EXERCISE SET 3.2
The temperature in the high-altitude desert of Ladakh is recorded
as 4 °C at noon. By midnight, it drops by 15 °C. What is the midnight
temperature?
2. A spice trader takes a loan (debt) of `850. The next day, he makes
a profit (fortune) of `1,200. The following week, he incurs a loss
of `450. Write this sequence as an equation using integers and
calculate his final financial standing.
3. Calculate the following using Brahmagupta’s laws:
(i) (–12) × 5 (ii) (–8) × (–7)
(iii) 0 – (–14) (iv) (–20) ÷ 4
4. Explain, using a real-world example of debt, why subtracting
a negative number is the same as adding a positive number
(e.g., 10 – (–5) = 15).
3.4 Filling the Spaces: Fractions and Rational
Numbers
As society grew more complex, measuring became just as important as
counting. If a farmer divides a field of wheat among his three children,
how much does each get? If a recipe calls for half a cup of ghee, how do
we represent that mathematically?
Numbers that represent parts of a whole are called fractions. Just as
every natural number has an additive inverse (3 has –3, 19 has –19, etc.),
we can conceive of additive inverses for every positive fraction: – 3
4 for
3
4 , –
19
7
for 19
7 , etc. Let us refer to such additive inverses of positivefractions as negative fractions. Note that in a negative fraction, the ‘–’
sign can also be placed with either the numerator or denominator, as
follows: – 1
5 =
–1
5 =
1
–5.
When we combine all integers and all fractions (both positive and
negative), we get the set of Rational Numbers, denoted by ℚ (for
quotient).
A rational number is defined as any number that can be expressed
in the form p
q , where p and q are integers and q ≠ 0.
Think and Reflect
Can you explain why we need q ≠ 0 in the definition of a rational number?
We must make some important observations at this stage.
* All rational numbers can be written in the form p
q , where p and
q are integers and q ≠ 0. For example, 5 and –10 can be written
as
5
1 and –10
1 , respectively. What this means is that the
rational numbers also include the natural numbers, whole
numbers and integers.
* Rational numbers do not have a unique representation in the
form p
q , where p and q are integers and q ≠ 0. For example, we have
– 1
3 = –
2
6 = – 3
9 = – 10
30 = – 2026
6078 and so on. These are equivalent
rational numbers (or equivalent fractions). This fact allows us
to freely divide any common factor between the numerator and
the denominator of a given fraction. The resulting fraction is
then equivalent to the original one. For example, 12
30 is equivalent
to 2
5. Here, we have divided both the numerator and the
denominator by 6.
* The common understanding is that when we say that p
q is a rational
number, or when we represent p
q on the number line, we assume
that q ≠ 0 and that p and q have no common factors other than1 (that is, p and q are co-prime). So, on the number line, among
the infinitely many fractions equivalent to 1
2 (i.e., the fractions 1
2,
2
4,
3
9,
6
12, … , 1013
2026 , …), we choose 1
2 to represent all of them.
As you will remember from Grades 6 and 7, Brahmagupta also gave
rules for addition, subtraction, multiplication, and division of fractions.
He noted that these rules also apply to both positive and negative
fractions which together constitute the rational numbers.
Here are the various laws:
1. Equality: Two rational numbers a
b
and c
d
are said to be equal if ad = bc.
2. Addition and subtraction of two rational numbers: We first
express the two rational numbers as fractions a
b
and c
b
with the
same denominator, b. Then we use the rules a
b
+
c
b
=
a + c
b
and
a
b
– c
b
=
a – c
b .
3. Multiplication and division of two rational numbers:
We use the rules a
b
c
d
ac
bd
× = provided that
(b ≠ 0, d ≠ 0) and a
b
c
d
a
b
d
c
ad
bc
÷ = × = provided that (b ≠ 0,
d ≠ 0, c ≠ 0).
4. With the arithmetic laws defined as above, addition and
multiplication are both commutative (i.e., a
b
c
d
c
d
a
b
+ = + and
a
b
c
d
c
d
a
b
× = × , and they follow the law of distributivity: if p, q and r
are rational numbers, then p (q + r) = pq + pr.
Rational numbers are closed under addition, subtraction, and
multiplication; that is, if one adds two rational numbers, or subtracts
two rational numbers, or multiplies two rational numbers, a rational
number is again obtained. Rational numbers are also closed under
division, provided that one does not divide by zero. That is, the quotient
of two rational numbers is again a rational number, as long as one does
not divide by zero.
Think and Reflect
1. While adding or subtracting two rational numbers having different
denominators, how will you make the denominators equal?
2. Verify the distributive law for rational numbers.