💻 Binary Coding
Master binary coding where numbers are represented using only two symbols — learn conversion techniques, position values, and solve arithmetic operations in symbol form
Binary Coding
Binary Coding is a Mains-level coding pattern where only two symbols are used to represent all numbers. One symbol stands for 0 and the other stands for 1. Every number, no matter how large, is written using just these two symbols. This is essentially the binary number system dressed up with symbols instead of digits.
This topic has appeared in SBI PO Mains, IBPS PO Mains, and RRB PO Mains in recent years. Once you understand the binary conversion, these questions become free marks.
Core Concept
In binary coding:
- Only two symbols exist in the entire system
- One symbol represents 0 (zero)
- The other symbol represents 1 (one)
- Numbers greater than 1 are expressed as combinations of these two symbols
For example, if * = 0 and $ = 1, then:
- The number 2 is written as $* (which is binary 10)
- The number 5 is written as (which is binary 101)
- The number 7 is written as $$$ (which is binary 111)
Position Values (Right to Left)
The key rule of binary coding is: the value of the 1-symbol doubles each time it shifts one place to the left.
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Binary Coding
Binary Coding is a Mains-level coding pattern where only two symbols are used to represent all numbers. One symbol stands for 0 and the other stands for 1. Every number, no matter how large, is written using just these two symbols. This is essentially the binary number system dressed up with symbols instead of digits.
This topic has appeared in SBI PO Mains, IBPS PO Mains, and RRB PO Mains in recent years. Once you understand the binary conversion, these questions become free marks.
Core Concept
In binary coding:
- Only two symbols exist in the entire system
- One symbol represents 0 (zero)
- The other symbol represents 1 (one)
- Numbers greater than 1 are expressed as combinations of these two symbols
For example, if * = 0 and $ = 1, then:
- The number 2 is written as $* (which is binary 10)
- The number 5 is written as (which is binary 101)
- The number 7 is written as $$$ (which is binary 111)
Position Values (Right to Left)
The key rule of binary coding is: the value of the 1-symbol doubles each time it shifts one place to the left.
Reading from right to left, the position values are:
| Position | 7th | 6th | 5th | 4th | 3rd | 2nd | 1st |
|---|---|---|---|---|---|---|---|
| Value | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
Each position with the 1-symbol contributes its position value. Positions with the 0-symbol contribute nothing.
Quick Reference: Powers of 2
Memorize these — they are your lifeline in binary coding questions:
| Power | 2^0 | 2^1 | 2^2 | 2^3 | 2^4 | 2^5 | 2^6 |
|---|---|---|---|---|---|---|---|
| Value | 1 | 2 | 4 | 8 | 16 | 32 | 64 |
Complete Conversion Table (0 to 15)
| Number | Binary | Symbol (* = 0, $ = 1) | Symbol (* = 0, & = 1) |
|---|---|---|---|
| 0 | 0 | * | * |
| 1 | 1 | $ | & |
| 2 | 10 | $* | &* |
| 3 | 11 | $$ | && |
| 4 | 100 | $** | &** |
| 5 | 101 | &*& | |
| 6 | 110 | $$* | &&* |
| 7 | 111 | $$$ | &&& |
| 8 | 1000 | $*** | &*** |
| 9 | 1001 | &**& | |
| 10 | 1010 | * | && |
| 11 | 1011 | $*$$ | &*&& |
| 12 | 1100 | $$** | &&** |
| 13 | 1101 | $$*$ | &&*& |
| 14 | 1110 | $$$* | &&&* |
| 15 | 1111 | $$$$ | &&&& |
How to Convert a Number to Binary Symbols
Step 1: Find the largest power of 2 that is less than or equal to the number.
Step 2: Subtract that power of 2 from the number. Mark that position as $ (or &).
Step 3: Repeat for the remaining value. All unfilled positions become *.
Step 4: Read the symbol code from left to right (highest power to lowest).
Example: Convert 22 to binary symbols (using $ = 1, * = 0)
- Largest power of 2 in 22: 16 (2^4). Mark position 5 as $. Remaining: 22 - 16 = 6
- Largest power of 2 in 6: 4 (2^2). Mark position 3 as $. Remaining: 6 - 4 = 2
- Largest power of 2 in 2: 2 (2^1). Mark position 2 as $. Remaining: 2 - 2 = 0
- Fill remaining positions with *
| Position | 5 (16) | 4 (8) | 3 (4) | 2 (2) | 1 (1) |
|---|---|---|---|---|---|
| Binary | 1 | 0 | 1 | 1 | 0 |
| Symbol | $ | * | $ | $ | * |
22 = $$$
How to Convert Binary Symbols to a Number
Step 1: Write down the symbol code.
Step 2: Replace each symbol: $ (or &) becomes 1, * becomes 0.
Step 3: Assign position values from right to left: 1, 2, 4, 8, 16, 32...
Step 4: Add up the position values where the digit is 1.
Example: Convert &&&& to a number (using & = 1, * = 0)
| Symbol | & | * | & | & | * | & |
|---|---|---|---|---|---|---|
| Binary | 1 | 0 | 1 | 1 | 0 | 1 |
| Position value | 32 | 16 | 8 | 4 | 2 | 1 |
Sum = 32 + 0 + 8 + 4 + 0 + 1 = 45
Solved Example 1
Q: If </mark> = 1 and <mark class="agridots-imp">*</mark> = 0, which number is represented by <mark class="agridots-imp">$$$*?
Solution:
| Symbol | $ | $ | $ | * | $ |
|---|---|---|---|---|---|
| Binary | 1 | 1 | 1 | 0 | 1 |
| Position (R to L) | 16 | 8 | 4 | 2 | 1 |
Value = 16 + 8 + 4 + 0 + 1 = 29
Answer: (c) 29
Solved Example 2
Q: What is the multiplication of && and &&&? (Using & = 1, = 0)
Solution:
Step 1 — Convert &*& to decimal:
| Symbol | & | * | & |
|---|---|---|---|
| Binary | 1 | 0 | 1 |
| Position | 4 | 2 | 1 |
Value = 4 + 0 + 1 = 5
Step 2 — Convert &&& to decimal:
| Symbol | & | & | & |
|---|---|---|---|
| Binary | 1 | 1 | 1 |
| Position | 4 | 2 | 1 |
Value = 4 + 2 + 1 = 7
Step 3 — Multiply: 5 x 7 = 35
Step 4 — Convert 35 back to binary symbols:
- 35 = 32 + 2 + 1
- Binary: 100011 (6 digits)
| Position | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|
| Binary | 1 | 0 | 0 | 0 | 1 | 1 |
| Symbol | & | * | * | * | & | & |
35 in symbols = &***&&
Answer: (a) &*&&**
Solved Example 3
Q: Which symbol code represents the value of 8 x 12 / 3 + 6? (Using & = 1, * = 0)
Solution:
Step 1 — Calculate the expression (follow BODMAS):
- 8 x 12 = 96
- 96 / 3 = 32
- 32 + 6 = 38
Step 2 — Convert 38 to binary:
- 38 = 32 + 4 + 2
- Binary: 100110
| Position | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|
| Binary | 1 | 0 | 0 | 1 | 1 | 0 |
| Symbol | & | * | * | & | & | * |
38 in symbols = &**&&*
Answer: (c) &&&***
Solved Example 4
Q: If # = 1 and @ = 0, what is the sum of #@@# and ##@#?
Solution:
Convert #@@#:
| Symbol | # | @ | @ | # |
|---|---|---|---|---|
| Binary | 1 | 0 | 0 | 1 |
| Position | 8 | 4 | 2 | 1 |
Value = 8 + 0 + 0 + 1 = 9
Convert ##@#:
| Symbol | # | # | @ | # |
|---|---|---|---|---|
| Binary | 1 | 1 | 0 | 1 |
| Position | 8 | 4 | 2 | 1 |
Value = 8 + 4 + 0 + 1 = 13
Sum: 9 + 13 = 22
Convert 22 to symbols:
- 22 = 16 + 4 + 2 = 10110
- Symbols: #@##@
Answer: #@##@
Solved Example 5
Q: If % - % in symbol form.
Solution:
Convert %%$%:
| Symbol | % | % | $ | % |
|---|---|---|---|---|
| Binary | 1 | 1 | 0 | 1 |
| Position | 8 | 4 | 2 | 1 |
Value = 8 + 4 + 0 + 1 = 13
Convert %:
| Symbol | % | $ | % | $ |
|---|---|---|---|---|
| Binary | 1 | 0 | 1 | 0 |
| Position | 8 | 4 | 2 | 1 |
Value = 8 + 0 + 2 + 0 = 10
Difference: 13 - 10 = 3
Convert 3 to symbols:
- 3 = 2 + 1 = 11 in binary
- Symbols: %%
Answer: %%
Step-by-Step Method Summary
For any binary coding question, follow this universal approach:
- Identify the two symbols and which represents 0 and which represents 1
- Symbol to Number: Replace symbols with 0s and 1s, then use position weights (1, 2, 4, 8, 16, 32...) reading right to left
- Perform the operation (addition, subtraction, multiplication, division) in decimal
- Number to Symbol: Convert the result to binary, then replace 1s and 0s with the appropriate symbols
Speed Tips for Exam
- Memorize powers of 2 up to 64 (1, 2, 4, 8, 16, 32, 64). This is non-negotiable.
- Start conversion from the left (highest power first) — it is faster than dividing by 2 repeatedly.
- Do arithmetic in decimal, not in binary. Convert symbols to numbers first, compute, then convert back.
- Common numbers to remember in binary: 10 = 1010, 15 = 1111, 20 = 10100, 25 = 11001, 30 = 11110.
- Check your digit count: The number of binary digits for a number N is roughly floor(log2(N)) + 1. For example, any number from 16 to 31 needs exactly 5 binary digits.
Common Traps
- Symbol reversal trap: The question may define = 1 (not 0). Always read the definition carefully. Do NOT assume is always 0.
- Reading direction: Binary is always read left to right (highest position first), just like normal numbers. Do not confuse with the right-to-left position assignment.
- Arithmetic before conversion: Some students try to add/multiply symbols directly. Always convert to decimal first, compute, then convert back.
- Leading zeros: The symbol code for 5 is (binary 101), NOT ** (binary 00101). Do not add extra 0-symbols on the left unless the question specifically shows a fixed number of positions.
- BODMAS: When the question gives an expression like 8 x 12 / 3 + 6, follow BODMAS strictly. Multiplication and division before addition and subtraction.