Decimal to Binary
- Introduction
- Defining Decimal Number System
- Defining Binary Number System
- Process to Convert Decimal to Binary
- Decimal to Binary Conversion Table
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
Converting a number from the decimal to binary number systems is known as decimal-to-binary conversion. Every number system has a base, which is established by adding up all of the digits the system uses. For instance, the binary number system, which only uses two digits to express a number, has a base of two. Similar to this, the decimal number system uses `10` digits for expressing a number, so its base is `10`. Before we proceed with the decimal to binary conversion, let us first understand the binary and decimal number systems.
Defining Decimal Number System
The decimal system, sometimes referred to as the base-`10` system, is a numeric system in which numbers are represented by ten different numbers: `0, 1, 2, 3, 4, 5, 6, 7, 8`, and `9`. The decimal system uses place values to indicate each digit's value within a number. For example, the digit on the right represents the unit place, the digit left to it represents the tens place, the next represents hundreds, and so on. Each place value is a power of `10`. For instance, in number `325`, the digit `5` stands for `5` units, the digit `2` for `2` tens, and the digit `3` for `3` hundreds. This method is widely used to represent and process numbers through arithmetic operations in mathematics, science, commerce, and daily life.
Defining Binary Number System
A base-`2` numeral system used in digital technology and mathematics is called the binary number system. The binary system only employs two digits—`0` and `1`—to represent numbers, as opposed to the decimal system's ten digits `(0–9)`. A "bit" is the abbreviation for a binary digit, which is each digit in a binary number.
Every place in binary represents a power of two, starting at the position farthest to the right, which is `2^0 (1)`. The position immediately to the left is `2^1 (2)`, followed by `2^2 (4), 2^3 (8)`, and so on. This pattern keeps on, where a higher power of two is represented by each new digit added to the left.
Process to Convert Decimal to Binary
Converting a decimal number to binary involves repeatedly dividing the decimal number by `2` and noting the remainders until the quotient becomes zero. The binary representation is obtained by reading the remainders in reverse order. Here are the steps to convert a decimal number to binary:
`1`. Start with the Decimal Number: Begin with the decimal number you want to convert to binary.
`2`. Divide by `2`: Divide the decimal number by `2`.
`3`. Note the Remainder: The remainder will be either `0` or `1`.
`4`. Record the Remainder: Write down the remainder.
`5`. Update the Quotient: Update the quotient to be the result of the division.
`6`. Repeat: Repeat steps `2-5` until the quotient becomes `0`.
`7`. Reverse the Remainders: Once the quotient becomes zero, read the remainders in reverse order. The series of remainders will be the binary representation of the original decimal number.
`8`. Pad with Zeros (Optional): If necessary, add leading zeros to the binary representation to make it the desired length.
Example: Convert the decimal number `11` to binary.
Solution:
- `11 ÷ 2 = 5` with a remainder of `1`. Record the remainder: `1`.
- Update the quotient to `5`.
- `5 ÷ 2 = 2` with a remainder of `1`. Record the remainder: `1`.
- Update the quotient to `2`.
- `2 ÷ 2 = 1` with a remainder of `0`. Record the remainder: `0`.
- Update the quotient to `1`.
- `1 ÷ 2 = 0` with a remainder of `1`. Record the remainder: `1`.
- The quotient is now `0`, so stop the division.
- The answer is `1011`.
Decimal to Binary Conversion Table
Numerous techniques exist for converting decimal to binary numbers. The base of a number changes from `10` to `2` when it is converted from decimal to binary. It should be remembered that every decimal number has a binary equivalent. The decimal-to-binary chart for the first `9` whole numbers is displayed in the following table. This table comes in handy for mental calculation while converting from decimal to binary.
Solved Examples
Example `1`: Convert decimal `10` to binary.
Solution:
- `10 ÷ 2 = 5` with a remainder of `0`. (Remainder: `0`)
- `5 ÷ 2 = 2` with a remainder of `1`. (Remainder: `1`)
- `2 ÷ 2 = 1` with a remainder of `0`. (Remainder: `0`)
- `1 ÷ 2 = 0` with a remainder of `1`. (Remainder: `1`)
Binary: Reading remainders in reverse order: `1010`
So, decimal `10` in binary is `1010`.
Example `2`: Convert decimal `27` to binary.
Solution:
- `27 ÷ 2 = 13` with a remainder of `1`. (Remainder: `1`)
- `13 ÷ 2 = 6` with a remainder of `1`. (Remainder: `1`)
- `6 ÷ 2 = 3` with a remainder of `0`. (Remainder: `0`)
- `3 ÷ 2 = 1` with a remainder of `1`. (Remainder: `1`)
- `1 ÷ 2 = 0` with a remainder of `1`. (Remainder: `1`)
Binary: Reading remainders in reverse order: `11011`
So, decimal `27` in binary is `11011`.
Example `3`: Convert decimal `42` to binary.
Solution:
- `42 ÷ 2 = 21` with a remainder of `0`. (Remainder: `0`)
- `21 ÷ 2 = 10` with a remainder of `1`. (Remainder: `1`)
- `10 ÷ 2 = 5` with a remainder of `0`. (Remainder: `0`)
- `5 ÷ 2 = 2` with a remainder of `1`. (Remainder: `1`)
- `2 ÷ 2 = 1` with a remainder of `0`. (Remainder: `0`)
- `1 ÷ 2 = 0` with a remainder of `1`. (Remainder: `1`)
Binary: Reading remainders in reverse order: `101010`
So, decimal `42` in binary is `101010`.
Example `4`: Convert decimal `75` to binary.
Solution:
- `75 ÷ 2 = 37` with a remainder of `1`. (Remainder: `1`)
- `37 ÷ 2 = 18` with a remainder of `1`. (Remainder: `1`)
- `18 ÷ 2 = 9` with a remainder of `0`. (Remainder: `0`)
- `9 ÷ 2 = 4` with a remainder of `1`. (Remainder: `1`)
- `4 ÷ 2 = 2` with a remainder of `0`. (Remainder: `0`)
- `2 ÷ 2 = 1` with a remainder of `0`. (Remainder: `0`)
- `1 ÷ 2 = 0` with a remainder of `1`. (Remainder: `1`)
Binary: Reading remainders in reverse order: `1001011`
So, decimal `75` in binary is `1001011`.
Example `5`: Convert decimal `99` to binary.
Solution:
- `99 ÷ 2 = 49` with a remainder of `1`. (Remainder: `1`)
- `49 ÷ 2 = 24` with a remainder of `1`. (Remainder: `1`)
- `24 ÷ 2 = 12` with a remainder of `0`. (Remainder: `0`)
- `12 ÷ 2 = 6` with a remainder of `0`. (Remainder: `0`)
- `6 ÷ 2 = 3` with a remainder of `0`. (Remainder: `0`)
- `3 ÷ 2 = 1` with a remainder of `1`. (Remainder: `1`)
- `1 ÷ 2 = 0` with a remainder of `1`. (Remainder: `1`)
Binary: Reading remainders in reverse order: `1100011`
So, decimal `99` in binary is `1100011`.
Practice Problems
Q`1`. What is the binary representation of the decimal number `14`?
- `1100`
- `1110`
- `1010`
- `1001`
Answer: b
Q`2`. Convert the decimal number `25` into binary.
- `11010`
- `10111`
- `101101`
- `11001`
Answer: d
Q`3`. What is the binary equivalent of the decimal number `63`?
- `100111`
- `111000`
- `101011`
- `111111`
Answer: d
Q`4`. Convert the decimal number `29` into binary.
- `11010`
- `11101`
- `10011`
- `10101`
Answer: b
Q`5`. What is the binary representation of the decimal number `91`?
- `1011011`
- `1101101`
- `1001011`
- `1110101`
Answer: a
Frequently Asked Questions
Q`1`. What is the significance of binary conversion in computer science?
Answer: Binary conversion is fundamental in computer science as it enables the representation of digital data and instructions in a format that can be processed by electronic devices. Computers use binary code internally to perform operations and store data, making it essential for programmers and computer engineers to understand binary conversion.
Q`2`. How do you convert a decimal number to binary manually?
Answer: To convert a decimal number to binary manually, you repeatedly divide the decimal number by `2` and note the remainders until the quotient becomes zero. The remainders, read in reverse order, represent the binary equivalent of the decimal number.
Q`3`. Are there any shortcuts or techniques for quickly converting decimal numbers to binary?
Answer: One common shortcut is to repeatedly divide the decimal number by `2`, while noting both the quotient and the remainder at each step. Reading the remainders in reverse order provides the binary representation. Additionally, some individuals find it helpful to memorize binary equivalents of common decimal numbers or use mental math techniques to expedite conversion.
Q`4`. What is the maximum number of bits needed to represent a decimal number in binary?
Answer: The maximum number of bits needed to represent a decimal number in binary depends on the range of values the binary representation needs to cover. Generally, for an \( n \)-bit binary number, the range of representable decimal numbers is \( 0 \) to \( 2^n - 1 \). For example, an `8`-bit binary number can represent decimal values from `0` to `255` (\( 2^8 - 1 \)).
Q`5`. How do you ensure accuracy when converting decimal numbers to binary?
Answer: To ensure accuracy when converting decimal numbers to binary, it's important to double-check each step of the conversion process, including the division and the recording of remainders. Additionally, verifying the final binary result by converting it back to decimal can help confirm the correctness of the conversion. Finally, using calculators or programming tools designed for binary conversion can reduce the likelihood of errors.