Number system (addition)

Sure! Let me explain the binary addition trick shown in your image step-by-step:

🔢 Binary Addition Section (Top Part of the Page)¶

You’re adding 4 binary numbers:

  110111   (55 in decimal)
+ 101101   (45 in decimal)
+ 111111   (63 in decimal)
+ 101101   (45 in decimal)
---------
10000000   (128 in decimal)

✅ Explanation:¶

Align All Numbers:

Carry →       0 0 0 0 0 0 0
            110111
            101101
            111111
          + 101101
          ----------

Column-wise Addition from Right to Left:
Start adding bits just like decimal addition, but using binary rules:
0 + 1 = 1
1 + 1 = 10 (which is 0 with a carry of 1)
1 + 1 + 1 = 11 (which is 1 with a carry of 1)
1 + 1 + 1 + 1 = 100 (which is 0 with a carry of 2 → propagate carry)
After adding all, you get 10000000 (which is 128 in decimal).

🧠 What’s the Trick Here?¶

When adding multiple binary numbers:

If sum at any column ≥ 2, you carry over to the next.
Use the idea: sum % 2 = result bit, carry = floor(sum / 2)

For example:

1 + 1 + 1 = 3
3 % 2 = 1 → result bit
3 / 2 = 1 (floor) → carry
You can keep adding carry to the next column as you would in decimal.

⚡️ Binary Addition Shortcut (Mental Math Style)¶

Let’s say you're adding:

🧠 Step-by-Step Shortcut:¶

Convert each binary number to decimal (fast if you know powers of 2):
110111 → 55
101101 → 45
111111 → 63
101101 → 45
Add them in decimal:

55 + 45 = 100  
100 + 63 = 163  
163 + 45 = **208**

Convert 208 back to binary:

208 ÷ 2 = 104 R0  
104 ÷ 2 = 52  R0  
52 ÷ 2  = 26  R0  
26 ÷ 2  = 13  R0  
13 ÷ 2  = 6   R1  
6 ÷ 2   = 3   R0  
3 ÷ 2   = 1   R1  
1 ÷ 2   = 0   R1
→ Binary: **11010000**

✅ You now have the answer without doing column-wise carry.

✅ Trick Summary:¶

Step	Tip
1. Convert binary → decimal	Use quick lookup or memorize small ones
2. Add in decimal	Much faster than manual binary sum
3. Convert result → binary	Use divide-by-2 shortcut

📸 Your Trick from the Image (Top part):¶

Carry →         0
             110111
             101101
             111111
           + 101101
          ----------
             10000000

⚡ What’s Happening Here?¶

You're adding four binary numbers, and the trick is:

✅ Whenever a column sum reaches 4 (in decimal), it triggers 3 steps:

Convert 4 to binary: 100

That means carry = 1 to the next left column

Final result = place a 0 in current column and propagate

In Bengali from your image:

"৪ বা তার বেশি হলেই ০ বসাও, ১ carry যাও, ৩-step হয়"

Translated:
“If column total is 4 or more → put 0, carry 1, 3 steps happen”

🔁 Column Addition Breakdown¶

Let's analyze column-by-column from right to left (just like decimal addition), tracking how many 1s you get:

Column #	Bits being added	Count of 1s	Action
1 (rightmost)	1 + 1 + 1 + 1	4	`4 = 100` → write `0`, carry `1`
2	1 + 0 + 1 + 0 + 1 (carry)	3	`3 = 11` → write `1`, carry `1`
3	1 + 1 + 1 + 1 + 1 (carry)	5	`5 = 101` → write `1`, carry `1`
4	0 + 1 + 1 + 0 + 1 (carry)	3	`3 = 11` → write `1`, carry `1`
5	1 + 0 + 1 + 1 + 1 (carry)	4	`4 = 100` → write `0`, carry `1`
6	1 + 1 + 1 + 1 + 1 (carry)	5	`5 = 101` → write `1`, carry `1`
7 (final carry)	1 (carry)	1	write `1`

Final answer: 10000000

🧠 Trick Summary (from your image):¶

Rule	Explanation
If total in column ≥ 4	Convert total to binary
When total = 4 (100 in binary)	Put 0 in result, carry 1
When total = 3 (11) or 5 (101)	Carry 1 and write 1 accordingly
Carry added just like extra input in next column	Like decimal carry logic

🧠 No-Memory Binary Trick for 0–9 (Use 8-4-2-1 Grid)¶

This is called the 8421 shortcut, also known as the binary weight method.

Just draw this grid in your mind (or on paper once) 👇

8	4	2	1

Now just fill in 1s to add up to the number!

✅ Examples:¶

Let’s say you want to convert:

5 → Binary¶

8	4	2	1
0	1	0	1	→ ✅ 0101

Because 4 + 1 = 5

7 → Binary¶

8	4	2	1
0	1	1	1	→ ✅ 0111

(4 + 2 + 1 = 7)

9 → Binary¶

8	4	2	1
1	0	0	1	→ ✅ 1001

(8 + 1 = 9)

✅ BONUS: Hand Trick (using 4 fingers)¶

Assign fingers:
Thumb = 8
Index = 4
Middle = 2
Ring = 1

Want to show 6? Raise Index (4) + Middle (2) → 6 = 0110

But if you forget?
👉 Just use 8-4-2-1 boxes.

🧠 Binary Column Addition Trick (Using Subtraction from Base)¶

Let’s say you're adding several 1s in a column.
Each time the sum ≥ base (2 in binary), you:

Write a 0 and carry 1
Keep subtracting base (2) to find how many times you carry

✅ Example: Adding 5 ones in a column¶

1 + 1 + 1 + 1 + 1 = 5

Now apply your trick:

5 − 2 = 3 → Carry 1
3 − 2 = 1 → Carry 1
1 < 2 → Stop (final bit = 1)

Final:¶

Result bit = 1
Carries = 2

So → 5 = 101 in binary
→ You write 1, and carry 2 steps

🔁 General Pattern:¶

Total 1s	Subtraction Steps (Carry)	Final Binary
2	1 carry	10
3	1 carry	11
4	2 carries	100
5	2 carries	101
6	3 carries	110

📌 How to Use in Column Addition:¶

Just keep subtracting 2 until ≤1, and count how many times.

Sum = 5  
→ 5 − 2 = 3 (carry 1)  
→ 3 − 2 = 1 (carry 1 more)  
→ Stop (bit = 1)

You write down the last remainder, and carry as many times as you subtracted 2.

✅ One-Line Rule:¶

"Keep subtracting 2 from sum, each subtraction is a carry. When remainder < 2, that’s your result bit."

Perfect! Let’s try a full example using the division & modulo trick to add multiple binary numbers column by column.

🧪 Example: Add these 5 binary numbers¶

Let’s stack them for clarity:

   1  1  0  1
 + 1  1  1  1
 + 1  0  1  1
 + 0  1  1  0
 + 1  0  0  1
 ----------------

We’ll go column by column, from right to left:

📍 Column 4 (rightmost):¶

Bits: 1 + 1 + 1 + 0 + 1 = 4

→ bit = 4 % 2 = 0
→ carry = 4 / 2 = 2

✅ Write 0, carry 2 to next column

📍 Column 3:¶

Bits: 0 + 1 + 1 + 1 + 0 = 3
Add carry 2 → total 5

→ bit = 5 % 2 = 1
→ carry = 5 // 2 = 2

✅ Write 1, carry 2

📍 Column 2:¶

Bits: 1 + 1 + 0 + 1 + 0 = 3
Add carry 2 → total 5

→ bit = 1, carry = 2