First, you know how we normally count: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, then we go to 10, 11, 12, and so on. That system is called the decimal (or base-10) system. It has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
A number system is just a way of writing numbers using a certain set of digits, and a certain base. The base tells us how many digits we use before “starting over” (carrying) to the next place. In base 10, when you count “9 + 1” you carry over to make 10, because there are ten digits (0 to 9).
The binary number system (also called base 2) is a special number system. In binary:
We only use two digits: 0 and 1.
After 1, we can’t go to 2 (because 2 is not allowed in base 2), so we carry over to the next place.
So binary looks like: 0, 1, 10, 11, 100, 101, 110, 111, 1000, … and so on.
Binary is very important for computers, because inside a computer, things are often “on” or “off” (which can be like 1 or 0).
Just like in decimal, every digit’s position has a value (units, tens, hundreds, etc.), in binary each position is a power of 2.
Let me show:
In decimal, the number “345” means
3 × 102 (that is 3 × 100)
plus 4 × 101 (that is 4 × 10 = 40)
plus 5 × 100 (that is 5 × 1 = 5).
In binary, the positions are like:
| place | name | binary weight |
|---|---|---|
| rightmost (first) | "20" place | 1 |
| next to left | "21" place | 2 |
| next | "22" place | 4 |
| next | "23" place | 8 |
| next | "24" place | 16 |
So if you have a binary number like 101102 (the “2” means base 2), you read it as:
Add them: 16 + 0 + 4 + 2 + 0 = 22.
So 101102 = 2210 (in decimal).
To change a normal decimal number into binary, there is a method:
Example: Convert 13 (decimal) to binary
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Now write remainders bottom to top: 1101.
So 1310 = 11012.
This is like we saw before: expand by powers of 2.
Example: convert 11012 to decimal
From right to left:
The rightmost digit is 1 in the 20 place → 1 × 1 = 1
Next is 0 in the 21 place → 0 × 2 = 0
Next is 1 in the 22 place → 1 × 4 = 4
Next is 1 in the 23 place → 1 × 8 = 8
Add them: 8 + 4 + 0 + 1 = 13.
So 11012 = 1310.
You can also add and subtract in binary, like you do in decimal, but you only have digits 0 and 1.
Addition rules in binary:
Example: Add 10112 + 1102
First, align them:
1011
+ 0110
(We put leading 0s to match the lengths.)
Add bit by bit from rightmost:
1 + 0 = 1
1 + 1 = 10 → write 0, carry 1
(carry 1) + 0 + 1 = 10 → write 0, carry 1
(carry 1) + 1 + 0 = 10 → write 0, carry 1
Now carry 1 goes to a new place:
So result is 100012.
You can check by converting both to decimal:
10112 = 1×8 + 0×4 + 1×2 + 1×1 = 8 + 0 + 2 + 1 = 11
01102 = 0×8 + 1×4 + 1×2 + 0×1 = 4 + 2 = 6
11 + 6 = 17
100012 = 1×16 + 0×8 + 0×4 + 0×2 + 1×1 = 16 + 1 = 17. It matches. Good!
Subtraction is similar, with borrowing, but using base 2 logic.
Computers use binary because their basic parts (transistors, circuits) can be in one of two states: ON (1) or OFF (0).
Everything inside a computer — numbers, letters, images — is eventually turned into strings (lots) of 0s and 1s.
Understanding binary helps with computer science, coding, digital electronics, etc.
Binary addition means adding two or more numbers that are written in the binary number system (that is, numbers made up of only 0s and 1s).
It is just like normal addition (in base 10), but here we only use two digits: 0 and 1.
Computers perform most of their calculations using binary addition because they only understand 0 (OFF) and 1 (ON).
When adding binary numbers, there are four simple rules to remember:
| Binary Addition | Result | What to do |
|---|---|---|
| 0 + 0 | 0 | Write 0 |
| 0 + 1 | 1 | Write 1 |
| 1 + 0 | 1 | Write 1 |
| 1 + 1 | 10 | Write 0 and carry 1 to the next column |
If you ever have 1 + 1 + 1, then:
1 + 1 + 1 = 11 → Write 1 and carry 1.
Think of binary addition like you are counting with only two fingers —
If you have 0 sweets and get 1 more, you have 1 sweet.
But if you have 1 sweet and get 1 more, you can’t say “2” (because “2” doesn’t exist in binary).
So you write 0 and carry 1 to the next group — just like how 9 + 1 in normal numbers gives you 10.
We add binary numbers from right to left, just like we do in normal addition.
Let us work through some examples
Example 1: Add 10₂ + 01₂
10 + 01 ------ 11
Answer: 11₂
(Explanation: 0+1=1, 1+0=1)
Example 2: Add 11₂ + 01₂
11 + 01 ------ 100
Answer: 100₂
(Explanation: Rightmost: 1+1=10 → write 0, carry 1.
Next: carry 1 + 1 + 0 = 10 → write 0, carry 1 again.
Write the carry at front → 100₂.)
Example 3: Add 10₂ + 10₂
10 + 10 ------ 100
Answer: 100₂
Example 4: Add 101₂ + 010₂
101 + 010 ------- 111
Answer: 111₂
Example 5: Add 111₂ + 001₂
111 + 001 ------- 1000
Answer: 1000₂
(Explanation: 1+1=10 → write 0 carry 1
carry 1 + 1 + 0 = 10 → write 0 carry 1
carry 1 + 1 + 0 = 10 → write 0 carry 1
write carry 1 → 1000₂)
Example 6: Add 101₂ + 011₂
101 + 011 ------- 1000
Answer: 1000₂
Example 7: Add 110₂ + 101₂
110 + 101 ------- 1011
Answer: 1011₂
Example 8: Add 111₂ + 111₂
111 + 111 ------- 1110
Let’s check:
1+1=10 → write 0 carry 1
1+1+carry1=11 → write 1 carry 1
1+1+carry1=11 → write 1 carry 1
So answer = 1110₂
Answer: 1110₂
Try these by yourself:
Binary subtraction means taking one binary number away from another — that is, subtracting numbers made up of 0s and 1s only.
It is just like normal subtraction (in base 10), but here, you can only use 0 and 1.
Binary subtraction is very important because computers use it when they perform operations like “difference,” “comparison,” or “minus.”
You already know how to subtract in normal numbers:
If you have 7 sweets and give 3 away, you have 4 left.
In binary, we do the same thing — but we only have two numbers: 0 and 1.
So sometimes we may need to borrow when we can’t subtract 1 from 0.
When that happens, we borrow 1 from the next column, just like we do in normal subtraction, but in binary the rule is a little simpler.
There are only four main rules you must know for binary subtraction:
| Binary Subtraction | Result | Explanation |
|---|---|---|
| 0 − 0 | 0 | Nothing left |
| 1 − 0 | 1 | One minus nothing is one |
| 1 − 1 | 0 | One minus one is nothing |
| 0 − 1 | 1 (with borrow 1 from next column) | We borrow 1 (which is 2 in binary), making 10₂ − 1 = 1 |
We subtract from right to left, just like normal subtraction.
If the top digit (minuend) is smaller than the bottom digit (subtrahend), we borrow 1 from the next left digit.
When we borrow in binary:
| Term | Meaning |
|---|---|
| Minuend | The number you are subtracting from (the top number). |
| Subtrahend | The number you are subtracting (the bottom number). |
| Difference | The answer after subtraction. |
Example 1: Subtract 01₂ from 10₂ (that means 10₂ − 01₂)
10 - 01 ---- 01
Answer: 01₂
Explanation: 0 − 1 → we borrow from the next left column.
So 10 − 1 = 1.
Then 0 (left column after borrowing) − 0 = 0.
Final answer = 01₂
Example 2: Subtract 10₂ from 11₂ (that means 11₂ − 10₂)
11 - 10 ---- 01
Answer: 01₂
Explanation: Right side: 1 − 0 = 1
Left side: 1 − 1 = 0
Answer = 01₂
Example 3: Subtract 01₂ from 11₂ (that means 11₂ − 01₂)
11 - 01 ---- 10
Answer: 10₂
Example 4: Subtract 001₂ from 010₂ (that means 010₂ − 001₂)
010 - 001 ----- 001
Answer: 001₂
Explanation: Rightmost: 0 − 1 → borrow from next column (becomes 10 − 1 = 1)
Middle: after borrowing, 0 becomes 0, 1 − 0 = 1
Leftmost: 0 − 0 = 0
Answer = 001₂
Example 5: Subtract 011₂ from 101₂ (that means 101₂ − 011₂)
101 - 011 ----- 010
Answer: 010₂
Example 6: Subtract 001₂ from 100₂ (that means 100₂ − 001₂)
100 - 001 ----- 011
Answer: 011₂
Explanation: Rightmost: 0 − 1 → borrow 1 from next column
Middle: after borrowing becomes 0 − 0 = 0
Leftmost: 0 (after borrow) − 0 = 0
Answer = 011₂
Example 7: Subtract 010₂ from 110₂ (that means 110₂ − 010₂)
110 - 010 ----- 100
Answer: 100₂
Example 8: Subtract 101₂ from 111₂ (that means 111₂ − 101₂)
111 - 101 ----- 010
Answer: 010₂
Example 9: Subtract 011₂ from 111₂ (that means 111₂ − 011₂)
111 - 011 ----- 100
Answer: 100₂
Binary multiplication means multiplying two numbers that are written in the binary number system — that is, numbers made up of only 0s and 1s.
It is just like the normal multiplication we do in base 10, but here we use only 0 and 1.
Computers use binary multiplication all the time when they do calculations, because they only understand 0 (OFF) and 1 (ON).
You already know that in normal multiplication:
0 × anything = 0
1 × anything = that same thing
It is the same in binary! When we multiply binary numbers, we just use those same simple rules — but we also have to add the results carefully, just like long multiplication in normal numbers.
There are only four simple rules to remember:
| Binary Multiplication | Result | Explanation |
|---|---|---|
| 0 × 0 | 0 | Nothing times nothing is nothing |
| 0 × 1 | 0 | Anything times zero is zero |
| 1 × 0 | 0 | One times zero is zero |
| 1 × 1 | 1 | One times one is one |
When multiplying binary numbers, we follow the same steps as normal multiplication:
Example 1: Multiply 10₂ × 1₂
10 × 1 ------ 10
Answer: 10₂
(Explanation: anything multiplied by 1 is itself.)
Example 2: Multiply 11₂ × 1₂
11 × 1 ------ 11
Answer: 11₂
Example 3: Multiply 10₂ × 0₂
10 × 0 ------ 00
Answer: 00₂
Example 4: Multiply 10₂ × 10₂
Let us work it step by step
10
× 10
------
00 (because 10 × 0 = 00)
+ 10 (because 10 × 1 = 10, shifted one place left)
------
100
Answer: 100₂
Check: 10₂ = 2 in decimal
10₂ × 10₂ = 2 × 2 = 4
100₂ = 4 in decimal correct!
Example 5: Multiply 11₂ × 10₂
11
× 10
------
00 (11 × 0 = 00)
+ 11 (11 × 1 = 11, shifted one place left)
------
110
Answer: 110₂
Check: 11₂ = 3, 10₂ = 2
3 × 2 = 6
110₂ = 6 correct!
Example 6: Multiply 11₂ × 11₂
11
× 11
------
11 (11 × 1 = 11)
+ 11 (11 × 1 = 11, shifted one place left)
------
1001
Answer: 1001₂
Check: 11₂ = 3, 3 × 3 = 9
1001₂ = 9 correct!
Example 7: Multiply 101₂ × 10₂
101
× 10
------
000 (101 × 0 = 000)
+ 101 (101 × 1 = 101, shifted one place left)
------
1010
Answer: 1010₂
Check: 101₂ = 5, 10₂ = 2, 5 × 2 = 10
1010₂ = 10 correct!
Example 8: Multiply 111₂ × 11₂
111
× 11
------
111 (111 × 1 = 111)
+ 111 (111 × 1 = 111, shifted one place left)
------
1110
Answer: 1110₂
Check: 111₂ = 7, 11₂ = 3, 7 × 3 = 21
1110₂ = 21 correct!
Example 9: Multiply 101₂ × 11₂
101
× 11
------
101 (101 × 1 = 101)
+ 101 (101 × 1 = 101, shifted one place left)
------
1111
Answer: 1111₂
Check: 101₂ = 5, 11₂ = 3, 5 × 3 = 15
1111₂ = 15 correct!
Example 10: Multiply 110₂ × 11₂
110
× 11
------
110 (110 × 1 = 110)
+ 110 (110 × 1 = 110, shifted one place left)
------
10010
Answer: 10010₂
Check: 110₂ = 6, 11₂ = 3, 6 × 3 = 18
10010₂ = 18 correct!
Binary division means splitting one binary number by another binary number.
It is the same as normal long division that you already know, but instead of numbers 0–9, we only use 0 and 1.
Just like in base 10, we call the numbers:
| Word | Meaning |
|---|---|
| Dividend | The number you want to divide (the big number). |
| Divisor | The number you are dividing by (the small number). |
| Quotient | The answer you get after dividing. |
| Remainder | What is left over after the division. |
When we divide in binary, we are simply asking:
“How many times does this small number fit into this bigger number?”
Since we only have 0 and 1, the answer (quotient digit) can only be 0 or 1.
If the divisor can fit into the dividend part → write 1.
If it cannot fit → write 0.
We use binary subtraction to remove (subtract) each part as we go, just like in normal long division.
| Division | Result |
|---|---|
| 0 ÷ 1 | 0 |
| 1 ÷ 1 | 1 |
| 0 ÷ 0 | Not possible |
| 1 ÷ 0 | Not possible |
Remember: dividing by zero is never allowed, even in normal numbers.
Example 1
Divide 10₂ by 1₂ (that means 10 ÷ 1)
1 0 ÷ 1
Now, 1 can go into 1 → write 1.
Multiply 1 × 1 = 1
Subtract → 1 − 1 = 0
Bring down next 0 → 0 ÷ 1 = 0
Answer: Quotient = 10₂, Remainder = 0
Example 2
Divide 10₂ by 10₂
10 ÷ 10
10 goes into 10 exactly once.
Write 1 on top.
Subtract 10 − 10 = 00.
Answer: Quotient = 1₂, Remainder = 0
Example 3
Divide 11₂ by 10₂
11 ÷ 10
10 goes into 11 once.
Write 1.
Subtract: 11 − 10 = 01.
Answer: Quotient = 1₂, Remainder = 1₂
Example 4
Divide 110₂ by 10₂
10 | 110
10 goes into 11 → yes, once (write 1).
Subtract 11 − 10 = 01.
Bring down next 0 → 010.
10 goes into 01 → no, write 0.
Bring down next → remainder 10.
Answer: Quotient = 11₂, Remainder = 0
Example 5
Divide 101₂ by 10₂
10 | 101
10 goes into 10 → once (write 1).
Subtract 10 − 10 = 00.
Bring down 1 → 01.
10 cannot go into 01 → write 0.
Answer: Quotient = 10₂, Remainder = 1₂
Example 6
Divide 111₂ by 11₂
11 | 111
11 goes into 11 → once (write 1).
Subtract → 11 − 11 = 00.
Bring down last 1 → 01.
11 cannot go into 01 → write 0.
Answer: Quotient = 10₂, Remainder = 1₂
Example 7
Divide 1001₂ by 10₂
10 | 1001
Step 1: 10 into 10 → once (write 1).
Subtract → 10 − 10 = 00.
Bring down next 0 → 000.
10 cannot go → write 0.
Bring down last 1 → 001.
10 cannot go → write 0.
Answer: Quotient = 100₂, Remainder = 1₂
It helps us to understand how computers share or split data.
It helps us to learn binary arithmetic, which is the language of computers.
It helps us to understand logic operations and programming.
Ada has 11₂ apples (that is 3 apples). Her friend gives her 11₂ more apples (another 3). How many apples does Ada have now?
Write the sum: 11₂ + 11₂. (3 + 3)
11 + 11 ----
Rightmost: 1 + 1 = 10₂ → write 0, carry 1.
Next column (left): 1 + 1 + carry1 = 1+1+1 = 11₂ → write 1, carry 1.
Put the final carry at the left: result = 110₂.
Decimal check: 11₂ = 3, so 3 + 3 = 6. And 110₂ = 6.
Answer: 110₂ (which is 6 apples).
Sam had 101₂ candies (5). He found 011₂ more (3). How many candies now?
101₂ + 011₂ (5 + 3)
101 + 011 -----
Rightmost: 1 + 1 = 0, carry 1.
Middle: 0 + 1 + carry1 = 0+1+1 = 10₂ → write 0, carry 1.
Leftmost: 1 + 0 + carry1 = 1+0+1 = 10₂ → write 0, carry 1.
Final carry gives new left digit → 1000₂.
Decimal check: 5 + 3 = 8, and 1000₂ = 8.
Answer: 1000₂ (which is 8 candies).
Tunde has 110₂ toy cars (6). He buys 101₂ more (5). How many toy cars now?
110₂ + 101₂ (6 + 5)
110 + 101 -----
Rightmost: 0 + 1 = 1.
Middle: 1 + 0 = 1.
Leftmost: 1 + 1 = 10₂ → write 0, carry 1.
Put the carry to leftmost → 1011₂.
Decimal check: 6 + 5 = 11, and 1011₂ = 11.
Answer: 1011₂ (which is 11 toy cars).
Lola had 101₂ cookies (5). She ate 011₂ cookies (3). How many cookies left?
101₂ − 011₂ (5 − 3)
101 - 011 -----
Rightmost: 1 − 1 = 0.
Middle: 0 − 1 → borrow 1 from leftmost. Leftmost becomes 0, middle becomes 10₂. Now 10₂ − 1 = 1.
Leftmost after borrow: 0 − 0 = 0.
Result = 010₂ (or 10₂).
Decimal check: 5 − 3 = 2, and 010₂ = 2.
Answer: 010₂ (which is 2 cookies left).
Aya has 100₂ balloons (4). She gives 001₂ balloon (1) to a friend. How many left?
100₂ − 001₂ (4 − 1)
100 - 001 -----
Rightmost: 0 − 1 → borrow. Middle is 0 so we borrow from leftmost: leftmost 1 → 0, middle becomes 10₂. Then middle lends 1 to rightmost: middle becomes 1, rightmost becomes 10₂.
Rightmost now: 10₂ − 1 = 1.
Middle: 1 − 0 = 1.
Leftmost: 0 − 0 = 0.
Result = 011₂.
Decimal check: 4 − 1 = 3, 011₂ = 3.
Answer: 011₂ (which is 3 balloons left).
Bola has 100₂ marbles (4). She loses 011₂ marbles (3). How many remain?
100₂ − 011₂ (4 − 3)
100 - 011 -----
Rightmost: 0 − 1 → borrow. Middle is 0, so we must borrow from leftmost 1. Leftmost becomes 0, middle becomes 10₂. Then middle lends to rightmost: middle becomes 1, rightmost becomes 10₂.
Rightmost: 10₂ − 1 = 1.
Middle: 1 − 1 = 0.
Leftmost: 0 − 0 = 0.
Result = 001₂.
Decimal check: 4 − 3 = 1, 001₂ = 1.
Answer: 001₂ (which is 1 marble remain).
In a garden, each row has 11₂ flowers (3). There are 11₂ rows (3). How many flowers in total?
11₂ × 11₂ (3 × 3)
11
× 11
----
11
+ 11
------
1001
Decimal check: 3 × 3 = 9, and 1001₂ = 9.
Answer: 1001₂ (which is 9 flowers).
One box can hold 101₂ toys (5). There are 10₂ boxes (2). How many toys can all boxes hold?
101₂ × 10₂ (5 × 2)
101
× 10
-----
000
+ 101
------
1010
Decimal check: 5 × 2 = 10, 1010₂ = 10.
Answer: 1010₂ (which is 10 toys).
A packet has 110₂ candies (6). There are 101₂ packets (5). How many candies total?
110₂ × 101₂ (6 × 5)
110
× 101
------
110
000
+ 11000
--------
11110
Decimal check: 6 × 5 = 30, and 11110₂ = 30.
Answer: 11110₂ (which is 30 candies).
There are 110₂ crackers (6). They are shared equally among 10₂ children (2). How many crackers each child gets and any remainder?
110₂ ÷ 10₂ (6 ÷ 2)
10 goes into 11 once → write 1, subtract 11 − 10 = 01.
Bring down last 0 → 010.
10 goes into 10 once → write 1, subtract 10 − 10 = 0.
Answer: Quotient = 11₂, Remainder = 0
Decimal check: 6 ÷ 2 = 3, 11₂ = 3.
Each child gets 11₂ (3) crackers, remainder 0.
There are 101₂ sweets (5). They are put into boxes of 11₂ sweets each (3). How many full boxes and how many sweets left?
101₂ ÷ 11₂ (5 ÷ 3)
11 does not fit into first 10, so look at 101.
11 fits into 101 once → write 1.
Subtract 101 − 011 = 010.
Nothing more to bring → remainder 010₂.
Answer: 1 full box (1₂) and remainder 10₂ (2) sweets.
Decimal check: 5 ÷ 3 = 1 remainder 2. 010₂ = 2.
There are 111₂ stickers (7). They are shared into groups of 10₂ (2). How many in each group and what's left?
111₂ ÷ 10₂ (7 ÷ 2)
10 goes into 11 once → subtract 11 − 10 = 01.
Bring down next 1 → 011.
10 goes into 011 once → write 1, subtract 011 − 010 = 001.
Answer: Quotient = 11₂, Remainder = 001₂
Decimal check: 7 ÷ 2 = 3 remainder 1. 11₂ = 3, 001₂ = 1.
Each group has 11₂ (3), remainder 1 (001₂).
