Converting Any Base to Any Other Base

To do this:

• First convert the given base to base ten. (using the expansion method)

• Then convert the result to the required base. (using the division method)

Worked Example 3.4.1:

Convert 321_four to base two 

Solution 

First convert 321_four to base 10

321_four = 3 × 4² + 2 × 4¹ + 1 × 4⁰ 

= 3 × 16 + 2 × 4 + 1 × 1 

= 48 + 8 + 1 

321_four = 57_ten 

Now convert 57_ten to base two  

321_four = 111001_two 

Worked Example 3.4.2:

Convert 101110_two to an octal number.  

Solution 

Note: Octal means eight 

Convert 101110_two to base ten 

5	4	3	2	1	0
1	0	1	1	1	0

101110_two = 1 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 1 × 2² + 1 × 2¹ + 0 × 2⁰ 

= (1 × 32) + (0 × 16) + (1 × 8) + (1 × 4) + (1 × 2)+ (0 × 1)

= 32 + 0 + 8 + 4 + 2 + 0 

= 46_ten 

Now convert 46_ten to octal number 

⇒ 101110_two = 56_eight