Introduction:
In Mathematics, we frequently come across simple equations to be solved. For example,
the equation x + 2 = 13 (1)
is solved when x = 11, because this value of x satisfies the given equation. The solution
11 is a natural number. On the other hand, for the equation
x + 5 = 5 (.. 2)
the solution gives the whole number 0 (zero). If we consider only natural numbers,
equation (2) cannot be solved. To solve equations like (2), we added the number zero to
the collection of natural numbers and obtained the whole numbers. Even whole numbers
will not be sufficient to solve equations of type
x + 18 = 5 ................ (3)
Do you see ‘why’? We require the number –13 which is not a whole number. This
led us to think of integers, (positive and negative). Note that the positive integers
correspond to natural numbers. One may think that we have enough numbers to solve all
simple equations with the available list of integers. Now consider the equations
2x = 3 ................(4)
5x + 7 = 0 .............(5)
for which we cannot find a solution from the integers. (Check this)
We need the numbers 3/2 to solve equation (4) and -7/5 to solve equation (5). This leads us to the collection of rational numbers.
We have already seen basic operations on rational
numbers. We now try to explore some properties of operations
on the different types of numbers seen so far.
Properties of Rational Numbers
(i) Whole numbers
Let us revisit the closure property for all the operations on whole numbers in brief.
Operation Numbers Remarks
Addition 0 + 5 = 5, a whole number Whole numbers are closed
4 + 7 = ... . Is it a whole number? under addition.
In general, a + b is a whole
number for any two whole
numbers a and b.
Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed
whole number. under subtraction.
Multiplication 0 × 3 = 0, a whole number Whole numbers are closed
3 × 7 = ... . Is it a whole number? under multiplication.
In general, if a and b are any two
whole numbers, their product ab
is a whole number.
Division 5 ÷ 8 =
5
8
, which is not a
whole number.
Check for closure property under all the four operations for natural numbers.
(ii) Integers
Let us now recall the operations under which integers are closed.
Operation Numbers Remarks
Addition – 6 + 5 = – 1, an integer Integers are closed under
Is – 7 + (–5) an integer? addition.
Is 8 + 5 an integer?
In general, a + b is an integer
for any two integers a and b.
Subtraction 7 – 5 = 2, an integer Integers are closed under
Is 5 – 7 an integer? subtraction.
Next part of rational number is naxt blog
