# Number Systems

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 1 What is a Number System? 2 Types of Number Systems in Maths 3 Difference Between Decimal and Binary Number Systems 4 Number Systems Chart 5 Number Systems Conversions 6 Important Notes on Number Systems 7 Number System Conversions Calculator 8 Solved Problems on Number Systems 9 Challenging Questions on Number Systems 10 Practice Questions on Number Systems 11 Important Topics of Number Systems 12 Maths Olympiad Sample Papers 13 Frequently Asked Questions (FAQs)

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## What is a Number System?

• A number system is a system of representing numbers.
• It is also called “the system of numeration”.
• It defines a set of values to represent a quantity.

For example, the system of numbers we use to represent the numbers in daily life use the $$10$$ digits $$0,1,2,3,4,5,6,7,8,$$ and $$9$$, and this system is known as the “Decimal Number System”. ## Types of Number Systems in Maths

We have different types of number systems in Maths. But there are $$4$$ main types of number systems:

1. Binary number system (Base - $$2$$)
2. Octal number system  (Base - $$8$$)
3. Decimal number system  (Base - $$10$$)
4. Hexadecimal number system  (Base - $$16$$)

We will study each of these systems one by one in detail.

### Binary Number System

• The binary number system uses only two digits: $$0$$ and $$1$$.

• Thus the numbers in this system have a base $$2$$.

• $$0$$ and $$1$$ are called bits.

• $$8$$ bits together make a byte.

• The data in computers is stored in terms of bits and bytes. Examples:

$$10001_2, 111101_2, 1010101_2$$ are some examples of numbers in the binary number system.

### Octal Number System

• The octal number system uses eight digits: $$0,1,2,3,4,5,6$$ and $$7$$.

• Thus the numbers in this system have a base $$8$$.

• The advantage of this system is it has less number of digits when compared to several other systems and hence there would be fewer computational errors. Examples:

$$35_{8}, 923_{8}, 141_{8}$$ are some examples of numbers in the octal number system.

### Decimal Number System

• The decimal number system uses ten digits: $$0,1,2,3,4,5,6,7,8$$ and $$9$$.

• Thus the numbers in this system have a base $$10$$.

• If we see any number without any base, it means that its base is $$10$$.

• This is the system that we generally use to represent the numbers in real life. Examples:

$$723_{10}, 32_{10}, 4257_{10}$$ are some examples of numbers in the decimal number system.

• The hexadecimal number system uses sixteen digits/alphabets: The digits $$0,1,2,3,4,5,6,7,8, 9$$ and the alphabets $$A,B,C,D, E, F$$.

• Thus the numbers in this system have a base $$16$$.

• Here, $$A-F$$ of the hexadecimal system means the numbers $$10-15$$ of the decimal number system respectively.

• This system is used in computers to reduce the large-sized strings of the binary system. Examples:

$$7B3_{16}, 6F_{16}, 4B2A_{16}$$ are some examples of numbers in the hexadecimal number system.

## Difference Between Decimal and Binary Number Systems

The differences between the decimal and binary number systems are as follows:

Decimal Number System Binary Number System
It uses $$10$$ digits:$$0,1,2,3,4,5,6,7,8$$ and $$9$$ It uses $$2$$ digits: $$0$$ and $$1$$.
Its base is $$10$$ and it doesn’t need to be written always to show that the number is in the decimal number system. Its base is $$2$$ and it has to be written to show that the number is in the binary number system.
It is the most general system used in daily life. It is the system used by the computers to store and process the data.

## Number Systems Chart

The information about number systems is summarised in the following chart: CLUEless in Math? Check out how CUEMATH Teachers will explain Number Systems to your kid using interactive simulations & worksheets so they never have to memorise anything in Math again!

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## Number Systems Conversions

A number can be converted from one number system to another number system.

We will see how to do this.

### Binary / Octal / Hexadecimal System to Decimal System

To convert a number from binary / octal / hexadecimal system to the decimal system, we use the following steps.

The steps are shown by an example of a number in the binary system.

Example:

Convert $$100111_2$$ into the decimal system.

Solution:

Step 1: Identify the base of the given number.

Here, the base of $$100111_2$$ is $$2$$.

Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.

The exponents should start with $$0$$ and increase by $$1$$ every time as we move from right to left.

Since the base here is $$2$$, we multiply the digits of the given number by $$2^0, 2^1, 2^2,...$$ from right to left. Step 3: We just simplify each of the above products and add them. Here, the sum is the equivalent number in the decimal number system of the given number.

Or, we can use the following steps to make this process simplified.

\begin{align} 100111 &= 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3+ 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\[0.3cm] &= 1 \times 32 + 0 \times 16 + 0 \times 8 + 1 \times 4 + 1 \times 2 + 1 \times 1 \\[0.3cm] &= 32 +0+0+4+2+1\\[0.3cm] &= 39 \end{align}
Thus,

 $$\therefore 100111_2 = 39_{10}$$

### Decimal to Binary / Octal / Hexadecimal System

To convert a number from decimal system to binary / octal / hexadecimal system, we use the following steps.

The steps are shown on how to convert a number from the decimal system to the octal system.

Example:

Convert $$4320_{10}$$ into the octal system.

Solution:

Step 1: Identify the base of the required number.

Since we have to convert the given number into the octal system, the base of the required number is $$8$$.

Step 2: Divide the given number by the base of the required number and note down the quotient and the remainder in the “quotient-remainder” form.

Repeat this process (dividing the quotient again by the base) until we get the quotient to be less than the base. Step 3: The given number in the decimal number system is obtained just by reading all the remainders and the last quotient from bottom to top. Thus,

 $$\therefore 4320_{10} = 10340_{8}$$

### One System to Another System

To convert a number from one of the binary / octal / hexadecimal systems to one of the other systems, we first convert it into the decimal system, and then we convert it to the required systems by using the above-mentioned processes.

Example:

Convert $$1010111100_2$$ to the hexadecimal system.

Solution:

Step 1: Convert this number to the decimal number system as explained in the above process. Thus,

$1010111100_2 = 700_{10} \rightarrow (1)$

Step 2: Convert the above number (which is in the decimal system), into the required number system.

Here, we have to convert $$700_{10}$$ into the hexadecimal system using the above-mentioned process. Thus,

$700_{10} = 2BC_{16} \rightarrow (2)$

From the equations (1) and (2),

 $$1010111100_2 = 2BC_{16}$$

More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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