Binary Digit Conversion: Step-by-Step Guide for Students
The binary digit, also known as the bit is a basic unit of information in computer and digital communication. A single binary digit is known as a bit. The binary digits represent logical code with one more two possible values. These two possible values are represented as either 0 or 1.
In Mathematics, binary numbers are made up of binary digits. In other words, binary numbers require only 2 digits to represent any number instead of 10 different symbols that are used in the decimal number system. Hence, the decimal numbers from 1 to 10 in binary are represented as
Also, binary numbers can be easily converted into other number systems like decimal number systems, hexadecimal number systems, octal number systems, and vice versa. Here, we will discuss binary number to decimal number conversion, and decimal number to binary number with a decimal point.
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Binary Number to Decimal Number Conversion
To convert binary numbers to decimal numbers, we use the multiplication method. In this method, if a base 2 has to be converted into base 10, then each digit of a given number is multiplied from the most significant digit to the least significant digit with reducing the power of the base. Following are the steps to convert binary numbers to decimal numbers.
Step 1: Write down the binary number and calculate the place value of each digit.
Step 2: Starting from the most significant digit to the least significant digit, multiply each digit of the given binary number by the corresponding power of 2.
Step 3: Add up the results obtained to convert the given binary number into the decimal number.
Let us understand with an example:
Convert the Binary Number 1110₂ into a Decimal Number.
Step 1: Calculate the place values
As 1110 has four digits, so we have four place values: 2º, 2¹, 2², and 2³
Step 2: Multiply each digit by the corresponding power 2
1 x 2³ = 8 1 x 2² = 4 1 x 2¹ = 2 0 x 2º = 0
Step 3: Sum up the result to get the given binary number into decimal numbers.
= 8 + 4+ 2 + 0
= 14
Decimal Number To Binary Number Conversion
Let us now understand how to convert decimal number to binary number through the following steps:
Divide the given decimal number by 2. By dividing the number by 2, the result will be obtained along with the remainder.
If the given decimal number is even, the result obtained will be the whole number, and the remainder will be 0.
If the given decimal number is odd, the result obtained will not be completely divided and will give the remainder 0.
The binary number will be obtained by placing all the remainders in order in such a way, the Least Significant Bit at the top and the Most Significant Bit at the bottom
Let us understand with an example:
Convert 294₁₀ into a Binary Number
Hence, 294.46₁₀ is 100100110₂
Decimal to Binary with Decimal Point Conversion
To convert decimal numbers to binary with a decimal point, we convert both integer and fractional parts individually and then add the values to get equivalent binary numbers. Let us understand with an example:
Convert 294.46₁₀ To Binary
To convert 98.46₁₀ to binary, we first convert the integer part that is 98 and then fractional parts that is 46. Further, we will add the values of both parts to get equivalent binary numbers.
Following are the steps to convert integer 294 to decimal:
Divide 294 by 2 and keep observing both quotient and remainder value. Continue dividing the quotient by 2 till you get the quotient value 0.
As 295 is an even number, the result will be the whole number and it gives the remainder 0.
Then just write the remainder value in reverse order to get an equivalent binary number.
Hence, 294.46₁₀ is 100100110₂
Following are the steps to convert decimal fraction 0.46 to decimal number
Step 1: Multiply the decimal fraction 0.46 by 2 and keep observing both integer and fractional values. Continue multiplying the decimal fraction by 2, till you get the resultant fractional values equal to 0.
Step 2: In this step, write down the integer part from the result of each multiplication to get an equivalent binary number.
Hence, the decimal number 0.46 in binary is 0.0111010111₂.
Therefore, the decimal number 292.46 in binary is 100100110. 0111010111₂.
Solved Examples
1. Convert the Binary Number 1011₂ in Decimal Number.
Step 1: Calculate the place values
As 1011 has four digits, so we have four place values: 2º, 2¹, 2², and 2³
Step 2: Multiply each digit by the corresponding power 2
1 x 2³ = 8 0 x 2² = 0 1 x 2¹ = 2 1x 2º = 1
Step 3: Sum up the result to get the given binary number into decimal numbers.
= 8 + 0+ 2 + 0
= 11
Hence, 1011₂ is 14₁₀
2. Convert the Decimal Number 16 into a Binary Number
Hence, 16₁₀ is 10000₂
FAQs on What Are Binary Digits?
1. What is a binary digit, and why is it also called a 'bit'?
A binary digit is the most basic unit of data in computing and digital systems. It can only have one of two possible values: 0 or 1. The term 'bit' is a portmanteau, or a blend of the words, 'bi'nary digi't'. These digits are the fundamental building blocks for all computer operations and data storage, representing 'off' and 'on' electrical states, respectively.
2. How do binary digits differ from the decimal digits we use in everyday maths?
The primary difference lies in the base of the number system. Our everyday decimal system is base-10, using ten distinct symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). The binary system, in contrast, is base-2 and uses only two symbols (0 and 1). This affects the place value of each digit; in decimal, place values are powers of 10 (1, 10, 100), while in binary, they are powers of 2 (1, 2, 4, 8).
3. How are the numbers from 1 to 10 represented using binary digits?
The first ten positive integers are represented in the binary system as follows:
1 in decimal is 1 in binary
2 in decimal is 10 in binary
3 in decimal is 11 in binary
4 in decimal is 100 in binary
5 in decimal is 101 in binary
6 in decimal is 110 in binary
7 in decimal is 111 in binary
8 in decimal is 1000 in binary
9 in decimal is 1001 in binary
10 in decimal is 1010 in binary
4. Why are binary digits so important for modern computers and electronic devices?
Binary digits are crucial because computer hardware, like processors and memory, operates using billions of tiny electronic switches called transistors. These switches have two stable states: on or off. These two states can be perfectly represented by the two binary digits, 1 (on) and 0 (off). This simple, two-state system is highly reliable, less prone to errors, and allows for the efficient design of complex logic circuits that perform all computational tasks.
5. How can you convert a binary number like 1101 into its decimal equivalent?
To convert a binary number to decimal, you multiply each binary digit by its corresponding place value (which is a power of 2) and then sum the results. For the binary number 1101, the process is as follows:
The rightmost digit '1' is in the 20 (or 1s) place: 1 × 1 = 1
The next digit '0' is in the 21 (or 2s) place: 0 × 2 = 0
The next digit '1' is in the 22 (or 4s) place: 1 × 4 = 4
The leftmost digit '1' is in the 23 (or 8s) place: 1 × 8 = 8
Adding these values together gives: 8 + 4 + 0 + 1 = 13. So, the binary number 1101 is equal to the decimal number 13.
6. What is a 'byte' and how does it relate to binary digits?
A byte is a standard unit of digital information that consists of a group of 8 bits (binary digits). Since each bit can be either 0 or 1, a byte containing 8 bits can represent 28, or 256, different possible values. This is significant because a single byte is typically used to encode a single character of text (like 'A' or '?'), a pixel's colour intensity in an image, or a small number.
7. Can a sequence of binary digits represent something other than a number?
Yes, absolutely. The meaning of a binary sequence is determined by the context in which it is used. For example, the 8-bit binary sequence 01000001 can represent:
The decimal number 65.
The uppercase letter 'A' in the ASCII character encoding standard.
A specific colour value within an image file.
A specific instruction for a computer's processor.
This versatility, managed through encoding schemes, allows binary to be the universal language for all types of digital data, not just numbers.
