Date: Apr 7, 2013 8:19 PM
Author: Jonathan J. Crabtree
Subject: Re: Proving a definition of multiplication (wrong) by induction

Recursive definitions often have the word 'times' in them. Being recursive, such definitions can be proven by the principle (axiom) of mathematical induction. 

Multiplication* an arithmetical operation, defined initially in terms of repeated addition, usually written a × b, a.b, or ab, by which the product of two quantities is calculated: to multiply a by positive integral b is to add a to itself b times.

ie ab = a added to itself b times

This can be restated recursively as ab = itself + a(b times)

When assessing the truth of a statement, the first step is to remove all pronouns.

In this case the word 'itself' is a third person singular reflexive pronoun.

So we first restate the definition ab = a added to itself b times.

ab = a added to itself b times


ab = a added to a b times.

The theorem is thus written algebraically...

ab = a + ab

...which straight away makes no sense!

We have an extra 'a' in the definition!

STEP 1) From the definition ab = a added to itself b times, the theorem is the proposition P(n): ab = a +ab

STEP 2) Show that P(base case) is true.

Let n = 1. Therefore for the proposition P(1) we let b = 1

a(1) = a + a(1)
a ? a + a
a ? 2a

The base case fails and we CANNOT prove ab = a added to itself by mathematical induction!

Therefore the commonly accepted and quoted dictionary definition of multiplication is FALSE.


Now let us prove another recursive repeated addition definition of multiplication known as an arithmetic progression true.

{a, a + a, a + a + a, a + a + a + a, ...}

Here our first term is 'a' and our common difference is also 'a'.

Our nth term in this arithmetic progression is defined algebraically as

an (Note: both the 'n' in 'an' and the '1' in 'a1' are subscript)

an = a1 + (n-1)a

STEP 1) From the definition an = a1 + (n-1)a the theorem is the proposition P(n): an = a1 + (n-1)a
STEP 2) Show that P(base case) is true.

Let n = 1. Therefore for the proposition P(1) we get
a(1) = a1 + (1-1)a
a1 = a1 + (0)a
a1 = a1, which is TRUE.

STEP 3) Inductive Step
Assume for some integer k, P(k) = a1 + (k-1)a1 (again note the '1' in 'a1' is subscript)

STEP 4) Show P(k+1) = a1 + (k+1-1)a1

STEP 5) Proof of 3) Inductive Step
a1(k+1) = a1 + (k+1-1)a1
a1(k+1) = a1 + (k)a1
a1(k+1) = a1(1+k)
This by commutative law of addition
a1(k+1) = a1(k+1), which is TRUE.

STEP 6) Therefore P(k+1) is TRUE when P(k) is true, and therefore P(n) is TRUE for all natural numbers and an = a1 + (n-1)a for all natural numbers 'n'.


ab = a added to itself b times
an = a added to itself n-1 times

Let b=n

ab = a added to itself b-1 times is TRUE
ab = a added to itself b times is FALSE

The definition of multiplication attributed to Euclid since 1570 has been FALSE and the contradiction is both revealed and proven.

^ Quote: 'Take the terminology. Does 'seven times three' mean 'seven added to itself three times'? Or 'three added to itself seven times'?

Five added to one three times is 16.
Seven added to itself three times is 28
Three added to itself seven times is 24

Seven multiplied by three, according to the true original Greek of Euclid, is simply... placed three times!

ab = a placed b times, or in more modern 16th century English...

ab = a taken b times

The 'addition bit' was an optional extra to be done AFTER the multiplication. Euclid was multiplying line segments, not numbers.

Multiplication NEVER WAS defined as repeated binary addition.

Euclid was observing magnitudes in action not counting multitudes. Euclid was undertaking unary actions and observing the result.

Quantity is a function of number and size. Multiplication varies number while scaling varies size.

So basically, our arithmetical pedagogy has been fundamentally flawed for 443 years.

In conclusion, this note can be viewed as a recall notice.

ab never was a added to 'itself' b times.

Teachers and parents would be well advised to demonstrate multiplication as the joining of equal line segments starting from zero on the number line.


an introductory explanation could become

ab = a added to zero b times.

This can then evolve into rationals and integers as required.

^ Source: Steven Strogatz, Professor of Applied Mathematics