Mathematics

Introduction to Number Bases with worked examples

Nini Ezeugo - Tue, 21st May, 2019 @ 15:42: PM



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:

Unary or Base 1

Base 10

Sumary

 

0

All base digits starts at 0

1

1

Next digit is 1

11

2

Next digit is 2

111

3

Next digit is 3

1111

4

Next digit is 4

11111

5

Next digit is 5

111111

6

Next digit is 6

1111111

7

Next digit is 7

11111111

8

Next digit is 8

111111111

9

Next digit is 9. Base 10 digit ends with 9

1111111111

10

After the last digit which is 9 above, we will start at zero again, but add 1 to the left

11111111111

11

We will add 1 digit to the right

111111111111

12

We will add 1 digit to the right

1111111111111

13

We will add 1 digit to the right

11111111111111

14

We will add 1 digit to the right

111111111111111

15

We will add 1 digit to the right

1111111111111111

16

We will add 1 digit to the right

11111111111111111

17

We will add 1 digit to the right

111111111111111111

18

We will add 1 digit to the right

1111111111111111111

19

We will add 1 digit to the right

11111111111111111111

20

After the last digit which is 9 above, we will start at zero again, but add 1 to the left

111111111111111111111

21

We follow the same procedure all through

 

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.

Answer is 10102

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. 

Thanks for reading!

Topics in Mathematics

Algebra Explained with examples Factorising Quadratic Equation Mathematics Scheme of Work, SS1, 1st Term Mathematics Scheme of Work, SS1, 2nd Term Mathematics Scheme of Work, SS1, 3rd Term Introduction to Number Bases with worked examples




Topics in Mathematics

Algebra Explained with examples Factorising Quadratic Equation Mathematics Scheme of Work, SS1, 1st Term Mathematics Scheme of Work, SS1, 2nd Term Mathematics Scheme of Work, SS1, 3rd Term Introduction to Number Bases with worked examples



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