# Exploring Integers

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## Integer Laws

Exploring Integers - Part 5

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 Properties of Integers: D.  Commutative Law of Addition The order makes no difference when adding two integers. Examples:     3 + 2       and   2 + 3  both equal  5.                    -2  + 4      and  4  + -2   both equal 2.                     6 - 4  (careful here...)                     6 - 4  IS   6 + -4    and -4 + 6  both equal 2.                    -5 - 10:   -5 + -10  and -10 + -5  both equal -15. E.  Commutative Law of Multiplication The order makes no difference when multiplying two integers. Examples:     3 * 2   and 2 * 3   both equal  6.                     -2 * 4  and 4 * -2  both equal -8. F.  Associative Law of Addition When adding 3 integers together, it makes no difference which two in sequence you add first. Example:  5 + 2 + 1                add 5 + 2  first, then add 1, that is, 7 + 1 = 8           or  add  2 + 1 first, then add 5, that is, 5 + 3 = 8 Grouping symbols (), {}, and [], are used to indicate which operation should be done first.  So, we can say for the preceding example,      (5 + 2) + 1   is the same as  5 + (2 + 1)     G.  Associative Law of Multiplication When multiplying 3 integers together, it makes no difference which two in sequence you multiply first. Example:   2 * 3 * 4                 multiply 2 by 3 first, then multiply by 4, that is, 6 * 4 = 24           or   multiply 3 by 4 first, then multiply by 2, that is  2 * 12 = 24 Using grouping symbols,  (2 * 3) * 4  is the same as 2 * (3 * 4). H.  Distributive Law The product of one integer with the sum of two or more integers is the sum of the products of the first integer with the other integers, taken one at a time. Example:   2 * (3 + 4)  is the same as 2*3 + 2*4 Let's take a closer look,                 2 * (3 + 4)    the parenthesis indicate 3+4 should be done first                so we have 2 * (7)     and we can drop the parenthesis since there is no other operation to do inside the parenthesis (THIS IS MOST IMPORTANT! there is no + or - sign inside the parenthesis, if there were, the parenthesis CANNOT be removed, yet.)  So we have 2 * 7 = 14 This distributive law states that we can distribute 2 across the + sign within the parenthesis and thereby remove the parenthesis and get    2*3  + 2*4.   So which operation comes first? the * or the +?  Since multiplication is a higher order operation than addition (recall that multiplication is defined in terms of addition), we first multiply then we add.  So, we have 6 + 8 = 14. This law is usually called the distributive law of multiplication over addition to emphasize that addition must be within the grouping symbols. In other words,  2 * (3 + 4)  = 2*3 + 2*4 = 14        We cannot distribute across multiplication, that is  2 * (3 * 4) is NOT 2*3 * 2*4             2 * 12         is NOT 6 * 2 * 4               24            is NOT 12 * 4 = 48!   Example:   5 * (6 - 2)                 5 * (6 + -2)                 5*6  + 5 * -2                 30   + -10                 20                       5*(6-2) = 5*(4) = 5*4 = 20.   (This law is heavily used, in reverse, when factoring terms into expressions.)   Practical use: Suppose you were told to multiply 34 by 7, knowing this distributive law and that 34 is 30 + 4  we have 30*7 + 4*7 = 210 + 28 = 238.  One can easily do such multiplications in one's head.   Example:  215 * 6 = 1200 + 60 + 30 = 1290               that is 200 * 6 +  10 * 6 + 5*6 Index Back Continue Last page This is another FREE ALGEBRA PRINTABLE presented to you from the Algebra section of K12math.com

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