Binary Place Value
The binary number system (base 2) is used by every digital computer and electronic device. Instead of the ten digits 0–9 used in decimal, binary uses only two digits: 0 and 1. Place values in binary are powers of 2 instead of powers of 10.
Binary Place Values
| Place | Power of 2 | Value in Decimal |
|---|---|---|
| 1st (rightmost) | 2⁰ | 1 |
| 2nd | 2¹ | 2 |
| 3rd | 2² | 4 |
| 4th | 2³ | 8 |
| 5th | 2⁴ | 16 |
| 6th | 2⁵ | 32 |
| 7th | 2⁶ | 64 |
| 8th | 2⁷ | 128 |
Converting Binary to Decimal
Multiply each binary digit by its place value, then add the results.
More Conversion Examples
| Binary | Calculation | Decimal |
|---|---|---|
| 0001 | 1 | 1 |
| 0010 | 2 | 2 |
| 0101 | 4 + 1 | 5 |
| 1000 | 8 | 8 |
| 1111 | 8+4+2+1 | 15 |
| 10000 | 16 | 16 |
Converting Decimal to Binary
Repeatedly divide by 2 and record the remainders. Read remainders from bottom to top.
Why Binary?
Electronic circuits can easily represent two states: on (1) and off (0). This maps perfectly to binary digits, making binary the natural language of computers.
- Binary uses only digits 0 and 1.
- Each place is a power of 2: 1, 2, 4, 8, 16, 32, …
- To convert binary to decimal: sum the place values where a 1 appears.
- To convert decimal to binary: divide repeatedly by 2 and read remainders upward.
- Computers store all data as binary (bits).
Quick Practice
- Convert binary 1010 to decimal.
- Convert binary 11001 to decimal.
- Convert decimal 9 to binary.
- Convert decimal 25 to binary.
- What is the highest value a 4-bit binary number can hold?
Summary
Binary place values work exactly like decimal place values but with powers of 2 instead of powers of 10. Every decimal number has a unique binary equivalent, and this conversion is the foundation of all digital computing. Understanding binary place value builds the bridge between pure mathematics and computer science.