Mathematics Dictionary Dr. K. G. Shih

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Axioms and Laws

Subjects

Read Symbol defintion Q01 | - Axiom, Theory and Law Q02 | - Axioms in algebra Q03 | - Laws in algebra Q04 | - Theory in algebra Q05 | - Q06 | - Q07 | - Q08 | - Q09 | - Q10 | -

Answers

Q01. What is Axiom ? Mathematical statements are obviously true in any cases without proof. Example for addition 1 + 2 = 2 + 1. What is Law ? Mathematical statements are true in any cases but need proof. Example : Pythagorean Law in right angle triangle c^2 = a^2 + b^2. What is Theory ? Mathematical statement has been tested & confirmed to be true for related case. Example : Factor theory of functions Go to Begin Q02. Axioms in algebra Additive Axiom If a = b abd c = d, then (a + c) = (b + d). Multiplicative Axiom If a = b abd c = d, then a*c = b*d. Commutative Axiom For addition : x + y = y + x. For multiplication : x*y = y*x. Note : This is not applied to subtraction and division. Associative Axiom For addition : (x + y) + z = x + (y + z). For multiplication : (x*y)*z = x*(y*z). Note : This is not applied to subtraction and division. Distributive Axiom For addition : (x + y)*z = x*z + y*z. For multiplication : x*y = y*x. Note : This is not applied to subtraction and division. Identity For addition : x + 0 = x. x - x = 0 For multiplication : x*1 = x. x*(1/x) = 1. Go to Begin Q03. Laws in algebra Go to Begin Q04. Theories in algebra 1. Factor thoery of functions F(x). If F(a) = 0, then (x-a) is a factor of F(x) Hence F(x) = (x-a)*G(x). If F(-a) = 0, then (x+a) is a factor of F(x) Hence F(x) = (x+a)*G(x). Example : Prove that x = 2 is a factor of F(x) = x^2 - 6*x + 8 F(2) = 2^2 - 6*2 + 8 = 0 Hence (x-2) is a factor of F(x) Hence F(x) = (x-2)*(x-4). 2. Remainder theory If F(x) is divided by (x-a), then remainder is F(a). If F(x) is divided by (x+a), then remainder is F(-a). Example : F(x) = x^2 - 6*x + 8 is divided by (x-3), find remainder. F(3) = 3^2 - 6*3 + 8 = -1. F(x) divide by (x-3), the remainder is -1. 3. Binomial theory (x+y)^n = x^n + C(n,1)*x^(n-1)*y + C(n,2)*(x^(n-2))*(y^2) + ..... + y^n. Where C(n,r) = n*(n-1)*(n-2)*...*(n-r+1)/n! Study subject | Binomial theorem. Where C(n,r) = n*(n-1)*(n-2)*...*(n-r+1)/n! Study subject | Binomial expansion and number sequences. 4. Equation theory of quadratic equation Go to Begin Q05. Laws in algebra 1. Laws of exponent Go to Begin Q06. Answer Go to Begin Q07. Answer Go to Begin Q08. Answer Go to Begin Q09. Answer Go to Begin Q10. Answer Go to Begin

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