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Exam (elaborations)

COS1501 MEMO

STUDY UNIT 1 NUMBER SYSTEMS INTEGERS 1. Z 2. Z = {…, -3, -2, -1, 0, 1, 2, 3, …} 3. See #2 4. No 5. For every integer m there exists an integer n such that m+n=0. eg.1. x+3 = 0, x = -3 eg.2. x2 = 9, x = 3 or x = -3 6. For any integer x, the absolute value of x is defined to be: |x| = x or |x| = -x. Eg.1. |2|<|-5| Eg.2. w 2 = |w| (same with squareroots) 7. A number where the only factors are one and itself 8. n! = n(n-1)(n-2)…(4)(3)(2)(1) Eg. 6! = 720 9. For all integers m, n, k:  If m=n, then m+k = n+k and mk = nk  If m<n, then m+k<n+k  If k>0, then mk<nk  If k<0, then mk>nk 10. The first difference: Monotonicity of + and * in Natural Numbers: For integers, there are 4 properties instead of 3. The 4th property being that if k<0, then mk>nk. The second difference: There exists a 10th Law: the existence of additive inverses. For every integer m there exists an integer such that m+n=0. 11. There is a difference between 0 and nothing. One would not know the difference between 3, 30 and 300 if zero did not exist. 0 is the additive identity element for nonnegative integers. 12. Let a and b be two real numbers. x = ab + (-a)(b) + (-a)(-b) 1. Factor out a x = ab+a(-b+b) = ab+a(0) = ab 2. Factor out –b x = (-a)(-b) + b(-a+a) = (-a)(-b) + b(0) = (-a)(-b) Therefore, x = ab = (-a)(-b) Therefore, + = (-)(-)

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