SOME PRINCIPLES OF ALGEBRA AND GEOMETRY 1. Classification of the Numbers of Algebra. Real Rational {Integral (1, 2, -3, etc.) Fractional (3, -, etc.) Irrational (√2, V3, π, etc.) Complex (or imaginary), which will not be considered in this course. 2. Laws of Addition and Multiplication. (a) Commutative laws: a + b = b + a. ab = ba. (b) Associative laws: a + (b + c) = (a + b) + c._a(bc) = (ab)c. (c) Distributive law: a(b+c) = ab + ac. (c) It is impossible to divide by zero, for the quotient of a by zero, if it existed, would be a number q such that q×0. = a. But as q x0 = 0, by (a), we have a contradiction, and hence division by zero must be excluded. 4. Fractions. (a) The value of a fraction is unchanged if numerator and denominator are divided by the same number not zero. This enables us to "cancel" a common factor of the numerator and denominator. (b) The value of a fraction is unchanged if numerato” and denominator This gives the rule: To simplify a given complex fraction, multiply numerator and denominator by the least common denominator of the fractions occurring in the numerator and denominator of the given fraction. Thus if we multiply numerator and denominator of the complex |
