Geometric Bites

Fractions | Algebra | First Principles The familiar rules for adding and subtracting fractions can be derived directly from the properties of real numbers. The key idea is that multiplying a number by 1 does not change its value. By writing 1 in a suitable fractional form, both fractions can be expressed with the same denominator. Assumptions Let a, b, c, d ∈ ℝ, with b ≠ 0 and d ≠ 0. Addition of Two Fractions Begin with two fractions whose denominators are not necessarily equal: a b + c d = a b · 1 + c d · 1 = a b ...

These workings derive the laws of base-10 logarithms from exponent laws by converting between exponential form and logarithmic form. Assume a > 0 , b > 0 , and n ≠ 0 . Product Rule log(ab) = log a + log b Let 10 x = a Let 10 y = b Therefore: log 10 a = x log 10 b = y 10 x 10 y = ab 10 x+y = ab Therefore: log 10 (ab) = x + y Substituting: log 10 (ab) = log 10 a + log 10 b Therefore: log(ab) = log a + log b Quotient Rule log(a / b) = log a - log b Let 10 x = a Let 10 y = b Therefore: log 10 a = x log 10 b = y 10 x / 10 y = a / b 10 x-y = a / b Therefore: log 10 (a / b) = x - y Substituting: log 10 (a / b) = log 10 a - log 10 b Therefore: log(a / b) = log a - log b Power Rule log(a n ) = n log a Let log(a n ) = x Therefore: 10 x = a n Taking the n-th root of both sides: (10 x ) 1/n = (a n ) 1/n 10 x/n = a Therefore: log 10 a = x / n n log 10 a = x Theref...