Green Induction Recursive Proof that carry less than 2 in addition of two numbers

We consider the addition of two numbers using place value in a tens unit system.

The ones place is 10^0 so we call it the zero exponent place.

If the carry out of a place is less than 2, we call the place green.

Lemma:  The ones place is green.

Proof

Let a, b be the two digits.

a <= 9

b <= 9

a+b <= 18

So the carry out is 0 or 1, which is less than 2.  So the ones place is green.

Lemma If a place is green, the next next place is green.

Proof

If a place is green, the carry out is at most 1.  Call the carry c.

Let a, b be the two digits in the next place.  We add a,b, and c.

a <= 9

b <= 9

a+b <= 18

c <= 1

a+b + c < 19

Thus the carry out of the next place is 0 or 1.  So it is less than 2.  Thus the next place is green.

Lemma:  All places are green.

Proof.

We have shown the first place is green. We have shown if a place is green, the next place is green.   By green induction, all places are green.

 

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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2 Responses to Green Induction Recursive Proof that carry less than 2 in addition of two numbers

  1. Pingback: The Principle of Tweet Induction | New Math Done Right

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