
Re: Proving a definition of multiplication (wrong) by induction
Posted:
Feb 4, 2013 9:21 PM


> > On Feb 4, 2013, at 5:51 PM, Dave L. Renfro > wrote: > > > In a formal treatment one would clear up any > possible ambiguity > > that arises in the use of a term such as this. To > me, it's a > > simple matter of writing > > > > a + a + a + ... + a > Indexing also shows up in board games, like do you start counting on the square you are on, or the next one. I remember my son's confusion (when he was 5). You actually NEED this ambiguity early on in order to distinguish differences in context. It is a critical and essential part of higher level reasoning that should be embraced. Bob Hansen
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Thank you Dave and Bob.
I suggest many parents would prefer the definitions taught to their children, and especially algorithmic definitions at that, be correct.
Surely a definition is not a 'word problem' given by teachers designed to be ambiguous in order to force parents and students to work out what it means?
After saying ab = a added to itself b times, mathematicians often write something like...
ab = a + a + a + ... + a .......\......................../ ...............b times
and this is where an ambiguity exists.
Are there b additions or b addends? There can't be BOTH b additions AND b addends as in this case, addition is a binary operation acting on two arguments.
The binary + sign results in n1 additions for n addends in multiplication.
In multiplication ab, the multiplicand a appears as an addend b times. In exponentiation a^b the base a appears as a factor b times.
A twostep pedagogical approach for the conjunct operation of multiplication may be...
STEP 1: MULTIPLICATION OF A, B TIMES ab = a a a a ... a
STEP 2: QUANTIFICATION OF AB
...........b addends of a ......./...........................\ ab = a + a + a + a +... a .......\.........................../ ..........(b  1) additions
Similarly in exponentiation, there are n1 multiplications for n factors.
.............b factors of a ........./.........................\ a^b = a x a x a x a x... a .........\........................./ .........(b  1) multiplications
Some mathematicians say 2^3 is two multiplied by itself three times while others say 2^3 is two multiplied by itself two times. It can't be both. Is there a risk this situation might make mathematicians look stupid?
The above diagrams are crudely drawn, yet you can imagine the horizontal braces { } above and below the depictions of ab and a^b.
You can then move onto linear growth from zero and exponential growth from 1 in subsequent lessons.
ab = 0 + a + a + a + a + ... + a for b a's
and
a^b = 1 x a x a x a x a x ... x a for b a's
Only when you model linear and exponential growth from their additive and multiplicative identities do you have the same number of binary operations as you have arguments. In both cases the a still appears, as an addend in multiplication or a factor in exponentiation, b times.
With the use of the identities, you reveal the essence of multiplication (when defined via repeated addition) and exponentiation (when defined via repeated multiplication) especially with respect to time.
In short, mathematicians must get definitions right so the pedagogy can follow. Using illdefined definitions such as ab = a added to itself b times just confuses children and their teachers.
All the above supports the case for math professors (who should both know better and lead by example) being sent to the naughty corner for repeating, without thinking, ab equals 'a added to itself b times'.
Mathematicians often hold a traditional view that definitions are to be accepted without proof on the assumption somebody would have checked
Yet for practical arithmetic it is reasonable to reject a definition that leads to contradictions.
A single counter example proves a theorem false. It follows from an axiom of real arithmetic that 1 x 1 = 1.
>From ab = a added to itself b times, 1 x 1 = 1 + 1 i.e. 1 added to itself 1 time, but 1 + 1 = 2 so 1 x 1 = 2 which is false, so the theorem ab = a added to itself b times is false.
I believe the logic of all the above is correct. I'm sure people will let me know if I am mistaken.
If you can't accept the definition of multiplication 'ab = a added to itself b times' is false, might you at least all agree this definition of multiplication is not well defined?
Thank you Jonathan Crabtree

