ALGEBRAIC PROOFS

The various multiplication etc. devices shown in this book can be proved by arithmetical and geometrical methods, but here for brevity we give algebraic proofs.

Chapter 3
Examples 1-5   (ax + 5)² = a(a + 1)x² + 25 x =10
6-9  (ax + b)(ax + 10-b) = a(a + 1)x² + b(10 - b) x=10
10    as above with x=100

Chapter 4
Examples 11-12   (ax + b)((10-a)x + b) = (a(10-a) + b)x²   x=10

Chapter 5
Examples 1-15   (x + a)(x + b) = x(x + a + b) +ab, x=10^n
Subtraction  x^n = (x-1)[sum from 1 to (n-1) of x^r] + x
Examples 16-19 (nx + a)(nx + b) = nx(nx + a + b) + ab
20-22 (x + a)(y + b) = (x + a)y + bx + ab, x=10^m, y=10^n
23-24 (x + a)(x + b)(x + c) = x²(x+a+b+c) + x(ab+ac+bc) + abc
25-26 (x + a)² = (x + 2a) + a²
27      (nx + a)² = n(nx + a)x + a²
28-29 (50 + a)² = 100(25 + a) + a²
30-31 a(xn - 1) = (a - 1)xn + (xn - a)

Chapter 6
(ax^n + bx^(n-1) + cx^(n-2) + . . . + zx^0)(Ax^m + Bx^(m-1) + Cx(m-2) + . . . + Zx^0)
= aAx^(n+m) + (aB+bA)x^(n+m-1) + (aC+bB+cA)x^(n+m-2) + . . . + zZ, x=10

For grouping 2, 3 etc. figures on the right of the numbers x=10², 10^3 etc.

Division near a base. Since bx = (x - a)b + ab
therefore bx/(x-a)= b remainder ab. x=10, a < x

Chapter 7
(a + b)(a - b) = a² - b², where a = average, b < a

Chapter 8
(a + p)² = a² + p(2a + p)
(a + 3p)(a + 2p) - (a + p)a = p(a+3p + a+2p + a+p + a)