Number System: Numbers are highly significant and play an essential role in Mathematics that will come up in further classes. In lower grades, we learned how to count numbers. The base 10 number system, also known as the decimal number system, is what we are used to. The number system is a method of expressing numerical values.

It is how we represent or name the numbers. Examples of representing numbers using the number system are – Binary Number System, Decimal Number System, Octal Number System and so on. A number system is used for counting numbers and hence finds application in arithmetic calculations. Read this article to learn more about Number systems. 

What is a Number System?

A number system is a method of expressing numbers by writing, that is mathematical notation for representing the numbers of a given set, by using the numbers or symbols in a consistent manner. Let’s see all the important numbers:

1. Imaginary and Complex Numbers

A number that does not exist on the number line is known as an imaginary number. For example, the square roots of negative numbers are imaginary numbers. It is denoted by \(‘i’\).

Complex numbers are the set \(\{ a + bi\} \), where, \(a\) and \(b\) are the real numbers and ‘\(i = \sqrt { – 1} \)’.

2. Real Numbers

A combination of rational numbers and irrational numbers is known as real numbers. Real numbers can be both positive or negative which can be denoted as \(‘R’\). Natural numbers, fractions, and decimals all come under this category.

3. Rational Numbers

Rational numbers are the numbers that are expressed in the form of  \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime integers and \(q≠0\).

Numerator and Denominator: In the given form \(\frac{p}{q}\), the integer \(p\) is the numerator and the integer \(q (≠0)\) is the denominator. So, in \(\frac{{ – 3}}{7}\) the numerator is \(-3\) and the denominator is \(7\).

4. Irrational Numbers

An irrational number is defined as the number that cannot be expressed in the form of  \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime integers and \(q≠0\).

There are so many irrational numbers that cannot be written in the form \(\frac{p}{q}\), and some of the examples are \(√8, √11, √50\) and Euler’s number \(e=2.718281…..\), Golden ratio \(φ=1.618034…..\).

5. Integers

All numbers which do not have the decimal place in them are known as integers. \(Z {\text{ }} = {\text{ }}\left\{ { – \infty \ldots \ldots . – 3, {\text{ }} – 2, {\text{ }} – 1, {\text{ }}0, {\text{ }}1, {\text{ }}2,~3 \ldots \ldots \ldots + \infty } \right\}\). Positive Integers: \(1, 2, 3, 4…..\) are the set of all positive integers. Negative Integers: \(-1, -2, -3…..\) are the set of all the negative integers. Non-Positive and Non-Negative Integer: \(0\) is neither the positive integer nor the negative integer.

6. Natural Numbers

Since our childhood, we are using numbers \(1, 2, 3, 3, 4, …\) to count and calculate.  When counting objects in a group, we begin with one and work our way up to two, three, four, and so on. Counting objects in this manner is a normal process. As a result, \(1,2,3,4,,….\) are known as natural numbers.

However, the fractional numbers like \(\frac{5}{ {11}},\,\frac{ {27}}{4},\,\frac{ {123}}{ {5948}}\) and decimal numbers like \(7.54, 36.98, 675.563\) are not natural numbers. By adding one to any natural number, we get the next natural number. Therefore, there is no last or greatest natural number. So, \(1\) is the first natural number and there is no last natural number.

Properties of Natural Numbers: Some of the properties of natural numbers are as follows:

1. The \(1\)^st and the smallest natural number is \(1\).
2. Every natural number (except \(1\)) can be obtained by adding \(1\) to the previous natural number.
3. The smallest natural number is \(1\).
4. There is no greatest natural number. 
5. We cannot complete the counting of the natural numbers. There are infinitely many natural numbers.

7. Whole Numbers

The numbers \(1, 2, 3, 4, ….\) etc. are natural numbers. These natural numbers along with the number zero \(0\) form the collection of whole numbers. That is numbers \(0, 1, 2, 3, …\) are called as whole numbers. Thus, a whole number is either zero or a natural number.

Properties of Whole Numbers

Following are some properties of whole numbers:

1. The number \(0\) (Zero) is the first and the smallest whole number.
2. There is no such thing as the last or greatest whole number.
3. There are an infinite number of whole numbers or an uncountable number of whole numbers. All the natural numbers are whole numbers.
4. All whole numbers are not natural numbers. For example, \(0\) is a whole number but it is not a natural number.

8. Even Numbers

A number which is exactly divisible by the number \(2\), is called an Even Number. For example: \(36, 88, 106\), etc. All even numbers will always end with the digits \(0, 2, 4, 6\) or \(8\).

9. Odd Numbers

A number that is not exactly divisible by the number \(2\), is called an Odd Number. For example: \(33, 49, 357\), etc. All odd numbers will always end with the digits \(1, 3, 5, 7\) or \(9\).

What are the Types of Number Systems in Maths?

The number system is used to represent the information in a quantitative way. Some of the important types of number systems are given below:

1. Decimal Number System
2. Binary Number System
3. Octal Number System
4. Hexadecimal Number System

The Decimal number system is used in usual cases in mathematics. The other three number systems are used at a higher level in Mathematics, Physics, Science & Technology.

Number System Chart

What are the Number System Formulas?

The Formulas of the number system are given below:

1. (a–b)(a+b)=(a2–b2)(a–b)(a+b)=(a2–b2)
2. (a+b)2=(a2+b2+2ab)(a+b)2=(a2+b2+2ab)
3. (a–b)2=(a2+b2–2ab)(a–b)2=(a2+b2–2ab)
4. (a+b+c)2=a2+b2+c2+2(ab+bc+ca)(a+b+c)2=a2+b2+c2+2(ab+bc+ca)
5. (a3+b3)=(a+b)(a2–ab+b2)(a3+b3)=(a+b)(a2–ab+b2)
6. (a3–b3)=(a–b)(a2+ab+b2)(a3–b3)=(a–b)(a2+ab+b2)
7. (a3+b3+c3–3abc)=(a+b+c)(a2+b2+c2–ab–bc–ac)(a3+b3+c3–3abc)=(a+b+c)(a2+b2+c2–ab–bc–ac). When a+b+c=0a+b+c=0, then a3+b3+c3=3abca3+b3+c3=3abc.

Digits: To represent any number, we use ten symbols \(0, 1, 2, 3, 4, 5, 6, 7, 8\) and \(9\). These symbols are called digits or figures.

Numeral: A group of digits representing a number is known as a numeral. For example: \(59, 456, 2986\) etc, are numerals.

Notation: The method of denoting a number in digits or figures is known as notation. 

Numeration: The method of representing a number in words is known as numeration.

What are Methods of Numeration?

While reading numbers it is always easy to use the words instead of reading individual digits. For example: instead of reading \(527\) as Five, two, seven, it is easy to read as Five hundred and twenty-seven.

There are two common methods of numeration and they are:

1. Indian system of numeration
2. International system of numeration

What is the Indian System of Numeration?

This is based on the Vedic numbering system. In this, we split up the given numbers into groups or periods. We start from the extreme right digit of the given number and move to the left. The steps are listed below:

• Step 1: First three digits on the extreme right for a group of ones. The ones are divided into hundreds, tens, and units.
• Step 2: The second group of the next two digits on the left of the group of ones form the group of thousands which is further split into thousands and ten thousand.
• Step 3: The third group of the next two digits on the left of the group of thousands form the group of lakhs which is split up into lakhs and ten lakhs.
• Step 4: Then two digits on the left of the group of lakhs form a group of crores which is split up into crores and ten crores.

Indian Place Value Chart

How to Identify Place Value and Face Value of a Digit in a Numeral?

In the Indian system of numeration, each digit of a number has a place value and a face value. The place value of a digit depends on its position, whereas the face value does not depend on its position.

For example: in the number Six thousand eight hundred forty-seven, that is \(6847\), the face value of \(7\) is \(7\). Similarly, the face value of \(4, 8\) and \(6\) are also \(4, 8\) and \(6\) respectively. However, the digit:

\(7\) has the place value \(7×1=7\), since it is in units place.
\(4\) has the place value \(4×10=40\), since it is in tens place.
\(8\) has the place value \(8×100=800\), since it is in hundreds place.
\(6\) has the place value \(6×1000=6000\), since it is in thousands place.

It is evident from this that a number is the sum of the place values of all its digits. Also, Place value of a digit \(=\) Face value \( \times \) Position value.

What is the International System of Numeration?

The international system of numeration is followed by most of the countries in the world. In this system also, a number is split up into groups or periods. We start from the extreme right digit of the number to form the groups. The groups are called ones, thousands, millions, and billions.

1. The ones, in turn, are divided into hundreds, tens and units.
2. The second of the next three digits on the left of the group of ones form the group of thousands which is further split up into thousands, ten thousand and hundred thousand.
3. The third group of the next three digits on the left of the group of thousands for the group of millions.
4. Three digits on the left of the group of millions form a group of billions which is split into billions, ten billion and a hundred billion.

It is represented below in the tabular form:

International System of Numeration Chart

Relation Between Indian and International System of Numeration

The difference between the Indian and the International system of numeration has been represented below in the table form:

Number	Indian System	International System
\(1\)	One \(=1\)	One \(=1\)
\(10\)	Ten \(=10\)	Ten \(=10\)
\(100\)	Hundred \(=100\)	Hundred \(=100\)
\(1000\)	Thousand \(=1,000\)	Thousand \(=1,000\)
\(10000\)	Ten thousand \(=10,000\)	Ten Thousand \(=10,000\)
\(100000\)	Lakh \(=1,00,000\)	Hundred Thousand \(=100,000\)
\(1000000\)	Ten Lakh \(=10,00,000\)	One Million \(=1,000,000\)
\(10000000\)	One crore \(=1,00,00,000\)	Ten Million \(=10,000,000\)
\(100000000\)	Ten Crore \(=10,00,00,000\)	Hundred Million \(=100,000,000\)
\(1000000000\)	Hundred Crore \(=100,00,00,000\)	One Billion \(=1,000,000,000\)

How to Compare Numbers?

To compare the numbers, follow the given steps:

1. Obtain the numbers and check the number of digits in each number. If the numbers have an unequal number of digits, then the number with less digits is less than the number with more digits. If the numbers have the same number of digits then go with the step \(2\).
2. Compare the digits at the leftmost place in both numbers. The number with a larger digit will be greater than the number with a smaller digit. If the left-most digits are equal, then compare the second digits from the left and go to step \(3\).
3. If the second digits from the left are unequal, then the number with greater digit is larger. If the second digits from the left are equal, then compare the third digits from the left and continue the above procedure.

What are Roman Numerals?

The numerals \(0, 1, 2, 3, 4, …. 9\) are used in writing numbers. These numerals are of Indian origin, and they were adopted by the Arabs and spread across Europe. As a result, the system is known as Hindu-Arabic numerals. One of the earliest systems of numeration, still in common use today, was developed by Romans and is known as the Roman numeral system. This is used in different places like the numbering of different volumes or parts of books, numbers of chapters, numbers of issues of magazines numbers on clock faces etc.

There are seven distinct symbols (numerals) in the Roman numerals system. It is given below:

Roman Numeral	\(I\)	\(V\)	\(X\)	\(L\)	\(C\)	\(D\)	\(M\)
Hindu-Arabic numerals	\(1\)	\(5\)	\(10\)	\(50\)	\(100\)	\(500\)	\(1000\)

Using these symbols, we can write any number by following certain rules which are given below:

Rule 1: If a symbol is repeated, its value is added as many times as it occurs. For example:

\(II=1+1=2\)
\(XX=10+10=20\)
It is to be noted that a symbol is never repeated more than three times but symbols \(V, L\) and \(D\) are never repeated. Only \(I, X, C\) and \(M\) can be repeated.

Rule 2: If a symbol with a lower value is written to the right of a symbol with a higher value, its value is added to the greater symbol’s value. For example:

\(VI=5+1=6, VII=5+2=7, VIII=5+3=8\)
\(XI=10+1=11, XII=10+2=12\)
\(LXV=50+10+5=65\) etc.

Rule 3: If a symbol of a smaller value is written to the left of a symbol of greater value, its value is subtracted from the value of the greater symbol. For example:

\(IV=5-1=4\)
\(IX=10-1=9\)
\(XL=50-10=40\)
\(XC=100-10=90\)

Rule 4: The symbols \(V, L\) and \(D\) are never written to the left of a symbol of greater value i.e. \(V, L\) and \(D\) are never subtracted.

The symbol \(I\) can be subtracted from \(V\) and \(X\) only.
The symbol \(X\) can be subtracted from \(L, M\) and \(C\) only.
The symbol \(C\) can be subtracted from \(D\) and \(M\) only.

Rule 5: If a smaller numeral is placed between two larger numerals, it is always subtracted from the larger numeral immediately following. For example:

\(XIV=10+5-1=14\)
\(XIX=10+10-1=19\)
\(CXIV=100+10+5-1=114\)

Rule 6: If a bar is placed over a numeral, it is multiplied by \(1000\).
For example:
\(\underline V  = 5000,\,\underline L  = 50000\)

What are Prime and Composite Numbers?

Prime numbers: A number is called a prime number if it has no factor other than 1 and the number itself.

Each of the numbers \(2, 3, 5, 7, 11……\) etc. no factor other than \(1\) and itself. So, they are prime numbers.

Composite numbers: A number is called a composite number it has a least one more factor other than 1 and the number itself. Each of the numbers \(4, 6, 8, 9, 10, 12, 14, 15\), etc. have more than two factors. Such numbers are called composite numbers as defined below.

The number \(1\) is not a prime number.

Note: It should be noted that the number \(1\) is neither prime nor composite. It is the only number with this property. Every number other than \(1\) is either a prime number or a composite number.

Some Important Facts:

1. \(2\) is the smallest prime number.
2. \(2\) is the only even prime number. All other even numbers are composite numbers.
3. If a number is not divided by one of the primes less than half of it, then it is prime. Otherwise, it is a composite number.

Co-prime Numbers:  Two numbers are said to be co-prime if they do not have a common factor other than \(1\).

\(2, 3; 3, 4; 5, 6; , 8, 13; 12, 23\) etc. are pairs of co-prime numbers.

Remark: Any two prime numbers are always co-prime, but two co-primes need not be both prime numbers. For example: \(14, 15\) are co-primes, while none of \(14\) and \(15\) is a prime number.

Rule to check whether a number between \(100\) and \(200\) is prime or not:

Examine whether the given number is divisible by any prime number less than \(15\) i.e., \(2, 3, 5, 7, 11\) and \(13\). If it is divisible, then it is not prime; otherwise, it is prime.

The Number Zero

Zero is a special number. It is neither positive nor negative. \(0\) is both a number and the numerical digit used to represent the number in numerals. The number zero is denoted with the \(‘0’\) symbol. It looks like the below,

Following are some of the important properties of zero:

1. Zero is considered the additive identity for all types of numbers. If zero is added to any number, the result remains the same.
2. Zero is an even number.
3. Zero is neither positive nor negative.
4. Zero is a whole number.
5. Zero is an integer.

Successor and Predecessor in Numbers

The concept of Successor and Predecessor in the number system is generally applicable for natural numbers, whole numbers, and integers.

Successor: The successor of a number is the number obtained by adding \(1\) to it. Clearly, the successor of \(0\) is \(1\); successor of \(1\) is \(2\) and so on.

Predecessor: The predecessor of a number is one less than the given number. Clearly, the predecessor of \(1\) is \(0\); predecessor of \(2\) is \(1\); and so on.

Note: The natural number \(1\) and the whole number \(0\) does not have any predecessor. We observe that if a is the successor of \(b\), then \(b\) is the predecessor of \(a\).

Solved Examples – Number System

Let us look at some of the solved examples to understand Number System:

Q.1. Find the difference of the place values of two \(8’s\) in the number \(578493087\).

Ans: By inserting commas to separate periods the given number can be written as:

So, the number is \(57,84,93,087\)
We observe that the \(8\) in the second place from right is at ten’s place.
So, place value of \(8\) in ten’s place \(=8×10=80\).
The second \(8\) is at the ten lakh’s place.
So, place value of \(8\) in ten lakh’s place \(=8×1000000=8000000\)
Hence, the required difference \(=8000000-80=7999920\).

Q.2. How many \(5\)-digit numbers are there in all?
Ans: The largest \(5\)-digit number \(=10000\)
\(∴\) Numbers of all \(5\)-digit numbers \(=99999-10000+1\)
\(=89999+1\)
\(=90000\)
Hence, the number of all \(5\)-digit number is ninety thousand.

Q.3. Which is greater \(24,37,58,923\) or \(6,47,89,235\) ?
Ans: We can see that \(24,37,58,923\) consists of \(9\) digits while \(6,47,89,235\) consists of \(8\) digits.
We know that the number with more digits is greater. Hence, \(24,37,58,923>6,47,89,235\).

Q.4. Write the Roman numeral for the given number:
Ans: We have \(4592\). So now we will write the expanded form of the number
\(4000 + 500 + 90 + 2 = \underline  {IV}  + D + XC + II = \underline  {IV} DXCH\)
Hence, the required answer is given above.

Q.5. Write the successor and the predecessor of the number \(400099\).
Ans: The successor of \(400099\) is \(400099+1=400100\)
And the predecessor of \(400099\) is \(400099-1=400098\).

Summary

In this article, the topics covered are the number system, explained about the number system, types of the number system and the chart. Then, we discussed the real, rational, irrational, imaginary, integers, complex, even, odd, natural, and whole numbers, and then explained the properties of natural and whole numbers. There is a glance at methods of numeration, Indian system, international system of numbers, place value, comparison of numbers, the number zero, successor and predecessor of whole numbers, Roman numerals, prime and composite numbers, and co-prime numbers. Later the solved examples are given followed by the frequently asked questions. Clearly, the understanding of the number system in computer is very useful in our daily life.

Frequently Asked Questions (FAQs) on Number System

Let us look at some of the frequently asked questions about Number System.

Q. What is the formula of the number system?
Ans: The important formula of number system are:

1. (a–b)(a+b)=(a2–b2)(a–b)(a+b)=(a2–b2)
2. (a+b)2=(a2+b2+2ab)(a+b)2=(a2+b2+2ab)
3. (a–b)2=(a2+b2–2ab)(a–b)2=(a2+b2–2ab)
4. (a+b+c)2=a2+b2+c2+2(ab+bc+ca)(a+b+c)2=a2+b2+c2+2(ab+bc+ca)
5. (a3+b3)=(a+b)(a2–ab+b2)(a3+b3)=(a+b)(a2–ab+b2)
6. (a3–b3)=(a–b)(a2+ab+b2)(a3–b3)=(a–b)(a2+ab+b2)
7. (a3+b3+c3–3abc)=(a+b+c)(a2+b2+c2–ab–bc–ac)(a3+b3+c3–3abc)=(a+b+c)(a2+b2+c2–ab–bc–ac) when a+b+c=0a+b+c=0, then a3+b3+c3=3abca3+b3+c3=3abc.

Q. What is the real number system?
Ans: Real numbers are a set of numbers that contains all the rational and the irrational numbers that can be represented on the number line.

Q. Describe the number system with an example.
Ans: We have several ways to represent counting numbers. The usual “base ten” or the “decimal” system. Number system examples include : 27,345,2905,……27,345,2905,……. All in the decimal number system.

Q. How do you study the number system?
Ans: A number system is a way to represent numbers. The Base ten or the decimal system is one common number system. Other number systems are Number System Binary (base 2), Number System Hexadecimal (base 16), and Octal (base 8).

Q. What are the 4 types of number systems?
Ans: The four types of number systems are given below:

1. Binary Number System
2. Decimal Number System
3. Octal Number System
4. Hexadecimal Number System.

Q. How many types of numbers are there?
Ans: There are six types of numbers given below:

1. Natural numbers
2. Whole numbers
3. Integers
4. Rational numbers
5. Irrational numbers
6. Real numbers.

Q. What is a Binary Number System?
Ans: A Binary Number System is one that has a base of 2 and uses just two binary digits: 0 and 1.

Q. What is a Decimal Number System?
Ans: A Decimal Number System is one that has a base 10 and uses ten digits, starting from 0 to 9.

Q. What is an Octal Number System?
Ans: A Number system that has a base of 8 and uses numbers from 0 to 7 is called an Octal Number System.

Q. What is a Hexadecimal Number System?
Ans: A Number System that has 16 as its base number is called a Hexadecimal Number System.

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