EXERCISE 3.1
1. A merchant in the port city of Lothal is exchanging bags of spices
for copper ingots. He receives 15 ingots for every 2 bags of spices. If
he brings 12 bags of spices to the market, how many copper ingots
will he leave with?
2. Look at the sequence of numbers on one column of the Ishango
bone: 11, 13, 17, 19. What do these numbers have in common? List
the next three numbers that fit this pattern.
3. We know that Natural Numbers are closed under addition (the sum
of any two natural numbers is always a natural number). Are they
closed under subtraction? Provide a couple of examples to justify
your answer.
*4. Ancient Indians used the joints of their fingers to count, a practice
still seen today. Each finger has 3 joints, and the thumb is used to
count them. How many can you count on one hand? How does this
relate to the ancient base-12 counting systems?
3.2 The Revolution of Śhūnya: When Nothing
Became Something
For millennia, the number line started at 1. If you had five apples and
gave all five away, you did not have a number to represent your state;
you simply had a void, a lack of apples. Civilisations like the Babylonians
and Mayans used placeholders — symbols to indicate an empty column
in a number — but they did not treat ‘nothing’ as a number that you
could add, subtract, and multiply.
It was in the work of Brahmagupta (628 CE) that the void was formally
transformed into a number, which truly transformed mathematics.
This monumental leap was in turn inspired by Indian philosophical
traditions.
3.2.1 From Philosophy to Mathematics: The Concept of
Śhūnyatā
In the Upanishads and in the vast Buddhist literature starting well before
the 7th century BCE, the concept of Śhūnyatā (emptiness or nothingness)was a profound state that was the goal of yoga and meditation. The
word śhūnya means zero, and the word śhūnyatā (zeroness) was used
extensively in these works to describe the state that one is trying to
reach during meditation — that of emptying one’s mind of all vṛttis
(fluctuations of the mind) to achieve the state of perfect stillness and
tranquility. Patanjali in the Yoga Sutras around the 3rd century BCE also
describes how śhūnyatā can help lead to control over one’s mind, body,
and senses.
Because Indian thinkers revered this state of ‘emptiness’, they
possessed the conceptual framework necessary to welcome ‘nothingness’
as a concept in philosophy. This concept of zeroness then found its
way into many fields, such as architecture, literature, linguistics, and
eventually made its way into mathematics in the works of Āryabhaṭa
and finally Brahmagupta.
Thus, the philosophical concept of emptiness crystallised into the
mathematical zero.
3.2.2 The Bakhśhālī Manuscript and Brahmagupta’s Rules
In the Hindu Number System that we use today, the physical transition
from a blank space to a symbol can be seen in the Bakhśhālī
Manuscript. Dated to the early centuries CE, it features a bold dot
(bindu) used to represent zero.
However, a symbol is just a mark on a page until it has rules.
The transformation of zero into a fully operational number
occurred in the work of Brahmagupta. In his seminal work, the
Brāhmasphuṭasiddhānta (628 CE), he explicitly defined zero as the
result of subtracting a number from itself (a – a = 0).
Brahmagupta then laid down the fundamental laws of arithmetic
with śhūnya:
Brahmagupta’s Rules for Zero
* When zero is added to a number, the number remains unchanged:
a + 0 = a.
* When zero is subtracted from a number, the number remains
unchanged: a – 0 = a.
* When any number is multiplied by zero, the result is zero: a × 0 = 0.
3.3 Integers: Expanding the Horizon
Brahmagupta did not stop at zero. He realised that if subtraction of a
number from itself can result in zero (5 – 5 = 0), then what would happen
if we subtracted a larger number from a smaller one (3 – 5 = )?
To answer this, Brahmagupta grounded his mathematics in the
reality of commerce and life.
He recognised two states:
* Fortunes (Dhana): Positive numbers, representing wealth or assets.
* Debts (Ṛiṇa): Negative numbers, representing debts.
By moving to the left of zero on the number line, Brahmagupta
formally introduced Negative Numbers to the world. The combination
of positive natural numbers, their negative counterparts, and zero
creates the set of Integers, denoted by the symbol Z (from the German
word Zahlen, meaning numbers).
3.3.1 The Arithmetic of Integers
Brahmagupta gave explicit rules for adding and multiplying these
integers, which we still use exactly as he wrote them over 1,300 years ago:
1. A fortune plus a fortune is a fortune: 5 + 4 = 9.
2. A debt plus a debt is a debt: (– 5) + (– 4) = –9. (If you owe `5 and
borrow `4 more, you owe `9.)
3. A fortune minus zero is a fortune, a debt minus zero is a debt:
7 – 0 = 7, and – 6 – 0 = – 6.
4. The product of a debt and a fortune is a debt: (– 3) × 4 = –12.
(If you take on 4 debts of `3, your total debt is `12.)
5. The product of two debts is a fortune: (–3) × (–4) = 12.
Think and Reflect
Why does a negative times a negative equal a positive? Think of it in
terms of action and debt. If a negative number represents a debt, then
multiplying by a negative number represents the removal of that debt.
(Hint: If someone takes away (–) four of your debts that are each
worth `3 (that is, –3), you are effectively `12 richer! Therefore,
(–3) × (– 4) = +12.)