Multiplying and dividing in binary follows the same logic as long multiplication and long division in decimal — the process is actually more forgiving in binary, since every digit is either 0 or 1, which removes most of the memorization decimal multiplication requires.
Binary Multiplication Rules
Binary multiplication only has two rules to remember: multiplying by 0 always gives 0, and multiplying by 1 gives back the original number unchanged. There’s no multiplication table to memorize the way there is in decimal, which makes the individual steps simpler even though there are usually more of them.
A Worked Multiplication Example
Multiplying 101 (5) by 11 (3) works like long multiplication: multiply 101 by the rightmost 1, giving 101. Multiply 101 by the next 1, shifted one place left, giving 1010. Adding 101 and 1010 together using binary addition gives 1111, which is 15 in decimal — matching 5×3=15.
Binary Division Basics
Binary long division follows the same repeated subtract-and-shift process as decimal long division, just with only two possible digits (0 or 1) to try at each step instead of ten. At each position, you check whether the divisor fits into the current portion of the dividend — if it does, the quotient digit is 1 and you subtract; if it doesn’t, the quotient digit is 0 and you move to the next position.
A Worked Division Example
Dividing 1100 (12) by 11 (3): the divisor 11 fits into the first two digits “11”, so the first quotient digit is 1, leaving a remainder of 0. Bringing down the next digit and repeating shows 11 fits again, giving a second quotient digit of 1. The result, 100, is 4 in decimal — matching 12÷3=4.
Where This Actually Gets Used
Processors handle multiplication and division in hardware using circuits built around exactly these repeated shift-and-add or shift-and-subtract steps, just executed far faster and with far more digits than anyone would do by hand. Understanding the manual process is mainly useful for grasping what the hardware is actually doing underneath a single multiplication instruction.
The Shift Shortcut
Multiplying by 2 in binary just moves every digit one place to the left and adds a 0 on the right. For example, 101 (5) becomes 1010 (10). Multiplying by 4 shifts two places, and by 8 shifts three places. Dividing by 2 does the opposite: it shifts every digit one place right. For instance, 1100 (12) becomes 110 (6). Processors use this trick constantly because shifting is faster than full multiplication.
More Multiplication Examples
110 × 10 = 1100 (6 × 2 = 12)111 × 101 = 100011 (7 × 5 = 35)1001 × 11 = 11011 (9 × 3 = 27)
For 111 × 101, write 111, then 000 shifted one place, then 111 shifted two places, and add them: 111 + 11100 = 100011, which is 35.
Division With a Remainder
Not every division comes out evenly. Dividing 1011 (11) by 10 (2) gives a quotient of 101 (5) with a remainder of 1. Check it: 5 × 2 + 1 = 11. Another example is 1111 divided by 101, which gives 11 (15 ÷ 5 = 3) with no remainder.
Why Learn This by Hand
You will rarely multiply binary numbers manually in real work, but understanding shifts explains several things: why memory sizes double, why multiplying by powers of two is so fast, and how low-level code can replace an expensive operation with a cheap one.
Common Mistakes
- Forgetting to shift each partial product one more place to the left.
- Using decimal addition when summing partial products. The sums must follow the binary carry rules.
- Dropping the remainder in division without checking it.
For a refresher on the carry rules, see binary addition and subtraction.
Practice the Basics First
Multiplication and division build directly on binary addition and subtraction, so it’s worth being comfortable with those first — our guide on binary addition and subtraction covers the carry and borrow rules these examples rely on. Our binary translator is also useful for checking any binary number’s decimal value while you practice.