What is the Binary Representation of the Decimal Number 84?
The decimal number 84 is represented as **1010100** in binary.Understanding Number Bases
Our everyday number system is the *decimal* system, also known as base 10. It uses ten unique digits (0-9) and assigns place values based on powers of 10 (e.g., 10^0, 10^1, 10^2, and so on). For instance, the number 84 in decimal means (8 × 10^1) + (4 × 10^0).In contrast, the binary system is base 2. It uses only two digits: 0 and 1. Each position in a binary number represents a power of 2 (e.g., 2^0, 2^1, 2^2, etc.). Binary is fundamental to digital electronics and computer systems because it directly corresponds to the "on" (1) and "off" (0) states of electrical signals.
How to Convert Decimal to Binary (The Division Method)
To convert a decimal number to its binary equivalent, one common method is the division method, also known as the "divide by 2" method. This process involves repeatedly dividing the decimal number by 2 and recording the remainder at each step.Here's how to convert 84 from decimal to binary using this method:
- Divide by 2: Divide the decimal number by 2.
- Record Remainder: Note down the remainder (which will be either 0 or 1).
- Use Quotient: Take the quotient from the division and use it as the new number for the next step.
- Repeat: Continue this process until the quotient becomes 0.
- Read Upwards: The binary equivalent is obtained by reading the remainders from bottom to top.
Let's apply this to the number 84:
| Division | Quotient | Remainder |
|---|---|---|
| 84 ÷ 2 | 42 | 0 |
| 42 ÷ 2 | 21 | 0 |
| 21 ÷ 2 | 10 | 1 |
| 10 ÷ 2 | 5 | 0 |
| 5 ÷ 2 | 2 | 1 |
| 2 ÷ 2 | 1 | 0 |
| 1 ÷ 2 | 0 | 1 |
Reading the remainders from bottom to top, we get 1010100.
You can also verify this by converting the binary number back to decimal:
- (1 × 2^6) + (0 × 2^5) + (1 × 2^4) + (0 × 2^3) + (1 × 2^2) + (0 × 2^1) + (0 × 2^0)
- (1 × 64) + (0 × 32) + (1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (0 × 1)
- 64 + 0 + 16 + 0 + 4 + 0 + 0 = 84
Practical Applications of Binary
Binary numbers are at the core of all modern computing and digital technology.- Computer Memory and Processing: Every piece of data—from text and images to software programs—is stored and processed as sequences of 0s and 1s within a computer's memory and processor.
- Digital Communication: Data transmitted across networks, such as the internet, is converted into binary signals.
- Electronics: Digital circuits operate using two distinct voltage levels, representing binary 0 and 1.
For more detailed explanations on number system conversions, you can refer to resources on decimal to binary conversion.
