# Introduction to Number Bases with worked examples

### Introduction to Number Bases:

Numbering begins with 0 (zero). Also, numbers are expressed in bases (sometimes called radix). A base 10 number for instance will comprise of 10 digits ranging from 0 – 9.

Note: We generally use base 10 in our numbering system.

Base 10 digits will comprise of: 0 1 2 3 4 5 6 7 8 9

Since the above numbers are composed of 10 digits, Base 10 may be referred to as decimals or denary digits.

The table below illustrates base 10 digits:

#### Apart from base 10 (Decimal), we do have other bases like:

Base 2 also called Binary: consists of 2 digits which are 0 and 1

Base 3 also called Ternary: consists of 3 digits which are 0, 1 and 2

Base 4 also called Quaternary: consists of 4 digits which are 0, 1, 2 and 3

Base 5 also called Quinary: consists of 5 digits which are 0, 1, 2, 3 and 4

Base 6 also called Senary: consists of 6 digits which are 0, 1, 2, 3, 4, 5 and 6

Base 7 also called Septenary: consists of 7 digits which are 0, 1, 2, 3, 4, 5, 6 and 7

Base 8 also called Octal: consists of 8 digits which are 0, 1, 2, 3, 4, 5, 6, 7 and 8

Base 9 also called Nonary: consists of 9 digits which are: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

Note: When we have a base number greater than 10; for instance base 11 or base 12 or base 13 and so on, the next digit will be an alphabet in its ascending order. Let see the examples below:

Base 11 also called Undecimal: consists of 11 digits which are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A

Base 12 also called Duodecimal: consists of 12 digits which are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B

Note: Base 16 is also known as hexadecimal digits. They prove useful in Internet Protocol address (IP address) and Media Access Control address (MAC address) in computer networking. Please read our article on OSI model here.

The digits for base 16 (Hexadecimal digits) are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

With regards to base 2, it will comprise of only 2 digits which are 0 and 1. Because we have just 2 digits here, base 2 is often referred to a binary digit. Binary digits are fundamental to how computers work. This is why you may see stuffs like "01001100101011011" in a programming or computer language. Remember, we have just two digits in binary and that’s 0 and 1. The table below shows what base 2 would look like

 Base 1 Base 2 Summary 0 All base digits starts at 0 1 1 Next digit is 1. Base 2 digit ends with 1 11 10 We start at zero again, but add 1 to the left 111 11 We will add 1 digit to the right 1111 100 Since the above digits is all 1’s and base 2 ends with digit 1, we will change all the 1’s to 0’s and add 1 to the left 11111 101 We will add one to the farthest 0 at the right

Note: Understanding base 2 is very vital for IP addressing in computer networking.

Ternary or Base 3 comprise of 3 digits which are 0, 1 and 2. The table below shows what base 3 would look like

 Unary/Base 1 Base 3 Summary 0 All base digits starts at 0 1 1 Next digit is 1. 11 2 Next digit is 2. Base 3 digit ends with 2 111 10 We will start back at 0 and add 1 to the left 1111 11 Next digit is 1 from the right 11111 12 Next digit is 2 from the right 111111 20 We will start back at zero and add one to the left 1111111 21 We will add 1 to the right 11111111 22 We will add 1 to the right 111111111 100 Since the above digits is all 2’s, we will change all  the 2’s to 0’s and add 1 to the left 1111111111 101 We will add 1 to the farthest right

The table below shows a summary of base 10, base 2, base 6, base 11 and base 16

 Unary Base 10 Base 2 Base 6 Base 11 Base 16 1 1 1 1 1 1 11 2 10 2 2 2 111 3 11 3 3 3 1111 4 100 4 4 4 11111 5 101 5 5 5 111111 6 110 10 6 6 1111111 7 111 11 7 7 11111111 8 1000 12 8 8 111111111 9 1001 13 9 9 1111111111 10 1010 14 A A 11111111111 11 1011 15 10 B 111111111111 12 1100 20 11 C 1111111111111 13 1101 21 12 D 11111111111111 14 1110 22 13 E 111111111111111 15 1111 23 14 F 1111111111111111 16 10000 24 15 10

Notice that we skipped the zero (0) value in the above table.

In order to convert a base to another base (e.g from base 10 to base 2), we will need to know the powers of the base we are converting to. We must start from power 0 since our numbering always start from zero (like I wrote at the beginning of this article). For instance:

The powers of 2 are:

2 raised to power 0 = 1. (That is; 20 = 1)

2 raised to power 1 = 2. (That is; 21 = 2) or (2 x 1 = 2)

2 raised to power 2 = 4. (That is; 22 = 4) or (2 x 2 = 4)

2 raised to power 3 = 8. (That is; 23 = 8) or (2 x 2 x 2 = 8)

2 raised to power 4 = 16. (That is; 24 = 16) or (2 x 2 x 2 x 2 = 16)

2 raised to power 5 = 32. (That is; 25 = 32) or (2 x 2 x 2 x 2 x 2 = 32)

And so on.

Note: Any value raised to the power of zero will always equal to 1.

Simply put, the power of 2 can be written as:

1, 2, 4, 8, 16, 32, 64, 128, 256 and so on.

Following the same steps as above, the power or 3 will be:

1, 3, 9, 27, 81, 243, 729 and so on

Still following the same steps as above, the powers of 4 are

1, 4, 16, 64, 256, 1024 and so on

#### Worked Example 1

• Convert the decimal 10 to base 2

Step 1: We need divide the decimal number by the base we want to convert to.

• decimal value = 10
• we are converting to base 2
##### 10/2 = 5 Remainder 0

Note: We must take into account the remainder from our division because it will be used as our answer. Every other step will be similar to step 1 but we will divide our resultant answer from the subsequent steps.

Step 2: Our previous answer from the above division was 5.

##### 5/2 = 2 Remainder 1

Step 3: Our previous answer from the above division was 2.

##### 2/2 = 1 Remainder 0

Step 4: Our previous answer from the above division was 1.

##### 1 divided by 2 = 0 Remainder 1

Once our division equals 0, we are done.

Next is to write the reminder from bottom - top; and that’s our answer.

OR

#### We can also solve this question by using a table, where we will divide the decimal number by the base we want to convert to:

 Base 2 Decimal value Remainder 2 10 - 2 5 0 2 2 1 2 1 0 2 0 1

Counting our remainder from below;

#### Our answer is 10102

Note: I will use different expressions for the questions below but they will all mean the same thing. (We will be converting a number from base 10 to another base).

Worked Example 2

• Convert 15 to base 2

For simplicity, we will use a table to solve this question.

 Base 2 Decimal value Remainder 2 15 - 2 7 1 2 3 1 2 1 1 2 0 1

Counting our remainder from below;

#### Our Answer 11112

Worked Example 3

• Convert 16 to 164
 Base 4 Decimal value = 16 Remainder 4 16 - 4 4 0 4 1 0 4 0 1

Counting our remainder from below;

#### Our Answer is 1004

Worked Example 4

• Convert 1610 to base 1616
 Base 16 Decimal value = 16 Remainder 16 16 - 16 1 0 16 0 1

Counting our remainder from below;

#### Our Answer is 1016

We will advance on this topic on another article. Alfred Ajibola is a Medical Biochemist, a passionate Academician with over 7 years of experience, a Versatile Writer, a Web Developer, a Cisco Certified Network Associate and a Cisco CyberOps Associate.