Bases

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Number Bases

A number base (also known as base) for short is a numeral system that tells us about the unique or different symbols and notations that can be used to represent a value.

For example, the base 2 number system tells that there are only 2 unique notations 0 and 1 to represent the value.

The most commonly used number base is base 10, also known as the decimal number system. The decimal number system uses ten different notations which are the digit 0-9 to represent a value Bases can be either positive, negative, 0, complex, or non-integer. The most frequently used bases are base 2 and base 16. They are also used for calculating and are known as binary, and hexadecimal respectively.


What is a Base Number?

A base number is a number raised to the power that represents the number of units of a number system. For example, the base number of the binary number system is 2.

For Example,

yx

Here, y is a base number.


Base 2 Number System

In Mathematics, the base 2 number system, also known as the binary number system uses 2 as the base and therefore requires only two digits i.e. 0 and 1 to represent any value, rather than 10 different symbols required in the decimal number system. The numbers from 0 to 10 in the binary number system are represented as “.” .The base 2 number system is widely used in Mathematics and Computer Science as bits are easy to create using physically logic gates (the logic gates are either open or closed meaning 0 or 1).


Counting in Different Bases

Counting in different bases substitutes the base 10 with a different bases. We often use Base 10. It is our decimal number system. It has 10 digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

We count numbers with base 10 as shown below:



0

Start at 0

1

Then 1

••

2

Then 2



•••••••••

9

Upto 9

••••••••••

10

Start counting back to 0 again, but add 1 to the left side

•••••••••••

11


••••••••••••

12




•••••••••••••••••••

19

Start counting back to 0 again, but add one on the left side.

••••••••••••••••••••

20


•••••••••••••••••••••

21

And So on


Let us Understand How to do Counting in Different Base

(Base 2) Binary Number System Has Only 2 Digits: 0 and 1

We count the base 2 like shown below:



0

Start at 0

1

Then add 1

••

10

Start back at 0 but add 1 to the left

•••

11


••••

100

Start back at 0 again, and add 1 to the number on the left side. As the number is already at 1 so it also goes back to 0 and 1 is added to the next place on the left side 

•••••

101


••••••

110


•••••••

111


••••••••

1000

Start counting back to 0 again (for all 3 digits) and add 1 on the left side

•••••••••

1001

And so on


(Base 3) Ternary Number System Has 3 Digits: 0,1, and 2

We count numbers with base 3 as shown below:



0

Start at 0

1

Then 1

••

2


•••

10

Start back at 0 but add 1 to the left

••••

11


•••••

12


••••••

20

Start back at 0 but add 1 to the left

•••••••

21


••••••••

22


•••••••••

100

Start back at 0 again, and add 1 to the number on the left side. As the number is already at 2 so it also goes back to 0 and 1 is added to the next place on the left side 

••••••••••

101



(Base 4) Quaternary Number System Has 4 Digits: 0, 1, 2, and 3

We count numbers with base 4 as shown below:



0

Start at 0

1

Then 1

••

2


•••

3


••••

10

Start back at 0 but add 1 to the left

•••••

11


••••••

12


•••••••

13


••••••••

20

Start back at 0 but add 1 to the left

•••••••••

21

And so on


(Base 5) Quinary Number System Has 5 Digits: 0, 1, 2, 3, and 4

We count numbers with base 5 as shown below:



0

Start at 0

1

Then 1

••

2


•••

3


••••

4


•••••

10

Start back at 0 but add 1 to the left

••••••

11


•••••••

12


••••••••

13


•••••••••

20

Start back at 0 but add 1 to the left

•••••••••••

21

And So On


(Base 6) Senary Number System Has 6 Digits: 0, 1, 2, 3, 4, and 5

We count numbers with base 6 as shown below:



0

Start at 0

1

Then 1

••

2


•••

3


••••

4


•••••

5


••••••

10

Start back at 0 but add 1 to the left

•••••••

11


••••••••

12


•••••••••

13


•••••••••••

14


•••••••••••

15


••••••••••••

20

Start back at 0 but add 1 to the left

•••••••••••••

21

And So On


(Base 7) Septenary Number System Has 7 Digits: 0, 1, 2, 3, 4, 5, and 6

We count numbers with base 7 as shown below:



0

Start at 0

1

Then 1

••

2

Then 2

•••


••••••

6


•••••••

10

Start back at 0 but add 1 to the left

••••••••

11


•••••••••

12


•••••••••


•••••••••••••

16


••••••••••••••

20

Start back at 0 but add 1 to the left

•••••••••••••••

21

And so on


(Base 8) Octal Number System Has 8 Digits: 0, 1, 2, 3, 4, 5, 6, and 7

We count numbers with base 8 as shown below:



0

Start at 0

1

Then 1

••

2

Then 2



•••••••

7

Up to 7

••••••••

10

Start back at 0 but add 1 to the left

•••••••••

11


••••••••••

12


•••••••••


•••••••••••••••

17


••••••••••••••••

20

Start back at 0 but add 1 to the left

•••••••••••••••••

21

And so on


Nonary (Base 9) Number System Has 9 Digits: 0, 1, 2, 3, 4, 5, 6, 7, and 8

We count numbers with base 9 as shown below:



0

Start at 0

1

Then 1

••

2

Then 2



•••••••

8

Up to 8

••••••••

10

Start back at 0 but add 1 to the left

•••••••••

11


••••••••••

12


•••••••••


•••••••••••••••••

18


••••••••••••••••••

20

Start back at 0 but add 1 to the left

•••••••••••••••••••

21

And so on


(Base 10) Decimal Number System Has 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 9, and 10

We count numbers with base 10 as shown below:



0

Start at 0

1

Then 1

••

2

Then 2



•••••••••

9

Upto 9

••••••••••

10

Start counting back to 0 again, but add 1 to the left side

•••••••••••

11


••••••••••••

12




•••••••••••••••••••

19

Start counting back to 0 again, but add one on the left side.

••••••••••••••••••••

20


•••••••••••••••••••••

21

And So on


Facts to Remember

In the number system, base, also known as radix, is the number of different digits or combinations of digits and letters that the number system uses to represent numbers.

FAQ (Frequently Asked Questions)

1. What is a Number System?

Ans: In Mathematics, a number system also known as the numeral system is a way of naming or representing numbers. It is a mathematical notation for representing numbers of a given set using digits or symbols. The same sequence of numbers may denote different numbers in different numeral systems. For example, 11 denotes number 11 in the decimal number system, number 3 in the binary number system, and number 2 in the unary number system.

2. What are the Characteristics of the Base 2 Number System?

Ans:

  • It is a 2 digit number system.

  • It is commonly known as the binary number system.

  • It uses only 2 digits i.e. 0 and 1.

  • The different position of the base 2 system shows the same value in power 2.

  • The left most position of binary representation shows 2 raised to the power y.

(Here, y denotes the extreme leftmost position).

3. What is the Base 1 Number System Known as?

Ans: The base 1 number system is known as the unary number system and is the easiest number system to represent natural numbers.

4. What are the Most Commonly Used Number Systems?

Ans: The most commonly used number system are:

  • Binary Number System.

  • Hexadecimal Number System.

  • Decimal Number System.

  • Octal Number System.