Mathematics Questions and Answers – Real Numbers and their Decimal Expansions

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Real Numbers and their Decimal Expansions”.

1. Which of the following is the correct way to write \(5.4\bar{12}\)?
a) 5.412412412…
b) 5.4121212…
c) -5.412
d) 5.4 – 12
View Answer

Answer: b
Explanation: 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.
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2. 0.3454545…..=_____________
a) 345/1000
b) 350/995
c) 355/900
d) 342/990
View Answer

Answer: d
Explanation: 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
View Answer

Answer: a
Explanation: 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.’
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4. 0.3333… is a/an ______________
a) natural number
b) whole number
c) rational number
d) irrational number
View Answer

Answer: c
Explanation: 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
View Answer

Answer: b
Explanation: 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.
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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….
View Answer

Answer: b
Explanation: 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
View Answer

Answer: a
Explanation: 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.
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8. √5 is a/an ___________
a) natural number
b) whole number
c) rational number
d) irrational number
View Answer

Answer: d
Explanation: √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.

Sanfoundry Global Education & Learning Series – Mathematics – Class 9.

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Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter