Introduction to Number Systems

Introduction

Numbers are nothing but a means to quantify. It could be counting or measuring distance , time, mass etc. As these quantifications get a little complex that is when the number system comes to our rescue.

What is Number System?

A number system defines a set of values used to represent a quantity. The collection of numbers is also called a number system. These numbers are of different types such as natural numbers, whole numbers, integers, rational numbers and irrational numbers.

Let us see the table below to understand with the examples.

Natural Number	N	1,2,3,4,5,...
Whole Number	W	0,1,2,3,4,5...
Integers	Z	..., -3,-2.-1,0,1,2,3
Rational Numbers	Q	p/q fom, where p and q are integers and q is not zero
Irational Numbers		Which can't be represented as rational numbers

Natural Numbers

All the numbers starting from 1 till infinity are natural numbers, such as 1, 2, 3, 4, 5, 6, 7, 8,…….infinity. These numbers lie on the right side of the number line and are positive. 

Whole numbers

All the numbers starting from 0 till infinity are whole numbers such as 0,1, 2, 3, 4, 5, 6, 7, 8, 9,…..infinity. These numbers lie on the right side of the number line from 0 and are positive. 

Integers

Integers are the whole numbers which can be positive, negative or zero.

Example: 3, 21, 0, -78 are integers.

Rational Numbers

A number which can be represented in the form of p/q is called a rational number. For example, 1/2, 4/5, 26/8, etc.

Irrational Numbers

A number is called an irrational number if it can’t be represented in the form of ratio p/q.

Example: \(\sqrt{3},\sqrt{5},\sqrt{11},\) etc.

Real Numbers

The collection of all rational and irrational numbers is called real numbers. Real numbers are denoted by R. Every real number is a unique point on the number line and also every point on the number line represents a unique real number.

Terminating Decimals

If the decimal expression of a/b terminates. i.e. comes to an end, then the decimal so obtained is called Terminating decimals.

Example: \(\frac{1}{4}=0.25\)

Repeating Decimals

A decimal in which a digit or a set of digits repeats repeatedly periodically is called a repeating decimal.

Example: \(\frac{2}{3}= 0.666…\)

Rationalisation

If we have an irrational number, then the process of converting the denominator to a rational number by multiplying the numerator and denominator by a suitable number is called rationalisation.

Example: \(\frac{3}{\sqrt{2}} = (\frac{3}{\sqrt{2}}) \times (\frac{\sqrt{2}}{\sqrt{2}}) = 3\frac{\sqrt{2}}{2}\)

Laws Of Radicals

Let a>0 be a real number, and let p and q be rational numbers, then we have:

i)  (a^p).a^q = a^(p+q)

ii)  (a^p)^q = a^pq

iii) a^p/aq = a^(p-q)

iv)  a^p x b^p = (ab)^p

Problems

Question 1:

Which of the following numbers have terminating decimal expansion?

(A) \(\frac{8}{225}\)

(B)  \(\frac{5}{18}\)

(C) \(\frac{11}{21}\)

(D) \(\frac{21}{150}\)

Step 1: Divide the numerators by the denominators.

Step 2: Check the remainder.

Step 3: If the remainder is ‘0’. The quotient obtained is a terminating decimal expansion.

Step 4: If the remainder is repeating and we are not able to get a finite number of digits after the decimal point in the quotient, then it is termed as non terminating decimal.

In Question 1 only \(\frac{21}{150}\) ie 21divided by 150 gives us 0.14.

Answer 1(D) 21/150

Question 2:

If 2^x−3. 3^2x−8=36, then the value of x is____.

(A) 2

(B) 5

(C) 3

(D) 1

Step 1: Simplify 2^x−3. 3^2x−8  to get it in the form of expression a^p x b^p = (ab)^p

2^x−3. 3^2x−8=2^x/2³.3^2x/3⁸ since a^p/a^q = a^(p-q)

2^x/2³.3^2x/3⁸ =2^x/2³.(3²)^x/3⁸ since (a^p)^q = a^pq

2^x/2³.9^x/3⁸=18^x /2³.3⁸ since a^p x b^p = (ab)^p

Step 2: As per the equation 18^x /2³.3⁸=36

18^x =36.2³.3⁸

Step 3: Now as 18^x =36.2³.3⁸, we need to expand 36.2³.3⁸such that we get it

In the form of 18^x.

36.2³.3⁸=9.4.2³.3⁸=9.2².2²2.(3²)⁴=9⁵.2⁵=18⁵since a^p x b^p = (ab)^p

Step 4: Since 18^x = 18⁵

Therefore x=5

Answer 2 (B) 5

 

Question3:

If \(N=\frac{\sqrt{\sqrt{5}+2+{\sqrt{5}-2}}}{{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}}\), then N equals_________

(A) 1

(B) 2√2−1

(C) 5√2

(D) 2

Step 1:  \(N+\sqrt{3-2{\sqrt{2}}}=\frac{{\sqrt{\sqrt{5+}2}} \,+ \,{\sqrt{\sqrt{5-}2}}}{\sqrt{\sqrt{5+}1}}\)

Step 2:
\(\begin{align}N+(\sqrt{2}-1)&=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}\quad\text{ since } {\sqrt{3-2\sqrt{2}}}=\sqrt{2}-1\\&\qquad\text{use }  (a-b)^2=a^2+b^2-2ab\\&\qquad(a-b)(a+b)= a^2-b^2\end{align}\)

Step 3:
\(\begin{align}(N+(\sqrt{2}-1)) 2 &=\frac{\sqrt{5}+2+\sqrt{5}-2+2(\sqrt{5}+2)(\sqrt{5}-2)}{\sqrt{5}+1} \\ &=\frac{2(\sqrt{5}+1)}{(\sqrt{5}+1)} \\ &=2 \end{align}\)

Step 4: \(N+\sqrt2-1=\sqrt2  \)

Step 5: \(N=\sqrt2- \sqrt2+1=1 \)

Answer 2 (A) 1

 

Question 4:

Express the mixed recurring decimal 4.322222 in the form p/q.

(A) \(\frac{389}{90}\)

(B) \(\frac{329}{90}\)

(C) \(\frac{29}{90}\)

(D) \(\frac{233}{990}\)

Step 1: Let \(x=4.322222\)

Step 2:
\(\begin{align} 100 x &=432.222 \\ 100 x-x &=432.2222-4.322222 \\ 99 x &=427.9 \\ x &=\frac{427.9}{99}=\frac{4279}{990}=\frac{389}{90} \end{align}\)

Answer 4 (A) 1

 

Question 5:

Rational number \(\frac{−18}{5}\) lies between consecutive integers____.

(A) −2 and −3

(B) −3 and −4o;ko;k

(C) −4 and −5

(D) −5 and −6

Step 1: Divide the numerator by the denominator ie -18÷5.

Step 2: Put the answer -3.6 on the number line. -5, -4, -3.6,-3,-2,-1,0.

Step 3: Check where the answer is on the number line. It is between -3 and -4.

So the answer is -3 and -4.

Answer 5 (B) −3 and −4

Question 6:

If \(x=1−\sqrt{2},\) then find the value of \((x−1/x)^2\)

(A) 2

(B) 3

(C) 4

(D) 5

Step 1: \((\frac{x-1}{x})^2= (\frac{x_2-1}{x})^2= (\frac{(x+1)(x-1)}{x})^2\) since \(a^2-b^2=(a-b)(a+b)\)

Step 2: Put the value of x ie. \(1-\sqrt{2}\) in the above equation.

Step 3: \(\left(\frac{(1-\sqrt{2}+1)(1-\sqrt{2}-1)}{1-\sqrt{2}}\right)^{2}=\left(\frac{(2-\sqrt{2)}(-\sqrt{2})}{1-\sqrt{2}}\right)^{2}=\left(\frac{(\sqrt{2}-2) \sqrt{2})}{1-\sqrt{2}}\right)^{2}=\left(\frac{\sqrt{2}(1-\sqrt{2}) \sqrt{2})}{1-\sqrt{2}}\right)^{2}=(\sqrt{2} \cdot \sqrt{2})^{2}=4\)

SUMMARY

We have learned how to perform operations on different types of number systems using various properties and laws which help us to solve complex problems.

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