Number Systems: Important Question

The number system, the numeral system, is referred to as the system of naming or representing numbers. We already know that a number is a mathematical value that helps to count or measure objects, it also helps in performing various mathematical calculations. There are various number systems in Maths like decimal number system, binary number system, octal number system, hexadecimal number system. Important questions from Number System are given below, in order to provide a better understanding to class 9 students so that they are familiar with the topic in a much simpler manner. Let us take a look at them.

Check also: NCERT Solutions for Class 9 Mathematics Chapter 1: Number Systems

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Very Short Answer Questions [1 Mark Questions]

Ques. A given expression lays down 20 + 81 + 64 + 224. what will be the square root obtained?

Ans. Solution:

Let us assume that x = 20 + 81 + 64 + 216

Now, x = 20 +81+216 

= 20+9+2 4

= 29 + 8

= 37

Now, upon squaring both the sides, we get 

(x)² = (37)²

x = 37

Ques. What will be the value of the expression 22/4 + 2-1/3?

Ans. According to the rule of powers in number system, we know that when am+ an = am+n

On applying the rule, we get 

22/4 + (-1/3) 

22/4 - 1/3 (taking LCM of 4 and 3)

2(6 - 4)/12

22/12

21/6

Ques. Simplify the following equation, (5 + 9) (5 - 9).

Ans. Solution: 

5(5 ) - 5 (9) + 9(5) - 9(9)

5 - 45 + 45 - 9 (both get cancelled out)

5 - 9

-4

Therefore, - 4 is the value of the given expression (5 + 9) (5 - 9). 

Ques. If x = 4 + 25, then find the value of x²?

Ans. Here, x = 4 + 25

Squaring upon both sides we get,

x² = (4+25)²

Now, by the formula of (a + b)² = a² + b² +2ab

42+ (25)²+ 2425

16 + 45 + 165

16 + 20 + 165

36 + 165

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Short Answer Questions [2 Marks Questions]

Ques. What is a rational number? Mention all rational numbers coming between 2 and 3. 

Ans. in Number System, a number that can be expressed in the form of a quotient, or a fraction of two integers like pq, where p is the numerator, and d plays the role of a denominator, is defined as a rational number. For example, in order to find out the rational numbers between 2 and 3, we will have to write down all the numbers which have the denominator 5+1=6. Now, we know that 2 = 2 6/6, and 3 = 3 6/6. This will give us 12/6, and 18/6 respectively. Therefore, 13/6, 14/6, 15/6, 16/6, 17/6 are all the rational numbers which fall between 2 and 3 in the number system. 

Ques. Which of the following values are equal?

1. 14
2. 40
3. 04
4. 41

Ans. a) and b) are equal in value

Solution: The values of the expression 14 and 40 are equal because, the former implies four times one, where the latter denotes 1. In simpler terms, we can write the former as 14=1111, on the other hand, we know by formula, anything to the power 0, will give us 1 in value. This proves it that 40=1.Therefore, option a) and option b) are equal upon being calculated. 

Ques. Given that, 0.5555 is a recurring decimal. Can this be expressed in the form of p/q?

Ans. Yes, the given decimal 0.55555 can definitely be expressed in p/q form. However it should be remembered that p and q, both must be integers, and q should not be equal to 0, or q 0. To do this, we need to multiply 10 with the given decimal, that is, 10x = 5.555…. and so on. Now, we may assume that 5.5555 = 5 + x, where x = 0.5555… . So now we have, 

10x = 5 + x

10x - x = 5

9x = 5

x = 5/9 

Thus from the above calculation, we can conclude that 5/9 is in p/q form where q is not equal to 0. 

Ques. What are known as irrational numbers in the number system? Write down some irrational numbers between 2/7 and 8/7. 

Ans. In simple words, all the real numbers which are not rational by nature are called irrational numbers. Irrational numbers cannot be laid down in the form of p/q form or expressed in ratio format. To calculate irrational numbers, at first, we would need to convert both fractions into decimal format. For example, 2/7 = 0.285714, and 8/7= 1.142857. Now, let's say we are finding out 4 irrational numbers. So 4 irrational numbers between 2/7 and 8/7 are given as, 

0.30010001000

0.42004200034200

1.0112314001

1.111230001

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Long Answer Questions [3 Marks Questions]

Ques. When one rational number is added to another irrational number, will the result be rational or irrational?

Ans. As we know that in the Number System, all rational numbers are always closed under addition, this would mean that, b = nm + (d - c), which by nature is rational. But according to assumptions, it has been deduced that b is irrational. It is evident that it is either of the two, and cant be both. Thus, it can be concluded that when a rational and an irrational expression come together to form their sum, it will always be irrational in nature. For example, we can say that, ¼ + 2 is irrational. 

Ques. What is the value of (7 + 4) (7 - 4). 

Ans. Solution:

By formula, we know that, (a + b)(a - b) = a²- b²

Therefore, upon applying the formula in the given expression, we get 

7(7 - 4) + 4 (7 - 4)

49 - 28 + 28 - 16 ( both of them get canceled due to opposite signs )

49 - 16

33

Hence, 33 is the value of the given equation (7 + 4) (7 - 4). 

Ques. If one rational number is added to another irrational number, what will be the nature of the resultant expression?

Ans. Let us assume that x is the rational number, and y is the irrational number. It is mandatory that x + y = irrational number. If we take x as 2, which is a rational number and y as √11 , which is an irrational number. x +y = 2 + √11. Let us consider p/q = 2 + √11. Upon squaring both sides, we get, 

p²/q² = 4 + 11 + 4√11

p²/q² = 15 + 4√11

p²/q² - 15 = 4√11

(p²/q² - 15)/ 4= √11

This proves that p/q is rational, whereas (p²/q² - 15)/ 4 is irrational. 

Ques. Is it necessary for a natural number to always be an irrational number in the number system?

Ans. No. A natural number has a different set of characteristics and they cannot be irrational numbers. It should be remembered that natural numbers are always present in the form of whole numbers. The reason behind this is, all integers are those numbers which begin from 0, going till infinity, whereas natural numbers start from 1 and are never ending. Therefore they are known as all positive integers. We cannot consider negative numbers as whole numbers. It must be noted that all natural numbers may be called whole numbers, but all whole numbers cannot be called natural, as 0 is a whole number by nature. 

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Very Long Answer Questions [5 Marks Questions]

Ques. Rationalise the denominator of 1/[7+3√3].

Ans. Solution:

Given that, 1/(7 + 3√3)

In the Number System, in order to rationalise the denominator consisting of square root, we know that first we need to multiply, and then divide the provided fraction, with the exact value of the square root. 

1(7 + 33)(7 - 33)(7 - 33)

Upon equating the expression, we get, 

(7 - 33) / (7 + 33)(7 - 33) [ solving it in the (a + b) (a - b) form which is a²- b² ]

(7 - 33) / 7 (7 - 33) + 33(7 - 33)

(7 - 33) / 49 - 213 + 213 - 9 3 (both get cancelled out)

(7 - 33) / 49 - 27

(7 - 33) / 22

Ques. Can zero be called a rational number? If yes, then explain how?

Ans. As we know, a rational number is a number which can be expressed in the p/q form. Upon putting zero into this format, it can be called a rational number. For this, we would need to make sure that p and q are integers, and q is not equal to 0. To make this possible, we can put 0 in the place of the numerator, like 03, 04, 07, and so on. Here, q is not equal to 0, so this satisfies the condition of being a rational number. Therefore in the Number System, 0 can be called a rational number. 

Ques. Is it possible to convert 0.22222 in the form of p/q?

Ans. Let x = 0.2222 - (i) first equation 

Multiplying 10 on both sides we get

10x = 2.22222 - (ii) second equation 

Upon subtracting equation i from ii, we get ,

10x = 2.2222

- x = 0.2222

9x = 2

Therefore, x = 2/9

Hence, on converting 0.22222 in the p/q form, we get, 29 as the answer. 

Ques. How to express 7.8 on the number line in a graph?

Ans. First, take a fixed point, let's say X, and mark 7.8 units starting from this point. Mark it in such a way on the line in the graph, so that it gives Y. this would mean, AB = 7.8 units. From the new point Y, mark a distance of 1 unit to call it to point Z. calculate the midpoint of XZ, and name it as P. Draw a semicircle, with P as its perfect center, such that it would give PZ 4.4 units as its exact radius. Draw a line which falls perpendicular to XZ, and passes through point Y, cutting the newly formed semicircle at Q. YQ = 7.8. 

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