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=10ⁿ

Subtraction

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=10ⁿ

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

(axⁿ + bx^n-1 + cx^n-2 + . . . + zx⁰)(Ax^m + Bx^m-1 + Cx^m-2 + . . . + Zx⁰)

= 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, 10³ 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)