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 mn, then m+kn+k  If k0, then mknk  If k0, then mknk 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 k0, then mknk. 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, + = (-)(-) 13. If you withdraw R5 from your account for 3 days, there will be a R15 decrease in your balance. (-5)(3) = 15. 14. One must understand properties of number systems in order to work with equations, functions and formulas in algebra, as they allow for the creation of equivalent expressions, allowing us to solve problems and justify solutions. POSITIVE INTEGERS 15. Z + 16. Z + = {1,2,3,4….} 17. Positive integers have a multiplicative identity, the number one, defined by m(1) = m. 18. Properties:  Commutative For all non-negative integers m and n m+n = n+m mn = nm  Associative For all non-negative integers m, n, and k (m+n)+k = m+(n+k) (mn)k = m(nk)  Distributive For all non-negative integers m, n and k m(n+k) = mn+mk 19. An even integer occurs when division results in a whole number. An odd integer occurs when division results in an answer that contains a fraction. NATURAL NUMBERS 20. N and Z≥ 21. N = {0,1,2,3,4 ….} 22. Natural numbers have a multiplicative identity of 1, m(1) = m. Importance of the number 0: There is an absence of zero-divisors, so m.n = 0, then either m=0 or n=0. They also have an additive identity of 0, m+0 = m. Describe the importance of the number zero. 23. 4 properties:  Linearity For all non-negative integers m and n, one of the following statements are true: mn or mn or m=n  Additive inverse m+(-m)=0  Transivity of = and - If m=n and n=k, then m=k - If mn and nk then mk  Monotonicity of + and * - For all non-negative integers m, n and k: if m=n then m+k = n+k and mk = nk if mn, then m+kn+k if k0, then mknk