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Binary to decimal conversion describes the conversion of a binary number to an equivalent decimal number and it can be done using a simple positional notation method. In mathematics, there are four types of number systems, which are used to represent and express numbers. These number systems differ in terms of the base they use. These four types of number systems are: Binary Number System (Base 2), Octal Number System (Base 8), Decimal Number System (Base 10) and Hexadecimal Number System (Base 16). Number system is an essential part used in computer architecture and most electronic gadgets also use it.
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Key Terms:Binary numbers, decimal numbers, binary to decimal conversion, positional notation method, number system
What is binary number system?
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Binary number system is the simplest kind of number system. Basically, it uses only two digits 0 and 1 to represent a number. Therefore, binary number is most preferred in modern computer engineering, networking and communication specialists as digital electronics only have these two states (0 or 1).
The entire number system like fractions,real numbers as well as large numbers can be represented through binary numbers. It is also called the base-2 numeral system in which each digit is known as a bit or binary digit.
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What is decimal number system?
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Decimal number system is the base 10 numeral system which uses ten digits from 0 to 9. It has two parts: whole number part and fractional part. These two parts are separated by a dot known as the decimal point. Digits lying on the left side of the decimal point represent the whole number and the digits on the right side represent the fractional part or decimal part. For example, in decimal number 25.2, 25 represent the whole number and 2 represent the fractional part.
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What is binary to decimal conversion?
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It is necessary to understand the binary to decimal conversion for computer programming applications. To convert a number in a binary system to a number in a decimal system, a multiplication method is used. There are two terms used while going through the conversion process, which are MSB (most significant bit) and LSB (least significant bit).
In this process of conversion, if a number with base n has to be converted into a number with base 10, then each digit of the given number is multiplied from the MSB to the LSB with reducing the power of base. This method of binary to decimal conversion is called the positional notation method.
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Methods of binary to decimal conversion
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There are two methods available to convert binary numbers to decimal numbers. Which are,
- Positional notation method
- Doubling method
Positional notation method
In positional notation method, value of a digit in a given number is determined by a weight based on its position. Let’s consider binary number (101101)2 to understand the steps of conversion using positional notation method. Steps for converting binary number in equivalent decimal number are as follow:
- Write down the given binary number and count the powers of 2 for all the digits starting from right to left. The first power will be 20 and as we move further, it will be 21 ,22 , 23… and so on.
There are 6 digits in the given example, therefore by starting from right to left, the weight of each position is 20,21,22,23,24,25.
- Now multiply each digit of binary number starting from rightmost side with its respective weight based on its position and evaluate the number such that the first binary digit multiplies with the greatest power of 2.
Now, add all the products obtained for all the digits of the binary number.
- Express the binary number as a decimal number.
In this example it’s (101101)2 = (45)10
Doubling method
The process of doubling or multiplying the digit by 2 is done to convert binary number to decimal. By considering the same example (101101)2, the steps for conversion are as follow:
- Write the given binary number and start evaluating from left to right. Double the previous number and do addition of the current digit. As evaluation is done from left to right, there is no previous digit to the leftmost digit. Therefore, the double of the previous digit is considered as 0.
For example, in 101101, the leftmost digit is 1. The double of the previous is considered as 0 and we get ((0×2) +1) = 1.
- Continue the same process in sequence for all digits. The sum at the end of process completion is the actual decimal value.
Binary to decimal conversion formula
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The formula for the conversion of binary number to decimal number is given by:
DecimalNumber= nthbit ×2n-1
Binary to decimal formula:
n=bnqn+bn-1qn-2+….+b2q2+b1q1+b0q0+b-1q-1+b-2q-2
Where, n is decimal equivalent
b is the binary digit
q is the base value.
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Binary to Decimal Conversion Chart
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Conversion chart for the first 20 decimal numbers is given in the table below:
| Binary | Decimal |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 10 | 2 |
| 11 | 3 |
| 100 | 4 |
| 101 | 5 |
| 110 | 6 |
| 111 | 7 |
| 1000 | 8 |
| 1001 | 9 |
| 1010 | 10 |
| 1011 | 11 |
| 1100 | 12 |
| 1101 | 13 |
| 1110 | 14 |
| 1111 | 15 |
| 10000 | 16 |
| 10001 | 17 |
| 10010 | 18 |
| 10011 | 19 |
| 10100 | 20 |
Things to remember
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- For a binary number with 'n' digits, the least significant bit has a weight of 20 and the most significant bit has a weight of 2n-1.
- Binary numbers are usually read digit-by-digit while speaking, in order to distinguish them from decimal numbers.
- Addition, subtraction, multiplication and division can be performed on binary numbers much like in other number systems.
- Decimal number system is also called as Hindu-Arabic number system or Arabic number system.
- To convert the fractional part of binary number to decimal, multiply it with 2-1,2-2…2-n from left to right.
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Sample Questions
Ques. To convert a binary number to decimal number, what do we multiply with every digit in the binary number? (1 Mark)
Ans: we multiply every digit in the binary number by a power of 2 from the rightmost side, to convert a binary number to its equivalent decimal number. For example, to convert (0010)2, we multiply each bit of this binary number to 20,21,22 and 23 from the right to left.
Ques. What will be the resulting decimal number when (100)2 are converted from binary to decimal? (2 Marks)
Ans: to convert binary to decimal,
(100)2 = (1×22) + (0×21) + (0×20)
(100)2 = 4 + 0 + 0
(100)2 = 4
Therefore, resulting decimal number for (100)2 will be 4.
Ques. What is the formula for converting the binary number to its equivalent decimal number? (2 Marks)
Ans: formula for converting the binary number to decimal number is:
n=bnqn+bn-1qn-2+…+b2q2+b1q1+b0q0+b-1q-1+b-2q-2
Where, n is decimal equivalent, b is the binary digit, q is the base value.
Ques. Convert the binary number 1011 to its equivalent decimal number. (2 Marks)
Ans: Given binary number is (1011)2
By using the binary to decimal conversion formula,
(1011)2 = (1×23) + (0×22) + (1×21) + (1×20)
= 8 + 0 + 2 + 1
= (11)10
Therefore, decimal equivalent of (1011)2 will be (11)10
Ques. Convert (11110110)2 into an equivalent decimal number. (2 Marks)
Ans: Given binary number is (11110110)2
By using the binary to decimal conversion formula,
(11110111)2 = (1×27) + (1×26) + (1×25) + (1×24) + (0×23) + (1×22) + (1×21) + (0×20)
= 1288 + 64 +32 + 16 + 0 + 4 + 2 + 0
= (246)10
Therefore, decimal equivalent of (11110110)2 will be (246)10
Ques. How is binary number different from decimal number? (2 Marks)
Ans:
| Binary number | Decimal number |
|---|---|
| It is the number system with base 2. | It is the number system with base 10. |
| Digits that are used in binary system are 0 and 1. | Digits that are used in decimal system are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. |
| All digital computers use binary number systems. | Decimal numbers are used every day in human life for calculation, conversion etc. |
Ques. Using the positional notation method, prove that (159)10 is decimal number of (10011111)2. (2 Marks)
Ans: By using positional notation method,
(10011111)2 = (1×27) + (0×26) + (0×25) + (1×24) + (1×23) + (1×22) + (1×21) + (1×20)
(11110111)2 = (1×128) + (0×64) + (0×32) + (1×16) + (1×8) + (1×4) + (1×2) + (1×1)
(11110111)2 = 128 + 0 + 0 + 16 + 8 + 4 + 2 + 1
(11110111)2 = 159
Hence, (159)10 is the decimal equivalent of given binary number.
Ques. What will be the equivalent decimal number of 1010.1010? (3 Marks)
Ans: The given binary number is (1010.1010)2
There is a binary point with fractional part in the given number. Therefore,
Now, by binary to decimal conversion formula,
(1010.1010)2 = (1×23) + (0×22) + (1×21) + (0×20) + (1×2-1) + (0×2-2) + (1×2-3) + (1×2-4)
(1010.1010)2 = 8 + 0 + 2 + 0 + 0.5 + 0 + 0.125 + 0
(1010.1010)2 = (10.6240)10
Therefore, decimal equivalent of (1010.1010)2 is (10.6240)10.
Ques. Find out the decimal number of (10101100)2 by using doubling method of conversion. (3 Marks)
Ans: here, 1 is the leftmost digit and there is no previous number. So, as per doubling method, doubled value is 0. Now, by adding it with the current digit 1, we get the value (0×2) + 1 = 1.
101011002→0×2+1=1
→1×2+0=2
→2×2+1=5
→5×2+0=10
→10×2+1=21
→21×2+1=43
→43×2+0=86
→86×2+0=172
Here, the final sum is 172.
Therefore, decimal number for the given binary number (10101100)2is (172)10.
Ques. Convert the following binary numbers into base-10 number system. (5 Marks)
a) (1101001)2
b) (11110111)2
Ans: (i) for the binary number (1101001)2,
(1101001)2 = (1×26) + (1×25) + (0×24) + (1×23) + (0×22) + (0×21) + (1×20)
(1101001)2 = (1×64) + (1×32) + (0×16) + (1×8) + (0×4) + (0×2) + (1×1)
(11110111)2 = 64 + 32 + 0 + 8 + 0 + 0 + 1
(11110111)2 = 105
Therefore, (105)10 is the decimal equivalent of given binary number.
(ii) for the binary number (01110111)2,
(01110111)2 = (0×27) + (1×26) + (1×25) + (1×24) + (0×23) + (1×22) + (1×21) + (1×20)
(01110111)2 = (0×128) + (1×64) + (1×32) + (1×16) + (0×8) + (1×4) + (1×2) + (1×1)
(01110111)2 = 0 + 64 + 32 + 16 + 0 + 4 + 2 + 1
(11110111)2 = 119
Therefore, (119)10 is the decimal equivalent of given binary number.
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