Math Concepts

# Introduction to Number Systems

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 1 Introduction 2 What is the Number system? 3 Introduction to Rational, Irrational, Real numbers 4 Rationalisation 5 Laws of Radicals 6 Example Problems 7 Summary 8 FAQs

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

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

i)  (ap).aq = a(p+q)

ii)  (ap)q = apq

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

iv)  ap x bp = (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.

Question 2:

If 2x−3. 32x−8=36, then the value of x is____.

 (A) 2 (B) 5 (C) 3 (D) 1

Step 1: Simplify 2x−3. 32x−8  to get it in the form of expression ap x bp = (ab)p

2x−3. 32x−8=2x/23.32x/3since ap/aq = a(p-q)

2x/23.32x/38 =2x/23.(32)x/3since (ap)q = apq

2x/23.9x/38=18x /23.38 since ap x bp = (ab)p

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

18x =36.23.38

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

In the form of 18x.

36.23.38=9.4.23.38=9.22.222.(32)4=95.25=185since ap x bp = (ab)p

Step 4: Since 18x = 185

Therefore x=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: $$N=\sqrt2- \sqrt2+1=1$$

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}

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.

Cuemath, a student-friendly mathematics and coding platform, conducts regular Online Live Classes for academics and skill-development, and their Mental Math App, on both iOS and Android, is a one-stop solution for kids to develop multiple skills.

## Who invented the number system?

There are many theories making rounds for this questions but the below link can broadly

Explain you about how the number system came into existence.

https://www.quora.com/Who-created-the-number-system?no_redirect=1

## What is the decimal number system?

Decimal number system is also called the base 10 number system.Its represented using 10 digits from 0 to 9. The reason why it's called so is because using this system any number can be represented using the 10 digits from 0 to 9. Decimal number system is also known as the positional value system wherein the value of the digit depends on its position in the number.Let us take a few examples.

Take 3 numbers with 8 in different positions --- 815, 781,158

In 815, value of 8 is 8 hundreds or 800 or 8$$×$$ 100

In 781, value of 8 is 8 tens or 80 or 8 $$×$$ 10

In 158, value 0f 8 is 8 units or 8 or 8 $$×$$ 1

## What are real numbers?

A real number is all the positive or negative numbers that includes all integers any natural number or whole number also is a real number. All rational and irrational numbers are also real numbers. Rational numbers can be represented as fraction and irrational numbers can be represented by an infinite decimal representation (3.1415926535...).

Written by Richa Khare, Cuemath Teacher

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