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 
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; 2^{0} = 1)
2 raised to power 1 = 2. (That is; 2^{1} = 2) or (2 x 1 = 2)
2 raised to power 2 = 4. (That is; 2^{2} = 4) or (2 x 2 = 4)
2 raised to power 3 = 8. (That is; 2^{3} = 8) or (2 x 2 x 2 = 8)
2 raised to power 4 = 16. (That is; 2^{4} = 16) or (2 x 2 x 2 x 2 = 16)
2 raised to power 5 = 32. (That is; 2^{5} = 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
Step 1: We need divide the decimal number by the base we want to convert to.
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.
Step 3: Our previous answer from the above division was 2.
Step 4: Our previous answer from the above division was 1.
Once our division equals 0, we are done.
Next is to write the reminder from bottom  top; and that’s our answer.
OR
Base 2 
Decimal value 
Remainder 
2 
10 
 
2 
5 
0 
2 
2 
1 
2 
1 
0 
2 
0 
1 
Counting our remainder from below;
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
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;
Worked Example 3
Base 4 
Decimal value = 16 
Remainder 
4 
16 
 
4 
4 
0 
4 
1 
0 
4 
0 
1 
Counting our remainder from below;
Worked Example 4
Base 16 
Decimal value = 16 
Remainder 
16 
16 
 
16 
1 
0 
16 
0 
1 
Counting our remainder from below;
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