Mathematics Multiple Choice Questions on “Real Numbers and their Decimal Expansions”.
1. Which of the following is the correct way to write (5.4bar{12})?
a) 5.412412412…
b) 5.4121212…
c) -5.412
d) 5.4 – 12
Answer: b
Clarification: Bar sign is used to show the blocks of numbers which are repeated. Here, the sign of ‘bar’ ( ̅ ) is only over two numbers, 1 and 2 which means block of these two numbers is repeated infinite times. There is no sign of bar over ‘4’ which shows that ‘4’ is not repeated.
2. 0.3454545…..=_____________
a) 345/1000
b) 350/995
c) 355/900
d) 342/990
Answer: d
Clarification: Let 0.3454545…=x
Then, 100x=34.54545…..
100x=34.2 + 0.3454545…
100x=34.2 + x
99x=34.2
Then, x=34.2/99
x=342/990.
3. The decimal expansion of rational numbers is either terminating or non-terminating and recurring (repeating).
a) True
b) False
Answer: a
Clarification: Let’s take a known rational number to understand this.
For example, 3/4
We know that 3/4 is a rational number because both 3 and 4 are natural
numbers and 3/4=0.75 which is terminating expansion.
Now, let’s take one example of non-terminating and recurring expansion.
For example 0.5787878…
Let x=0.5787878…
Then 100x=57.878787…
100x=57.3 + 0.5787878…
100x=57.3 + x
99x=57.3
Then, x=57.3/99
x=573/990
This is of the form of p/q where p and q are natural numbers. Hence we can say that .5787878… is rational number.
Hence, we can conclude that the decimal expansion of rational numbers is either terminating or non-terminating and recurring (repeating).
We can also conclude that ‘The decimal expansion of irrational numbers is non-terminating and non-recurring.’
4. 0.3333… is a/an ______________
a) natural number
b) whole number
c) rational number
d) irrational number
Answer: c
Clarification: Let 0.333…=x
Then, 10x=3.33333…
10x=3 + 0.333…
10x=3 + x
9x=3
Then, x=3/9=1/3
This is of the form p/q where p and q are natural numbers. Hence by definition, 0.3333…. is a rational number.
Another method: Decimal expansion of rational numbers is either terminating or non-terminating and recurring (repeating).
As we can see that 0.3333… is non-terminating and recurring so it is a rational number.
5. 0.238765563246… is a/an __________
a) rational number
b) irrational number
c) natural number
d) whole number
Answer: b
Clarification: Given number is not an integer, so it does not belong to natural number and whole number set.
It is not-terminating and non-recurring, so we can say that 0.238765563246…is an irrational number and not a rational number.
6. Which of the following number is an irrational number between 1/3 and 2/3?
a) 0.3333….
b) 0.343403400…
c) 0.55555….
d) 0.67670671….
Answer: b
Clarification: 1/3=0.3333… and 2/3=0.6666…0.333… and 0.555… are non-terminating and recurring, so they can be represented in the form p/q where p and q are integers. Hence they are rational numbers. 0.343403400 and 0.67670671… are non-terminating and non-recurring. Hence they are irrational numbers. But, among 0.343403400…and 0.67670671, only 0.343403400 is between 1/3 and 2/3. So option 0.343403400… is correct.
7. There are infinite irrational numbers between two numbers.
a) True
b) False
Answer: a
Clarification: Let’s take any two numbers to understand this.
For example, 3/4 and 4/5
We know that 3/4=0.75 and 4/5=0.8
For a number to be irrational, it has to be non-terminating and non-recurring.
We can find infinite numbers between 0.75 and 0.8 which are non-terminating and non-recurring.
Like 0.756757758…., 0.78754729… and so on.
8. √5 is a/an ___________
a) natural number
b) whole number
c) rational number
d) irrational number
Answer: d
Clarification: √5=2.236067…
This is not an integer, so it doesn’t belong to natural and whole number set by definition. Moreover, expansion of √5 is non-terminating and non-recurring. Hence, it is an irrational number.