Multiplication

Theorem 30 (Landau 28).

To every pair of numbers, A, B, the product of A and B, written A•B (or just AB), is such that

a). A•1 = A, for every A;

b). A•B’ = A•B + A, for every A and B.

Proof:

I. For A = 1, the number A•B = B is as required, since A•1 = 1 = A and A•B’ = B’ = B+ 1 = A•B + A.

II. Let this be true of some number A so that there is an A•B for all B. Thus, A’•1 = A•1 + 1 = A + 1 = A’ and

A’B’ = AB’ + B’ = (AB + A) + B’= AB + (A + B’) = AB + (A + B)’ = AB + (A’ + B) = AB + (B + A’) = (AB + B) + A’ = A’B + A’.

Theorem 31 (Landau 29; Euclid VII.16). Commutative Law of Multiplication.

AB = BA

Proof: A. We have B•1 = B, and from Theorem 30 we also have 1•B = B, and 1•B = B•1.

B. If this is true of some A and B, then AB = BA; and AB’ = AB + A = BA + A = B’A, by the construction of Theorem 30.

Theorem 32 (Landau 30). Distributive Law

A(B + C) = AB + AC

Proof: A. Fix A and B and let C be 1.

A(B + 1) = AB’ = AB + A = AB + A•1.

B. Let C be any number for which this is true and now consider C’.

A(B + C’) = A((B + C)’) = A(B + C) + A = (AB + AC) + A = AB + (AC + A) = AB + AC’.

Theorem 33 (Landau 31). Associative Law of Multiplication

A(BC) = (AB)C

Proof: Fix A and B and consider 1.

(AB)•1 = AB = A(B•1).

Now consider A for which the assertion holds:

(AB)C = A(BC), and so:

(AB)C’ = (AB)C + AB = A(BC) + AB = A(BC + B) = A(BC’), so that the assertion also holds of C’.

Theorem 34 (Landau 32).

If A > B, A = B, or A < B, then AC > BC, AC = BC, or AC < BC, respectively.

Proof: If A > B, then A = B + D, AC = (B + D)C = BC + DC > BC.

If A = B, then AC = BC.

If A < B, then B > A, and hence as above, BC > AC and AC < BC.

Theorem 35 (Landau 33).

If AC > BC, AC = BC or AC < BC, then A > B, A = B, or A < B.

This follows from Theorem 34. The three cases exhaust all possibilities.

Theorem 36 (Landau 34).

If A > B and C > D, then AC > BD.

From Theorem 34 we have that AC > BC, and BC = CB > DB = BD, and hence AC > BD.

Theorem 37 (Landau 35).

If A ≧ B, C > D or A > B, C ≧ D, then AC > BD.

Proof: This follows from Theorems 34 and 36.

Theorem 38 (Landau 36).

If A ≧ B, C ≧ D, then AC ≧ BD.

Proof: If two equality signs hold, then it is clearly true; otherwise, it follows from Theorem 37.

Theorem 39 (Euclid IX.28). If an odd number by multiplying an even number make some number, the product will be even.

Proof: Let the odd A by multiplying the even B make C. C is made up of as many numbers equal to B as there are units in A. B is even, so C is a sum of even numbers, and is even (Theorem 23).

Theorem 40 (Euclid IX.29). If an odd number by multiplying an odd number make some number, the product will be odd.

Proof: Let the odd A by multiplying the odd B make C. C is made up of as many numbers equal to B as there are units in A: an odd multitude of odd numbers. So C is odd (Theorem 25).