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PROPERTIES OF INTEGERS I = { ...-3, -2, -1, 0 , 1, 2, 3, ...} Commutative Properties: For all integers, a and b, Associative Properties: For all integers, a, b, and c, Identity Properties: For all integers, a, Closure Properties: For all integers, a and b, Distributive property of multiplication over addition: For all integers, a, b, and c, Distributive property of multiplication over subtraction: For all integers, a, b, and c, Property of multiplication by zero: For all integers, a, a+b=b+a a•b=b•a (a + b) + c = a + (b + c) (a • b) • c = a • (b • c) a+0=a a•1=a a + b is an integer a • b is an integer a – b is an integer a(b + c) = a • b + a • c (b + c)• a = b • a + c • a a(b – c) = a • b – a • c (b – c)• a = b • a – c • a a•0=0 Property of opposites: For all integers, a, there exists an integer, –a, such that a + (–a) = 0 Double opposite generalization: For all integers, a, –(–a) = a Addition generalization: For all integers, a and b, –(a + b) = (–a) + (–b) Subtraction generalization: For all integers, a and b, a – b = a + (–b) The Symbol “–” The symbol – is used to mean several different, but related, ideas in mathematics. (i) We are used to seeing it as a subtraction sign. Subtraction is a binary operation--that is, the operation of subtraction involves two numbers. (ii) We have also been using – to indicate a negative number. In this case, we use the – to indicate direction. On which side of 0 do we look for a negative number? (iii) Finally, – can be used to mean “the opposite of.” This is a unary operation- that is, it is something we can do to a single number. When we find the opposite of a number, we change its sign either from negative to positive or from positive to negative. –x means “the opposite of x.” NOTE that we cannot say whether x is negative or positive. Neither can we say whether –x is negative or positive. What happens if you take the opposite of the opposite of a number? Write the following in English and then perform the computation. – [ ( –3) – ( –2)] Absolute Value When we think about Integers on the number line, we consider two ideas: • How far is the number from 0? • On which side of 0 is the number? For instance, we know that – 3 is 3 units from zero, and it is on the left side of zero. + 3 is ALSO 3 units from zero, but it is on the right side of zero. When we consider the absolute value of a number, we are simply asking how far from zero the number is without regard to whether it is on the right or the left. means “the absolute value of negative six equals six.” The absolute value symbols also indicate groups, just as parentheses and brackets do. Any computation that is inside absolute value symbols must be calculated first. Write each of the following in English and then perform the computation. (a) 3 4 (b) 3 4 If the statement is true, tell what property justifies it. If it is false, tell why. 1. ___________________________ –(–7) = 7 2. ___________________________ (-4) 3 = 3 (-4) 3. ___________________________ -7 + -5 = - (7 + 5) 4. ___________________________ (-3 + 7) + (5 + 11) = [(-3 + 7) + 5] + 11 5. ___________________________ (-3 + 7) + (5 + 11) = (5 + 11) + (-3 + 7) 6. ___________________________ – 7. ___________________________ 3 45 3 5 4 5 8. ___________________________ –6 ÷ 0 = 0 9. ___________________________ (-3)(-7) = 3 7 10. ___________________________ (7 – 11)1 = 7 – 11 11. ___________________________ 4(8 3) (4 8)3 12. ___________________________ 9 9 is an integer 13. ___________________________ (3 + –8)(4 + 2) = (3 + –8)4 + (3 + –8)2 14. ___________________________ – 6 – 5 = –6 + –5 15. ___________________________ – 7 + -(–7) = 0 16. ___________________________ – 3 + 0 = –3 17. ___________________________ 91 is an integer 18. ___________________________ 8 + –3 = 8 – 3 19. ___________________________ - [ - (–7)] = –7 20. ___________________________ – 3 – 4 = 3 + –4 7 + 0 = –7