The Lost Logic of Elementary Mathematics

Thanks Jonathan
and Podo. What’s
on the agenda?
The Lost Logic of
Elementary Mathematics
JonathanCrabtree
La TrobeUniversity
Melbourne Campus, Australia
December2016
www.jonathancrabtree.com/LLEM research@jonathancrabtree.com

Background | Zero | Negative Numbers | Multiplication | Exponentiation?

Hello. I’m Jonathan.
Podo is my super
puppy. My story is...
Background | Zero | Negative Numbers | Multiplication
Copyright © 2016 Jonathan Crabtree All Rights Reserved

The Lost Logic
of Elementary
Mathematics!
Copyright © 2016 Jonathan Crabtree All Rights ReservedCopyright © 2016 Jonathan Crabtree All Rights Reserved

My name is Little
Math. So what’s
on the agenda?

It’s elementary
what’s on!
The simple
ideas of...

Euclid, Liu Hui
Brahmagupta,
Newton and
others, have…

Mathematics most hated subject in school
Science 149 10.9%
Math 499 36.6%
History 218 16.0%
English 282 20.7%
Phys.Ed 214 15.7%
TOTAL 1362 100.0%
Source: www.quibblo.com/quiz/1lE5Q15/Whats-your-most-hated-subject-in-school (Nov. 2016)

Two multipliedby
three is two added
to itselfthree times
and that equals six.
But that’s crazy.
Two added to itself
three times is eight.
Grade 2C
1968

Of course2 multiplied
by 3 equals 2 added
to itself3 times.
It’s Euclid’smultiplicationdefinition
from 300 BC!

…to multiply a by integral b is to add a to itself b times
Collins Dictionary of Mathematics
In book VII, Euclid defines multiplication as ‘when that which is
multiplied is added to itself as many times as there are units in the other’
The Development of Multiplicative Reasoning
in the Learning of Mathematics

How can 2 added to 1
three times be seven?
1 + 2 + 2 + 2
and 2 added to itself or 2
three times be six?
2 + 2 + 2 + 2

With 2 multiplied
by 3, the three hops
of 2 drawn on the
number line start at
zero.

So 2 multiplied
by 3 equals two
added to zero three
times, not itself.
0 + 2 + 2 + 2

Jonathan had seen
the lost logic of
elementary
mathematics!
Euclid had been
translated incorrectly!

Yet Jonathan felt
stupid. Later, he
failed mathematics
and stayed down a
year at school.

Then, in 1983, on
Friday March 18,
he broke his back!

Let me walk
and I promise
to make maths
simpler!
A few years later...

Next? Zero...

Zero is what you get
when you subtract a
number from itself.

4 – 4 = 0
123 – 123 = 0

–3 – –3 = 0
– – – = 0

The person most responsible for modern
mathematics is the Indian, Brahmagupta.
In 628 CE he gave rules for adding and
subtracting integers. In his rules for
subtraction, Brahmagupta does not
define zero as any number subtracted
from itself!

Brahmagupta defined zero, not in his
vyakalana rules for subtraction, but
in his saṅkalana rules for addition, as:
the sum of a positive number and negative
number of equal magnitude, सम-ऐक्यम् खम् (Brāhma
Sphuta-siddhānta, Chapter 18:30a).
So zero was defined as:
+n + –n

So zero in mathematics is
also what you get when you
add any equal number of
opposites together.
Why is a –ve subtracted from zero equal to a +ve?
+n + –n

For example, start at zero,
go right n steps, then left n
steps, (or vice-versa) and
you’re back where you
started, at zero!

So, North/South, East/West,
Forwards/Backwards and
Left/Right are all opposites
that cancel each other out.

Assets/Debts,
Revenues/Expenses,
Surpluses/Deficits and so
on, can also cancel each
other out to make zero.

If you use the correct unit,
from any two opposing
units, you don’t need to use
the ‘adjective’ negative.
(Nouns are simpler!)
Get ready...

Zero was split and the real
number line was the result!

The real number line is
symmetrical about zero.
The mirror-image numbers
and magnitudes on either
side of zero sum to ZERO.
Next? Negative numbers...

Collins Dictionary
“A negative number, quantity,
or measurement is less than zero.”
Negative numbers are
complex abstract ideas,
involving numbers less than
zero. So we avoid negatives
until kids are age 12 to 13.

Oxford English Dictionary
“A number or amount that is lower
than zero; a negative quantity;
spec. (with of) the quantity obtained
from a given (positive) quantity by
subtracting it from zero or
multiplying it by −1.”

Now I’m
more confused
than ever!

What’s negative
seven minus
negative four?
Let’s dig deep for the
answer from ground
level zero!

Ground Level Zero
No Bumps and No Holes!

From Ground Level Zero, Podo makes ...
... bumps and holes!

True or false?

True or false? (Podo won’t look!)

Q1-5 Answers & discussion time
So, is
Podo
real?

Podo found
the Lost Logic of
Negative
Numbers!

Chinese maths
had positive and
negative 2000 years
before Europe!
Copyright © 2016 Jonathan Crabtree All Rights ReservedCopyright © 2016 Jonathan Crabtree All Rights Reserved

Yet the first
Chinese math
text with a zero
was in 1247.
The Mathematical Treatise in Nine Sections, by Qin Jiushao

So negative
numbers were
not less than zero
in China or India!

The ancient
Chinese and Indians
had numbers for
positive things...

... and numbers
for negative things.

Positive numbers
just count or
measure things...

...and negative
numbers just
count or measure
opposite things.

Opposite things
in mathematics
cancel each
other out.

Like holes
and bumps, or
go away and
come here!

All opposite things that
cancel each other out
can get counted with
the same numbers!

DIRECTION
FINANCIAL POSITION
POPULATION
TEMPERATURE
SUFFICIENCY
SEA LEVEL
TIME
North/South, East/West, Left/Right, Up/Down
Assets/Debts, Profit/Loss
Births/Deaths, Immigration/Emigration
Hot/Cold, Above Zero/Below Zero
More Than Enough/Less Than Enough
Above/Below
To the hour/Past the hour
Q. How +ve result bad & –ve result good?
A. Your cancer test!
So, in mathematics, ‘negative’ is not ‘bad’ and ‘positive’ is not ‘good’!

I can imagine
lots of negatives as
holes and lots of
positives as bumps.

Separate, or
altogether, all my
bumps and holes
give me zero!

‘Negative three
and positive seven’
are hard to imagine.

Yet three holes
for –3 and seven
bumps for +7 is
lots of fun!

3 holes
7 bumps

3 negatives
7 positives

So –3 + +7 = +4
3 holes (–3) and 7 bumps (+7 )
leads to 4 bumps (+4 )

3 negatives!
What’s 7 negatives
minus 4 negatives?
Next? Multiplication...

OK, so what’s –2
multiplied by -3?
I’m going home!

“Brahmagupta (598 – 670 CE) was a Hindu
mathematician and astronomer who lived in
the first century.”
“He used negative integers to represent debts
and positive integers to represent assets.”
“The product ... of two debts is one fortune.”

“Brahmagupta... defined zero as the result of
subtraction of a number from itself.”
“He also gave the following rules for operations
on what he called ‘fortunes’ (positive numbers)
and ‘debts’ (negative numbers).”
“The product... of two debts is one fortune.”

Debt × Debt = Fortune! Really?
Credit Card Debt × Mortgage Debt = Fortune! Really?
If it looks like nonsense it probably is!
Yet authors keep writing it and teachers keep teaching it!
No Indian mathematician would have been that stupid.
“The product or quotient of two negatives is one positive.” – Brahmagupta!

Brahmagupta
"The product of a positive and a negative (number) is negative; of two
negatives is positive; positive multiplied by 'positive is positive.“
Mahāvīra
"In the multiplication of two negative or two positive numbers the result is
positive; but it is negative in the case of (the multiplication of) a positive and a
negative number."
Śrīpati
"On multiplying two negative or two positive numbers (the product is)
positive; in the multiplication of positive and negative (the result is) negative."
Bhāskara II
"The product of two positive or two negative (numbers) is positive; the product
of positive and negative is negative.” The same rule is stated by Nārāyaṇa.
SOURCE: History of Hindu Mathematics: A Source Book, Part II, Algebra, Bibhutibhusan Datta
and Bidyāraṇya Avadesh Narayan Singh, pp. 22-23, Motilal Banarsidass, Lahore, 1938.

John Wallis Paraphrase (Note: cipher = zero)
But in case the multiplier is a negative number;
suppose –2; then instead of adding the multiplicand
to cipher 2 times, it will signify so many times to
subtract the multiplicand from cipher. For as A × +2
implies twice adding A to cipher; 0 + A + A, to arrive at
+ 2A, so A × –2 implies twice subtracting A from
cipher; 0 – A – A, to arrive at –2A.
A Treatise of Algebra, both Historical and Practical, John Wallis,
p. 74, Printed by John Playford, for Richard Davis, Bookseller,
in the University of Oxford, 1685.

Colin Maclaurin Paraphrase
Multiplication by a positive number implies a
repeated addition of the multiplicand to cipher: But
multiplication by a negative number implies a
repeated subtraction of the multiplicand from cipher.
And when positive a is to be multiplied by negative n,
(+a × –n), the meaning is that +a it to be subtracted as
many times from cipher as there are units in n:
Therefore the product is negative, being –na.
A treatise of algebra : in three parts, Colin Maclaurin, pp. 12-13,
Printed for A. Millar, and J. Nourse, London, 1748.

In 263 CE, Liu Hui wrote a commentary on the ancient
The Nine Chapters on the Mathematical Art
(九章算术 Jiǔzhāng Suànshù circa 100 CE).
Liu Hui said:
I read the Nine Chapters as a boy, and studied it in full
detail when I was older. I observed the division between
the dual natures of Yin and Yang [the negative and positive
aspects] which sum up the fundamentals of mathematics.
The Nine Chapters on the Mathematical Art: Companion and Commentary,
Shen Kangshen, John N. Crossley and Anthony W. C. Lun, Oxford University Press, 2000.
Now we reveal what –n × –n REALLY means!

Positive Integer Multiplied by Minus Multiplier
+2 x –3

+2 x –3 = –6
Positive Integer Multiplied by Minus Multiplier
+2 x –3

Negative Integer Multiplied by Minus Multiplier
– 2 x –3

– 2 x –3 = +6
– 2 x –3
Use plastic bottle caps for classrooms

If you want to interpret the Sanskrit of Brahmagupta
with a financial analogy for –n × –n , it is
A debt repeatedly subtracted makes a fortune!
If Bill Gates paid your $1000 mortgage debt for the next 10
months, it would have the same net effect on your financial
position as being given a $1000 fortune for the next 10 months.
Both debt repeatedly subtracted AND fortune repeatedly added
result in a fortune, which is the reason why –n × –n = +n × +n
– 2 x –3

Plastic bottle caps work great to model integer arithmetic.
HINT! If you ever run out of caps to subtract, just ‘add zero’
eg. 5 white caps and 5 black caps and keep going!

So…
For integral multiplication, a × ±b, according
to the sign of b, we can either add a to zero b
times in succession or subtract a from zero b
times in succession.

Returning the identity element zero reveals patterns not often seen.
a × +4 = 0 + a + a + a + a
a × +3 = 0 + a + a + a
a × +2 = 0 + a + a
a × +1 = 0 + a
a × 0 = 0
a × –1 = 0 – a
a × –2 = 0 – a – a
a × –3 = 0 – a – a – a
a × –4 = 0 – a – a – a – a
So, integral multiplication involves
either repeated addition or repeated
subtraction, from zero, depending on
the sign of the multiplier.

Background | Zero | Negative Numbers | Exponentiation!
‘The Binary Bug’
BV1570 spread to
exponentiation!
“ab = a multiplied
by itself b times”

I wish mathematicians
would be rigorous and
consistent!

Returning the identity element one into the pattern of exponentiation.
a+4 = 1 × a × a × a × a
a+3 = 1 × a × a × a
a+2 = 1 × a × a
a+1 = 1 × a
a 0 = 1
a–1 = 1 ÷ a
a–2 = 1 ÷ a ÷ a
a–3 = 1 ÷ a ÷ a ÷ a
a–4 = 1 ÷ a ÷ a ÷ a ÷ a
So, integral exponentiation
involves either repeated
multiplication or repeated
division, from one, depending
on the sign of the exponent.
Idea extension via Disquisitiones
Arithmeticae, Carl F. Gauss, 1801.
Copyright © 2016 Jonathan Crabtree All Rights Reserved Download the conference paper next...

Full conference proceedings https://issuu.com/julieallen35/docs/2016_mathematical_association_of_vi/98
Just this paper:
http://bit.ly/LostLogicOfMath

Thanks Jonathan
and Podo. What’s
on the agenda?
Thatisjustsomeof…
The Lost Logic of
Elementary Mathematics
Thank you!
Feedback? ⇒ www.jonathancrabtree.com/LLEM/

The Lost Logic of Elementary Mathematics

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The Lost Logic of Elementary Mathematics