Basic Operations in Number Bases

Once you're familiar with different number bases, the next step is learning how to carry out basic arithmetic operations like addition, subtraction, multiplication, and division in those bases. This is similar to what you already do in base 10 — just using the digits and rules of the new base.

 

1. Addition in Number Bases

Just like in base 10, you add digits column by column and carry over if needed — but you carry over when the sum reaches the base.

Example: Add $1011_2 + 1101_2$

Step-by-step addition in base 2:

• 1 + 1 = 10 → write 0 and carry 1
• 1 + 0 + carry 1 = 10 → write 0 and carry 1
• 0 + 1 + carry 1 = 10 → write 0 and carry 1
• 1 + 1 + carry 1 = 11 → write 1 and carry 1
• Carry 1 → bring down

$$1011_2 + 1101_2 = 11000_2$$

 

2. Subtraction in Number Bases

Subtract just like in base 10, borrowing when necessary — but remember the digits available in that base.

Example: Subtract $1001_2 - 0110_2$

Step-by-step:

$$\begin{align*} &\quad\ 1001_2 \\ -&\quad 0110_2 \\ \hline &\quad\ 0011_2 \\ \end{align*}$$

You may need to borrow, just like in base 10, but borrow a full unit of the base (e.g., borrow a '2' in base 2).

 

3. Multiplication in Number Bases

Use the same method as in base 10 — multiply each digit and shift left as needed — but remember the multiplication must follow the rules of that base.

Example: Multiply $11_2 \times 10_2$

In base 2:

• $11_2 = 3_{10},\ 10_2 = 2_{10}$
• So, $3 \times 2 = 6_{10}$
• Convert 6 to base 2: $$6_{10} = 110_2$$

$$11_2 \times 10_2 = 110_2$$

 

4. Division in Number Bases

Divide as you would in base 10 — find how many times the divisor fits, and write the remainder if needed — but stay within the rules of the base.

Example: Divide $1100_2 \div 10_2$

• $1100_2 = 12_{10},\ 10_2 = 2_{10}$
• $12 \div 2 = 6$, and $6_{10} = 110_2$

$$1100_2 \div 10_2 = 110_2$$

 

Practice Problem