The Relaunch of Indian Mathematics

© 2018 J. J. Crabtree | www.jonathancrabtree.com
2

In 300 BCE the Greek Euclid of Alexandria
defined multiplication as repeated addition.
a to the power of b (ab) equals a into itself b times.
a into b (a × b ) equals a added to itself b times.
3

Zero is defined as a – a
‘a number subtracted from itself’.
The Arabic world embraced India’s
ideas on Zero and negative numbers.
Arabic maths was then adopted by Leonardo Pisano
who brought India’s ideas on Zero to Europe.
4

5
Your ‘facts’ are false!
Everyone is entitled to their own opinion, yet not to their own facts.

Extraordinary Claims
Require
Extraordinary Evidence
6
✔

•India’s definition of ZERO never made it to
either the ancient Arabic world or Europe.
© 2018 J. J. Crabtree | www.jonathancrabtree.com 7
•In the Arabic world, India’s ZERO only
came to exist as a placeholder, not as the
power tool to solve simple problems like
+3 minus +4, or –2 minus –4, or –4 minus +2
Extraordinary Claims
Extraordinary Evidence...

Al-Khwārizmī (c. 780-850)
“So they made 9 symbols, which are
these: 9 8 7 6 5 4 3 2 1.
...
Because one is the root of all number
and is outside number. It is the root
of number because every number is
found by it.
...
But it [one] is outside number
because it is found by itself, I mean,
without any other number.”

Al-Uqlīdisī (c. 920-980)
“Why is zero multiplied by zero equal to zero
and zero multiplied by any letter zero?
We say that by multiplying zero by zero the
aim is only to occupy the place; the same
applies for multiplying the letter by zero.
We multiply the letter by zero ... to occupy
the place, and tell that there is a place and
that it is empty.

200 years after Brahmagupta,
al-Khwārizmī did not accept 1 as a
number. Zero as a number? Never!
300 years after Brahmagupta,
al-Uqlīdisī accepted India’s ZERO as a
placeholder, yet not a number. Why?

Al-Uqlīdisī means ‘the Euclidist’. He
was known for his skill in studying
the Greek geometry of Euclid and
translating it into Arabic.
Around 300 BCE, Euclid defined ‘number’ as a
multitude of units. So Euclid’s definition of
number came before 0 and 1 were numbers.

As we will see, India defined zero as the sum of
opposing negative and positive numbers with
the same multitude or magnitude.
If Arabic and European writers in medieval times
really understood India’s zero, where are all the
negative numbers in their writings?

“I have read a few dozen medieval Arabic books on arithmetic
and algebra, and there is no hint of negative numbers in any
of them. Zero, too, was not regarded to be a number, but was
merely the place holder for an empty place in the
representation of a number in Arabic (Indian) notation.”
“All numbers in Arabic arithmetic were positive. No Arabic
author to my knowledge ever even contemplated the existence
of negative numbers.”
By email courtesy of Dr. Jeffrey Oaks, Professor of Mathematics on:
Medieval Arabic algebra and the mathematics of Greece and medieval Europe

Back to the future...

Images courtesy of
the British Library.
For this talk,
Brahmagupta’s
Laws of Positives
Negatives and
Zero have been
freshly analysed.
20© 2018 J. J. Crabtree | www.jonathancrabtree.com

Brahmagupta’s 5
Addition Laws
AL
(saṅkalana)

positive plus positive is positive
AL1

negative plus negative is negative
AL2

positive plus negative is the difference
between the positive and the negative
AL3

when positive and negative
are equal the sum is…
AL4

ZERO
AL4
when positive and negative
are equal the sum is…

“…sunya (ZERO) is neither
positive nor negative
but forms the boundary line
between the two kinds,
being the sum of two equal
but opposite quantities.”
Joseph, G. G. (2016). Indian mathematics:
Engaging with the world, from ancient to modern
times. World Scientific. p. 208

positive plus zero is positive
AL5 part 1

negative plus zero is negative
AL5 part 2

zero plus zero is zero
AL5 part 3

AL5

Brahmagupta’s 5 Addition Laws
positive plus positive is positiveAL1
negative plus negative is negativeAL2
positive plus negative is the difference between the
positive and negativeAL3
when positive and negative are equal the sum is zeroAL4
AL5
34

Following text from: Plofker, K. (2009).
Mathematics in India: 500 BCE-1800 CE. p.
151, Princeton, N.J: Princeton University Press.
Brahmagupta’s 5
Subtraction Laws
SL
(vyavakalana).

[If] a smaller [positive] is to be
subtracted from a larger positive,
[the result] is positive.
SL1

[If] a smaller negative (is
subtracted) from a larger negative,
[the result] is negative.
SL2

[If] a larger [negative or positive is to be
subtracted] from a smaller [negative or
positive, the algebraic sign of] their
difference is reversed - negative
[becomes] positive and positive negative.
SL3

A negative minus zero is negative, a positive
[minus zero] positive; zero [minus zero] is zero.
SL4

When a positive is to be subtracted from a
negative or a negative from a positive, then
it is to be added.
SL5

Brahmagupta’s 5 Subtraction Laws
A smaller positive subtracted from a larger positive is positive.SL1
A smaller negative subtracted from a larger negative is negative.SL2
If a larger negative or positive is to be subtracted from a smaller
negative or positive, the sign of their difference is reversed –
negative becomes positive and positive negative.
SL3
A negative minus zero is negative,
a positive minus zero is positive,
zero minus zero is zero.
SL4
When a positive is to be subtracted from a negative
or a negative from a positive, then it is to be added.SL5
41

Seeing is Believing!
Applying Brahmagupta’s
5 Addition Laws
AL

AL1
positive plus positive is positive

AL2
negative plus negative is negative

positive plus negative is the difference
AL3

3 holes
7 bumps
Podo the Puppy by
AFX Animation

3 negatives
7 positives
Podo the Puppy by
AFX Animation

So –3 + +7 = +4
3 holes (–3) and 7 bumps (+7 )
leads to 4 bumps (+4 )
Watch the AFX Animation
video of Podo the Puppy
squaring a circle with rope!

ZERO
ZERO
when positive and negative are equal the sum is ZEROAL4

plus ZERO
AL5
positive
plus ZERO
negative
ZERO + positive = positive, ZERO + negative = negative, ZERO + ZERO = ZERO
is positive
is negative
52

Seeing is Believing!
Applying Brahmagupta’s
5 Subtraction Laws
SL

If a smaller positive is subtracted from a
larger positive the result is positive.
SL1
larger positive minus smaller positive is positive

If a smaller negative is subtracted from a
larger negative the result is negative.
SL2
larger negative minus smaller negative is negative

If a larger positive is to be subtracted from a
smaller positive, the sign of their difference is
reversed – positive becomes negative.
SL3
We don’t have enough positives. So use AL4 and AL5.
3 positives minus 4 positives?

If a larger positive is to be subtracted from
a smaller positive… e.g. positive 3 – positive 4
SL3
AL4 “when positive and negative are equal the sum is zero”
AL5 “positive with zero is positive”
minus
Add ZERO

SL3
Add ZERO
Subtractminus
58

SL3
Add ZERO
Subtract
Simplify
59

SL3
Add ZERO
Subtract
Simplify
60

SL3
Add ZERO
Subtract
Simplify
61

a smaller positive, the sign of their difference is
reversed – positive becomes negative.
SL3
3 positives minus 4 positives is 1 negative

If a larger negative is to be subtracted from
a smaller negative, the sign of their difference is
reversed – negative becomes positive.
SL3
We don’t have enough negatives. So use AL4 and AL5.
2 negatives minus 4 negatives?

a smaller negative… e.g. negative 2 – negative 4SL3
AL5 “negative with zero is negative”
minus
Add ZERO

SL3
Add ZERO
Subtractminus
a smaller negative… e.g. negative 2 – negative 4

SL3
Add ZERO
Subtract
Simplify

If a larger negative is to be subtracted from a
smaller negative… the sign of their difference is
reversed – negative becomes positive.
SL3
2 negatives minus 4 negatives is 2 positives

negative minus zero is negative
SL4
4– minus 0– is 4–
positive minus zero is positive
zero minus zero is zero
3+ minus 0+ is 3+

negative … then it is to be added. –4 – +2
SL5
Add ZERO
–4 – +2 = ?
71

SL5
Add ZERO
Subtract
–4 – +2 = ?
72

SL5
Add ZERO
Subtract
–4 – +2 = –6
For you. Show how a negative
subtracted from a positive
is to be added with +3 – –5
73

Brahmagupta’s 4
Multiplication Laws
ML
(guṇana)
Text from: Plofker, K. (2009). Mathematics in India
74

ML1
The product of a negative
and a positive is negative
Brahmagupta multiplies positives and negatives by either
adding to zero multiple times or subtracting from zero multiple times.
–a × +b
–a added to zero b times
75
False from 1570: a into b, (a × b), is a added to itself b times

Negative Multiplicand Multiplied by Positive Multiplier
–a × +b
×
added to zero times
Next, from pictures to equations…
–2 × +3
–2 added to zero 3 times

Negative Multiplicand Multiplied by Positive Multiplier
–2 × +3 = –2 × (0 + 1 + 1 + 1)
–2 × +3 = 0 + –2 + –2 + –2
–2 × +3 = 0 + –6
–2 × +3 = –6
–a × +b
–2 × +3
–2 added to zero 3 times

Positive Integer Multiplied by Negative Multiplier
+2 × –3
Brahmagupta did not say “The product of a positive and a negative
is negative”. However, we can demonstrate this via his Laws.
+a × –b
+a subtracted from zero b times

Positive Multiplicand Multiplied by Negative Multiplier
×
+a × –b
+2 subtracted from zero 3 times
+2 × –3
As we don’t have any positive
twos to subtract, we first add
zero in the form +6 + –6

Positive Multiplicand Multiplied by Negative Multiplier
×
Next, from pictures to equations…
+a × –b
+2 subtracted from zero 3 times
+2 × –3

Positive Integer Multiplied by Negative Multiplier
+2 × –3
+2 × –3 = +2 × (0 – +1 – +1 – +1)
+2 × –3 = 0 – +2 – +2 – +2
+2 × –3 = 0 – +6
+2 × –3 = –6 + +6 – +6
+2 × –3 = –6 + 0
+2 × –3 = –6

ML2
The product of two
negatives is positive
–1 × –1 = +1
A demonstration goes like this…
Crabtree,JonathanJ.Snippet:Anew
reasonnegativemultipliedbynegativeis
positiveVinculum,Vol.52,No.3,Jul2015:
82

ML2
Negative Multiplicand Multiplied
by Negative Multiplier
–a × –b
–a subtracted from zero b times
–1 × –1
–1 subtracted from zero 1 times
83

Brahmagupta Defined ZERO in Law AL4
when positive and negative are equal the sum is ZERO
–1 × –1
84
–1 subtracted
from zero 1 times
+ –1+1

Brahmagupta Defined ZERO in Law AL4
when positive and negative are equal the sum is ZERO
+ –1
–1 × –1
–1 subtracted
from zero 1 times
now prove it equals +1
∴ –1 × –1 = +1
85
+1

ML3 The product of two
positives is positive
ML4 The product:
- of zero and a negative,
- of zero and a positive,
- or of two zeros
is zero.
86

Brahmagupta’s 4 Multiplication Laws
The product of a negative and a positive is negative.ML1
The product of two negatives is positive.ML2
The product of two positives is positive.ML3
The product of zero and a negative,
of zero and a positive, or
of two zeros is zero.
ML4
87

MY ASSESSMENT OF THE WORLD’S
PEDAGOGICAL EVOLUTION (628 to Now)

Brahmagupta’s 4
[Non-Zero]
Division Laws
DL
(haraṇa)

Brahmagupta’s 4 Division Laws
A positive divided by a positive is positive.DL1
A negative divided by a negative is positive.DL2
A positive divided by a negative is negative.DL3
A negative divided by a positive is negative.DL4
90

NOTE: Why Non-Zero Division Laws Only?
Whilst zero divided by any non-zero number (0/±n),
is zero, Brahmagupta says a non-zero number
divided by zero(i.e. ±n/0), remains ‘zero divided’.
Commentators such as Bhāskara II (1114 –1185)
suggested n/0 is infinite or indeterminate.
91

MODELS OF DIVISION
Partitive Model Equal Groups
+12 ÷ +4
Q. If you have 12 things and 4 equal
groups, how many things go in each group?
A. 3 things go in each group.
12 ÷ 4 = 3 things
Partitive Model ✔
92

MODELS OF DIVISION
Quotitive Model Repeated Subtraction
Q. If you have 12 things, how many times
can you subtract 4 things?
A. You can subtract 4 things
3 times until you get 0 things.
12 ÷ 4 = 3 times
Partitive Model ✔
Quotitive Model ✔
+12 ÷ +4
93

METHODS OF DIVISION
Equal Groups
Repeated Subtraction
Equal Groups
Equal Groups
Equal Groups
✔
✔
✔
✔
✘
✘
✘
✘
94

MODELS OF DIVISION
Partitive Model Equal Groups
+12 ÷ –4
Q. If you have 12 positive things and negative 4
groups, how many things go in each group?
A. You can’t have a negative
number of groups!
Equal Groups ✘
Model Fails
95

MODELS OF DIVISION
Quotitive Model Repeated Subtraction
Q. If you have 12 positive things, how many
times can you take away 4 negative things?
A. You can’t subtract 4 negative
things from positive things. Try...
+12 – –4 = +16, – –4 = +20, – –4 = +24. i.e.
You go the wrong way and never get to zero.
+12 ÷ –4
Equal Groups✘
Repeated Subtraction✘
96
Model Fails

PROPORTIONAL COVARIATION, PCV
Equal Groups✘
Numbers are counts or measures
of standard units. We choose 1.
All numbers are measured by
the unit called 1.
1 is the multiplicative identity
and the divisional identity.
97

Equal Groups✘
Our goal is to convert
12/4 into the form n/1
DENOMINATOR
To make –4 be +1 we take one of
the 4 parts of –4, which is –1 and
subtract it from zero to get +1.
98

Whatever we have done to the
denominator, we must do to the
numerator to keep the number (ratio)
NUMERATOR
We take one of the 4 parts of
+12, which is +3 and subtract it
from zero to get –3.
Equal Groups✘
99

Whatever we have done to the
denominator, we must do to the
numerator to keep the number (ratio)
Equal Groups✘
100
(Zero Minus Numerator) by
(Zero Minus Denominator)
–12
+4

Multiplication:
Whatever we do to the Unit 1 to make the Multiplier b
we do to the Multiplicand a to make the Product c.
Division:
Whatever we do to the Divisor b to make the Unit 1
we do to the Dividend a to make the Quotient c.
101

As –4 is to +1 +12 is to...

3 MODELS OF DIVISION
Equal Groups
Proportional Covariation Proportional Covariation
Proportional Covariation
Equal Groups
Repeated Subtraction✔
✔
✔
✔
✔
✔
103
Proportional Covariation ✔
+12 ÷ –4
✔

104

–2 x –3 = ? (i.e. negative two subtracted three times from zero)
What do we do to the standard Unit 1 to
make the Multiplier –3?
We place three Units and subtract them
from zero to make the Multiplier –3.

–2 x –3 = ? (i.e. negative two subtracted three times from zero)
What do we do to the Multiplicand –2
to make the Product ?
We place three Multiplicands and
subtract
them from zero to make the Product +6.

+2 –3 ?
+2 × –3 = ?
a × b = ?
We start off with
+1 in this square

Then we put b the
Multiplier here
+2 × –3 = ?
a × b = ?
We start off with
+1 in this square
+2 –3 ?

+2 –3 ?
Then we put a the
Multiplicand here
+2 × –3 = ?
a × b = ?
Then we put b the
Multiplier here
We start off with
+1 in this square

To go from +1 to –3 we
took 3 Units & changed
their sign by subtracting
from 0.
So, we take 3 a’s and
change their sign by
subtracting from 0 to
make c.
+2 × –3 = ?
a × b = ?
Then we put b the
Multiplier here
We start off with
+1 in this square
+2 –3 ?
Then we put a the
Multiplicand here

+2 × –3 = ?
a × b = ?
+2 –3 ?

+2 –3 ?
+2 × –3 = ?
a × b = ?

PCVMultiMat
PCVDiviMat

–2 x –3
As +1 is to –3 so –2 is to +6
PROPORTIONAL COVARIATION PCV
b
a
c
1

René Descartes
1596 – 1650
Brahmagupta
598 – 668
Brahmagupta’s ideas were not applied 1000 years later, yet should have been.

Copyright © 2016 Jonathan Crabtree All Rights Reserved
“For example, let AB be taken as unity, (1), and let it be required to
multiply BD (the multiplicand) by BC (the multiplier), I have only to join
the points A and C, and draw DE parallel to AC; and BE is the product
of this Multiplication.”
Applying Indian Logic to Descartes’s Multiplication

https://www.geogebra.org/m/wprjkwam
Let point B be 0 and line segments on the other side of 0 be negative...

https://www.geogebra.org/m/wprjkwam
A Negative Multiplicand and a Negative Multiplier result in a Positive Product.

https://www.geogebra.org/m/je3SEyQr
www.geogebra.org/m/ZBTZd6AF

www.bit.ly/New-x-Model

Discussion on Brahmagupta’s Laws
A. Even though the numbers are different, both
numerators are smaller than their denominators.
Q. How can the following fractions be equal?
+
1
+
2
=
+
4
+
8
127

If one side has a smaller number on the top and the other side
has a larger number on the top, even if the same numbers
are used as before, the two sides can’t be equal.
+
1
+
2
≠
+
8
+
4

Q. Is the following equation correct?
+
1
+
2
=
−
4
−
8
A. Yes. Therefore, just as +1 < +2 we must accept –4 < – 8, to be
consistent with Brahmagupta’s laws and the laws of proportion.
The British system has –4 > – 8, which is illogical given positives
and negatives are equal and opposite. Brahmagupta’s laws
of integer ordering are intuitive logical and correct!
129

The same number of positives and negatives
sum to zero as they are equal and opposite.
Therefore today’s integer inequality laws, based
on Greek foundations 1000 years older than India’s
are out-of-date and sub-optimal.

Which numbers are greater?
or
or
or
or

or
or
or
or
x x
x xxx xxxx xx
xxxx x xxx
xx xx xxxxx x

or
or
or
or
xx xx x x
xx xx x xxx
x x
xx xx x xxx xx
5+ < 7–
9+ > 4–
1+ < 3–
5+ ≹ 5–
xx xx

1+< 3+ 1–< 3– 2–= 2– 3–>1+ 3+>1–
1–< 3+ 1+< 3– 2+= 2+ 3–>1– 3+>1+
2+ 2–o=2+ are equal and opposite to 2–
134

…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
1 × 1 = 2?

136
Paper @ www.bit.ly/LostLogicOfMath

Returning India’s zero reveals unseen patterns.
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
Integral multiplication involves
either repeated addition or repeated
subtraction of the multiplicand from
zero, depending on the sign of the
multiplier.
The Billingsley ‘virus’of 1570 (BV1570) spread widely! CLICK HERE FOR MORE

When explaining a3, Sir Isaac Newton used the Latin word ‘bis’,
meaning ‘twice’, so a3 is a twice into itself. Yet because of Henry
Billingsley’s mistranslation of Euclid’s definition of multiplication, upon
translating from Newton’s Latin into English, the translator changed
Newton’s explanation.
To match Billingsley’s approach, (thought to be Euclid’s), Newton now
reads ‘… the Number 3 in the Quantity a3bb, does not denote that bb is to
be taken thrice, but that a is to be thrice multiplied by itself.’ So, today,
we find, nonsense definitions of exponentiation that simply do not work!
© 2018 J. J. Crabtree | www.jonathancrabtree.com© 2018 J. J. Crabtree | www.jonathancrabtree.com

23 ≠ 16
Cube ‘the result of multiplying a number,
quantity, or expression by itself three times’.
Collins Dictionary of Mathematics
If a3 is a into itself three times, we get the sequence
a into a, into a, into a.
If 23 is 2 into itself three times, we get the sequence
2 into 2, into 2, into 2, which is 16.
Just as India’s 0 went missing from definitions of multiplication,
we now know 1 went missing from definitions of exponentiation!

Returning the identity element one into 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
Integral exponentiation involves either
repeated multiplication or repeated
division of the base from one,
depending on the sign of the exponent.
140
a+b = 1 into a, b times
2+3 = 1 into 2, 3 times, 1 × 2 × 2 × 2
a–b = 1 by a, b times
2–3 = 1 by 2, 3 times, 1 ÷ 2 ÷ 2 ÷ 2
There is much confusion about cubes and cube roots. CLICK HERE FOR MORE

South
West
Left
Down
Debts
Loss
Deaths
Emigration
Cold
Decay
Below Zero
Less Than Enough
Below Ground
To the hour
Deceleration
Head Wind (knots)
Under Par (golf)
North
East
Right
Up
Assets
Profit
Births
Immigration
Hot
Growth
Above Zero
More Than Enough
Above Ground
Past the hour
Acceleration
Tail Wind (knots)
Over Par (golf)
Counts or measures
of negative units
Counts or measures
of positive units
Simple Symmetries of Quantity
©2018J.J.Crabtree|www.jonathancrabtree.com
141

It has been said…
“God created the universe from
nothing, from Shunya, from Zero”
Planet Positron
Planet Negatron
Wherever opposing quantities or forces are equal you will find zero.
142

It’s as if ZERO was split, creating
infinite real number lines from ZERO.

• China had positives and negatives in the 2nd Century BCE,
yet their first maths text with India’s zero, (with a symbol 0)
came in 1247 CE in The Mathematical Treatise in Nine
Sections, by Qin Jiushao.
• China used positives and negatives in their mathematics for
about1400 years without any concept of a number zero.
• The Chinese did not define negatives as less than zero. Their
positives and negatives were simply equal and opposite,
consistent with science and the philosophy of Yin and Yang.
144

Is zero defined as ‘a number subtracted from itself’ (n – n)?
No. Brahmagupta AL4 says zero is the sum of equal positive and
negative quantities. Zero is defined by addition, (–n + +n) not
subtraction, for use with Indian Laws of positives negatives and zero.
145

Are lessons on inequalities correct? e.g. (–4 > –7)
No. SL3 says if a larger negative is subtracted from a smaller negative
the sign of the difference is reversed and negative becomes positive.
We prove – 4 > –7 false by contradiction. Assume –4 > –7 true. We
subtract larger from smaller as –7 – –4 and the difference is –3. As the
sign of the difference is NOT reversed we cannot assume –4 > –7 true.
Thus –4 < –7 as seems obvious. (A debt of ₹400 is less than a debt of
₹700.) Brahmagupta’s SL3 also holds for greater positive from smaller
positive as +4 – +7 = –3 and the sign is reversed as SL3 states.
146

Are negative numbers ‘less than zero’?
No. Numbers represent count or measures of quantities, even abstract
units. The big bang created matter and antimatter. Equal in magnitude,
electrons have negative charge and positrons have positive charge.
When antimatter and matter meet, they annihilate each other, producing
energy. It is said “God made the Integers”. Those integers are consistent
with physics. NOTE asymmetry between matter versus antimatter in universe = Baryon asymmetry
“If God made the Integers, when it comes to fixing the foundations of
elementary mathematics, the Devil is in the detail!” Jonathan J. Crabtree
147

Are negative numbers ‘less than zero’?
No. Before the Chinese adopted India’s zero, they used positives and
negative for around 1400 years, with no need to move away from the
idea negative and positive are simply equal and opposite.
The idea negative numbers are less than zero emerged in 1685 via
Englishman John Wallis. If after marching East 5 yards, a man was
forced back 8 yards, Wallis said in total he advanced 3 yards less then
nothing . Chinese and Indians might have said the man retreated 3 yards
West. (We don’t say negative East.)
148

In our universe, the least quantity you will
ever have of anything is ZERO quantity.
Numbers are only ever counts or measures
of quantities, even if the quantity is a ‘unit’.
Maths is about relationships between quantities
and the quantities don’t have to be the same kind.

There’s also quantity of fuel and distance, which generate
the maths called fuel consumption.
For example, if people drive a car,
maths looks at the relationships
between quantities of distance and
time to get average speed.
151

ZERO (the equilibrium point where opposing
quantities are equal) is the starting point
of relations between quantities.
Numbers count or
measure quantities
and the study of …
… relations between
quantities is called
mathematical science.
159

Illustration by AFX Animation
Ryunosuke Satoro

161

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The Relaunch of Indian Mathematics

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The Relaunch of Indian Mathematics