PROOF 74

Prove.

In decimal multiplication the decimal places add

On the rational numbers ℚ.

A way to read this proof

First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.

Definition D1

A decimal with m places after the point is a number that can be written A · 10⁻ᵐ, with A an integer.

Proof.

(1)10⁻ᵐ is the multiplicative inverse of 10ᵐ, and A2 guarantees its existence.A2

(2)That the number of places is exactly m means that A is not a multiple of 10.convention

∎

Theorem T1

If a is a decimal with m places and b a decimal with n places, then ab = (AB)·10⁻⁽ᵐ⁺ⁿ⁾ and AB is an integer.

Proof.

(1)a = A·10⁻ᵐ, b = B·10⁻ⁿ. A and B are integers.D1

(2)ab = (A·10⁻ᵐ)(B·10⁻ⁿ)(1), E

(3)　 = (AB)(10⁻ᵐ · 10⁻ⁿ)A3, A4

(4)　 = (AB)·10⁻⁽ᵐ⁺ⁿ⁾A1

(5)AB is an integer.A5

(6)Therefore ab can be written with m+n places after the point.(4), (5), D1

∎

Corollary C

If AB is a multiple of 10, the number of decimal places is less than m+n.

Proof.

(1)Suppose there is an integer C with AB = 10C.assumption

(2)ab = (10C)·10⁻⁽ᵐ⁺ⁿ⁾T1, (1), E

(3)　 = C(10 · 10⁻⁽ᵐ⁺ⁿ⁾)A3, A4

(4)　 = C·10⁻⁽ᵐ⁺ⁿ⁻¹⁾A1

(5)ab can be written with m+n−1 places after the point.(4), A5, D1

(6)Example: if a = 0.5, b = 0.2, then A = 5, B = 2, AB = 10, and 0.5 × 0.2 = 0.10 = 0.1.(5)

(7)Even when neither A nor B is a multiple of 10, AB can be. What T1 gives is that the number of places is at most m+n.T1, (1), (5)

∎

Remark

Can A1 be removed from this proof?

Check whether the rule of counting and adding decimal places comes from multiplication or from exponents.