placed underneath As with multiplication, parentheses are placed around complicated expressions when they are divided by each other

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When you see 8/2, it means 8 divided by 2 When you see 4/a, it means 4 divided by a When you see a/( 4), it means a divided by 4 When you see a/b, it means a divided by b When you see a/(b c), it means a divided by (b c) When you see (a + b)/(c + d ), it means (a + b) divided by (c + d )

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The identity element You can multiply or divide any integer by 1, and it won t change the value Because multiplying or dividing by 1 always gives you the same number again, 1 is called the multiplicative identity element (For some reason I ve never heard it called the divisive identity element, but technically this term is okay) For any integer a

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a1 = a

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Identity, Grouping, and Signs

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and a/1 = a

The sign-changing element When you multiply or divide any integer by 1, you reverse the sign but do not change the absolute value A positive integer becomes negative, and a negative integer becomes positive, but the distance from 0 on the integer line stays the same For any integer a

a( 1) = a and a/( 1) = a Conversely, a( 1) = a and a/( 1) = a Note that a( 1) here means a times 1, not a minus 1 Those parentheses are important! The integer 1 can be called the multiplicative sign-changing element or the divisive sign-changing element

Parentheses in simple products and quotients Look at these expressions:

3 ( 5) = a and 15/( 3) = b If you don t write the parentheses, the above expressions are 3 5 = a and 15/ 3 = b Expressions like these might be clear enough to you, but they would confuse some people Don t be afraid to add parentheses to an expression if you think they will prevent ambiguity Just be sure that for every opening parenthesis you put in, you include a corresponding closing parenthesis later in the expression