Date: Feb 5, 2013 10:49 PM
Author: Jonathan J. Crabtree
Subject: Re: Proving a definition of multiplication (wrong) by induction

Re: Charles Smith
"A Treatise on Algebra" (1896)

The following is from p. 14, the beginning of Article 28:

In Arithmetic, multiplication is first defined to be the taking one number as many times as there are units in another. Thus, to multiply 5 by 4 is to take as many fives as there are units in four.

As soon, however, as fractional numbers are considered, it is found necessary to modify somewhat the meaning of multiplication [...]

Dave L. Renfro

Thank you Dave.

Here Charles Smith is quoting Euclid's original correct definition of multiplication BEFORE the incorrect phrase 'added to itself' entered the mindset of 16th century speakers of English.

Charles Smith and Euclid would be turning in their graves if they thought their correct definitions of multiplication were no longer being followed by mathematicians and those mathematicians even believed a wrong 'added to itself' definition to be true!

Smith continues on to state;

"To multiply one number by a second is to do to the first what is done to unity to obtain the second."

Let's use the example two times three where two is the multiplicand and three is the multiplier.

"what is done to unity to obtain the second" means 1 1 1
"do to the first" means 2 2 2

Smith is paraphrasing Sir Isaac Barrow (Isaac Newton's math teacher). Barrow clarified the definition of multiplication in 1660 in his edition of Euclid's Elements in which he morphed Henry Billingsley's edition of the Elements into a new version replete for the first time with Recorde's = sign and Oughtred's x sign.

Barrow explained Euclid's definition of multiplication as;

"Hence in every multiplication, unity is to the multiplier as the multiplicand is to the product."

Therefore with two times three the product is 2 2 2.

"Unity is to the multiplier" 1 is to 1 1 1
"Multiplicand is to the product" 2 is to 2 2 2

With Smith, Barrow and Euclid, the multiplication of 2 x 3 stops with 2 2 2. The taking, placing, putting, or setting of the multiplier as many times as the multiplier contains unity is a sequence of unary operations.

In visual and/or geometric environments, that's where the multiplication ends. ie an array or a set of line segments.

Here's a story before I return to mathematics!

(Advertising at a market stall a long time ago)

"Hello, I'd like a free multiplication please." said a shepherd to an abacist. "I'd like the number two multiplied by three!"

"Certainly!" said the abacist. "That is two two two."

"Thank you, yet I need just a single number as I fear I may forget what you say and the number of times you say it." said the shepherd.

"Oh, you want a single number do you?" said the abacist. "That will be extra because I will now have to add all these numbers for you! Adding is not multiplying you know."

"How much extra?" asked the shepherd cautiously.

"I charge $2 per addition" replied the abacist.

"OK, you're the expert." said the shepherd.

"Your answer to two times three is six!" said the abacist.

"Wonderful!" said the shepherd. "how much do I owe you?"


"Well others might charge you six dollars for that is what two times three equals! Yet today, as a special offer I will charge you the discounted fee of just 4 dollars."

The abacist painted his work on a piece of papyrus and so the shepherd paid the $4 and said, "Wonderful! Thank you so much! May your camel not give you fleas."

As the shepherd walked away, he realised he'd been conned by the abacist. The fee should never have been six dollars as there were only two additions in the calculation!

The abacist had written two addition two is four. Then four addition two is six. 2 + 2 = 4 thus 4 + 2 = 6 and the series 2, 4, 6.

Being not one to complain and not one to argue with a clever abacist, the shepherd simply came to distrust the abacict's sales scam of saying he should charge for as many additions as there were in the second number.

The shepherd worked out the truth for himself as he made a sandwich. Just as you need one slice of cheese between two slices of bread, you need two slices of cheese between three slices of bread. For a sandwich, the rule was you always needed one more slice of bread than the number of slices of cheese you had. (The bread being needed to keep the flies off the cheese.)

The abacicts always advertised their work of multiplication at a discount. They said a number multiplied by another was added to itself as many times as the latter number contained unities.

However they quoted by the addition and yet always performed one less addition then the number of unities in the multiplying number. They did in fact charge correctly yet their advertising always made them appear generous by undercharging by the amount of one addition.

Mathematics is the science of patterns.

{1} (one arument and zero binary additions)

{1 + 1} (two arguments and one binary addition)

{1 + 1 + 1} (three arguments and two binary additions)

{1 + 1 + 1 + ... 1} (n arguments and n-1 binary additions)

OK from the story about multiplication and additions I've given to kids, we turn back to the algebra text of 1896 by Charles Smith our friend Dave L. Renfro quotes from.

On P. 255 in article 219 Smith defines an arithmetical progression (A.P.) as,

"A series of quantities is said to be in Arithmetical Progression when the difference between any term and the preceding one is the same throughout the series."

The multiplication of two times three is

2 + 2 + 2

which becomes the arithmetical series

2, 4, 6.

The common difference (d) is always the multiplicand.

Then on p. 257 in article 220 Smith writes,

"If the first term of an arithmetical progression be 'a' and the common difference 'd' then by definition...

...the nth term (in an arithmetical progression) will be a + (n - 1)d."

All sequences of repeated addition are taught to children via skip counting. So the times tables are reinforced by converting repeated counting into sequential inline additions that accumulate the product of our multiplication.

Let us look at 3 x 7 where 3 is the multiplicand and 7 the multiplier

3 was arrived at via
1 + 1 + 1

and then the A.P. becomes
1, 2, 3,

As 3 is to unity, so is the multiplicand, hence 3 x 7 is
3 + 3 + 3 + 3 + 3 + 3 + 3

and then the arithmetical progression (A.P.) becomes
3 6 9 12 15 18 21

Multiplication viewed as an A.P. means exactly what Smith said

"...the nth term (in an arithmetical progression) will be a + (n - 1)d."

The nth term of an A.P. is the multiplier.

The common difference (d) is by definition our multiplicand being repeated in the serial addition.

a + (n - 1)d resolves to
a + (n - 1)a
which by commutativity is
a + a(n - 1)

As mentioned above the nth term in any A.P. is the multiplier in any multiplication so n = b in ab where a is the multiplicand.

Therefore as
a + a(n - 1)
so to is
a + a(b - 1)

ab = a added to itself (b - 1) times.

Dave provided a correct definition of multiplication, consistent with Euclid, being

"taking the multiplicand as many times as there are units in the multiplier OR ab = a taken as many times as b has units"

I provide a correct definition of multiplication, when viewed as repeated binary addition, being

"ab = a added to itself b - 1 times ...

which is what Smith is stating in ab = a + (b - 1)a."

Mathematicians go on and on about the linkage between addition and multiplication via the distributive law, yet often fail to appreciate repeated addition of a common difference is an arithmetical progression without all the plus signs in the repeated addition!

The nth term in an A.P. is your multiplier and the first term is the multiplicand.

I may have taught more children basic arithmetic than others in this forum. So I can assure you when I ask a child how many numbers there are in 3 + 5 + 2 + 1 they answer correctly, "Four!"

Then when I ask, "How many plus signs are there?" they answer, "Three!"

After we do this a few times with different numbers of addends, almost straight away kids recognise the pattern.

There is always one less plus sign compared to how many numbers there are to be added!

So how many plus signs are needed for 63 numbers written as an addition in a row? "Sixty-two!"

That is teaching logic and patterns.

In 2 + 2 + 2 there are three addends and two additions and 2 has been added two times whether with or without the word 'itself'.


ab = a added to itself b - 1 times is correct


ab = a taken b times is correct

(or the traditional Euclidean way is correct, ab = a taken as many times as there are units in b)

ab is NOT a added to itself b times <<<< this is a stuff-up from 1570 which almost no mathematicians bothered to correct for fear of being seen to correct a definition of Euclid! Had they corrected the definition, they would have been fixing a translation done by a haberdasher with no mathematical qualifications.

I simply care about math being a precise and correct language that is not ambiguous. As a class of professionals, we should do what we say and say what we do. When we say 2 x 3 is 2 + 2 + 2 correctly then wrongly as 2 added three times, any seven year old can see the hypocracy on display. You say it is added three times, yet the visual truth reveals it to be added twice by the expression '2 plus 2 plus 2'.

If we want children to understand arithmetic and learn to love and trust it, we must be precise with our language just as we expect them to be precise with their equations.

As for Bob (an American) raising the issue of whether of not the incorrect 'ab = a added to itself b times' is in the public domain, take a look at a recent Scientific American article about a math book.

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

Source: Steven Strogatz, Professor of Applied Mathematics

For those interested in the algebra text Dave L Renfro is quoting from, I have attached the relevant pages.

I also attach the relevant page from an 1825 treatise translating Lacroix, a French mathematician, who provides an example of 16 x 4 being 16 taken 4 times and added to itself three times or as many times as it is repeated 'wanting one'.

ie 16 x 4 is 16 repeated 4 times and added to itself 3-1 times, as per the logic of arithmetical progressions written about by Smith.

So the definition of multiplication 'ab = a added to itself b times' is wrong.

It would be nice if mathematicians would agree, rather than merely provide alternative true definitions or arithmetical expressions that do not match the false definition in the dictionary.

I thought I had proven the definition ab = a added to itself b times wrong by letting b = 1.

Then ab becomes a x 1 which by definition is a + a (a added to itself 1 time).

Others disagree and say I have not proven the definition false.

Therefore the first professor of mathematics to provide a formal proof why ab is NOT a added to itself b times will be well regarded and sourced in future articles.

Be brave professors! History will judge you kindly and children will agree with you.

Jonathan Crabtree