# Binary to Decimal Conversion: Conversion Methods With Solved Questions

A number system is represented as a system of addressing numbers. It is the mathematical notation for expressing numbers of a provided set by consistently applying digits or other symbols. In other words, the number system can be taken as a system of writing or representing numbers in different forms and vice versa.

A number is a numerical value applied for counting and measuring objects, and for performing arithmetic calculations. Numbers have various classes like whole numbers, natural numbers, rational and irrational numbers, and so on.

With this article on Binary to Decimal Conversion, you will learn about the types of number systems along with how to convert binary to decimal and terms like the decimal equivalent of binary numbers with solved examples. Binary to decimal conversion is done to transform a given binary number to its equivalent in the decimal number.

If you are reading Binary to Decimal Conversion then also read about Linear Inequations here.

**Classification of Number System**

There are various types of number systems for example the binary number system, the decimal number, the octal number system, the system, and the hexadecimal number system. Along with this, there are various conversions within the number system.

**Decimal Number System**

The decimal number system is also known as the base 10 number system because it uses ten digits from 0 to 9. The decimal number system is the one we use frequently in our day-to-day life. In the decimal number system, the position progressive is towards the left of the decimal point represented by units, tens, hundreds, thousands, and so on.

The decimal number system is the system that we frequently practice to represent numbers in real-life instances. If any number is represented without a base, it indicates that its base is 10.

**Binary Number System**

The binary number system or the base 2 system constitutes only two digits that are 0 and 1. Digits 0 and 1 are termed as bits and eight bits together form a byte. The data in computers is saved in terms of bits and bytes. The binary number system does not work with other numbers such as 2,3,4,5 and so on.

**Octal Number System**

The octal number system is represented with the base of 8, that is it uses numbers from 0 to 7 to represent numbers.This states that the octal number system employs eight digits: 0,1,2,3,4,5,6 and 7 with the base of 8. The benefit of this system is that it has lesser digits in comparison to various other systems, hence, there would be fewer computational failures.

Numbers like 8 and 9 are not covered in the octal number system. Exactly as the binary, the octal number system is applied in minicomputers but with digits from 0 to 7.

**Hexadecimal Number System**

A hexadecimal system is represented with base 16. This implies in the hexadecimal system there are 16 hex numbers. The hexadecimal number system utilises sixteen digits plus alphabets together: 0,1,2,3,4,5,6,7,8, 9 and A,B,C,D, E, F with the base number as 16. Here, A-F indicates the numbers 10, 11, 12, 13, 14, and 15 of the decimal number system respectively. This hexadecimal number system is used in computers to decrease the large-sized strings of the binary system.

Hexadecimal number | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |

Decimal equivalent | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

Also, read about Sequences and Series here.

**How to Convert Binary to Decimal**

A number can be transformed from one number system to another. For example, octal numbers can be converted to decimal numbers and vice versa, binary numbers can be converted to octal numbers and vice versa, binary number systems can be converted to hexadecimal number systems, and so on. In this article, we will mainly focus on binary to decimal conversion and decimal to binary conversion.

**Binary to Decimal Conversion: Using Positional Notation Method**

In any binary number, the rightmost digit is termed the ‘Least Significant Bit’ or the LSB bit, and the left-most digit is named the ‘Most Significant Bit or the MSB bit.

- In this method, the value of the digit is determined by a weight based on its position.
- This is obtained by multiplying each digit of the given data by the base(2) raised to the corresponding power depending upon the position of that digit in the number.
- The summation of all these values received for all digits provides the equivalent value of the given binary number in its respective decimal system. Let us understand this method by taking a few examples:

**Question:** Convert the binary (11011) number to the decimal equivalent.

**Solution:**

\(\left(11011\right)_2=\left(?\right)_{10}\)

\(Considering\ the\ above\ formula:\)

\(\Rightarrow\left(1\times2^4\right)+\left(1\times2^3\right)+\left(0\times2^2\right)+\left(1\times2^1\right)+\left(1\times2^0\right)\)

\(\Rightarrow\left(16\right)+\left(8\right)+\left(0\right)+\left(2\right)+\left(1\right)\)

\(\Rightarrow27\)

\(\left(11011\right)_2=\left(27\right)_{10}\)

**Question:** 10101 (binary) to decimal equivalent.

**Solution:**

\(\left(10101\right)_2=\left(?\right)_{10}\)

\(Considering\ the\ formula:\)

\(\Rightarrow\left(1\times2^4\right)+\left(0\times2^3\right)+\left(1\times2^2\right)+\left(0\times2^1\right)+\left(1\times2^0\right)\)

\(\Rightarrow\left(16\right)+\left(0\right)+\left(4\right)+\left(0\right)+\left(1\right)\)

\(\Rightarrow21\)

\(\left(10101\right)_2=\left(21\right)_{10}\)

**Question:** Convert 10011 to the decimal equivalent.

**Solution:**

\(\left(10011\right)_2=\left(?\right)_{10}\)

\(Considering\ the\ formula:\)

\(\Rightarrow\left(1\times2^4\right)+\left(0\times2^3\right)+\left(0\times2^2\right)+\left(1\times2^1\right)+\left(1\times2^0\right)\)

\(\Rightarrow\left(16\right)+\left(0\right)+\left(0\right)+\left(2\right)+\left(1\right)\)

\(\Rightarrow19\)

\(\left(10011\right)_2=\left(19\right)_{10}\)

Check out this article on Probability.

**Binary to Decimal Conversion: Using Doubling Method**

As the name implies, the method of doubling or multiplying by two is done to transform a binary number to its decimal equivalent.

- Here we start with the left-most digit.
- In the next step, double the previous number and add it with the current digit. As we are beginning from the left-most digit and there is no previous digit to the LSB digit, we estimate the double of the previous digit as zero(0).
- Continue the same step throughout the sequence for all the digits. The final sum that is obtained in the last step is the exact decimal value.

Question:1000 (binary) in decimal form?
1000→(0x2)+1=1 this one drops down The obtained 1 is taken in the next step as the previous digit and is doubled. 1000→(1×2)+0=2, Similarly the next step is carried out. 1000→(2×2)+0=4 1000→(4×2)+0=8 1000(binary)=8(decimal) |

Consider the below images for understanding purposes.

Question: 100(binary) in decimal is represented as.

Solution:

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**Binary to Decimal Conversion Formula**

Binary to decimal conversion can be done most simply by summing the products of each binary digit with its respective weight and is in the form** – binary digit × 2 raised to a power of the position of the digit** beginning from the right-most digit.

**Formula**

\(\left(Decimal\ number\right)_{10}=n^{th}\ bit\times2^{n-1}\)

\(n=b_nq^n+b_{n-1}q^{n-1}+……b_2q^2+b_1q^1+b_0q^0+b_{-1}q^{-1}+b_{-2}q^{-2}+….\)

\(We\ generally\ start\ with\ the\ conversion\ from\ the\ right\ most\ digit.\)

\(Here\ ,n\ is\ the\ decimal\ equivalent,\ b\ is\ the\ digit\ and\ q\ is\ the\ base\ value.\)

**Question:** What is the binary to decimal equivalent of 11111?

**Solution:**

\(\left(11111\right)_2=\left(?\right)_{10}\)

\(Considering\ the\ above\ formula:\)

\(\Rightarrow\left(1\times2^4\right)+\left(1\times2^3\right)+\left(1\times2^2\right)+\left(1\times2^1\right)+\left(1\times2^0\right)\)

\(\Rightarrow\left(16\right)+\left(8\right)+\left(4\right)+\left(2\right)+\left(1\right)\)

\(\Rightarrow31\)

\(\left(11111\right)_2=\left(31\right)_{10}\)

If you have mastered Binary to Decimal Conversion, you can learn about Integral Calculus here.

**Question:** 11110 in binary to decimal.

**Solution:**

\(\left(11110\right)_2=\left(?\right)_{10}\)

\(Considering\ the\ above\ formula:\)

\(\left(Decimal\ number\right)_{10}=n^{th}\ bit\times2^{n-1}\)

\(n=b_nq^n+b_{n-1}q^{n-1}+……b_2q^2+b_1q^1+b_0q^0+b_{-1}q^{-1}+b_{-2}q^{-2}+….\)

\(\Rightarrow\left(1\times2^4\right)+\left(1\times2^3\right)+\left(1\times2^2\right)+\left(1\times2^1\right)+\left(0\times2^0\right)\)

\(\Rightarrow\left(16\right)+\left(8\right)+\left(4\right)+\left(2\right)+\left(0\right)\)

\(\Rightarrow30\)

\(\left(11110\right)_2=\left(30\right)_{10}\)

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**Binary to Decimal Conversion FAQs**

**Q.1 How do you convert binary to decimal?**

**Ans.1**A binary number can be converted into its respective decimal equivalent using the positional notation method and doubling method.

**Q.2 How do you write 13 in binary?**

**Ans.2**The binary equivalent of 13 can be written as 1101.

**Q.3 What are the four types of number systems?**

**Ans.3**The four types of number systems are as follows: Binary number system, Octal number system, Decimal number system and the Hexadecimal number system.

**Q.4 How do you convert 110 in binary to decimal?**

**Ans.4**The decimal equivalent of the binary number 110 is as follows: 4 + 2 + 0 = 6.

**Q.5 What does 13 mean in binary?**

**Ans.5**13 in decimal is equivalent to 1101 in binary.

**Q.6 What is the easiest way to convert decimal to binary?**

**Ans.6**The simplest way to convert decimal to binary is; for the given value of the digit, determine the weight based on its position. This is obtained by multiplying each digit of the given data by the base(2) raised to the corresponding power depending upon the position of that digit in the number. The summation of all these values received for all digits provides the equivalent value of the given binary number in its respective decimal system.

**Q.7 How do you convert a number to binary?**

**Ans.7**To transform an integer to its binary equivalent, start with the integer given in the question and divide it by two keeping notice of the quotient and the remainder. Continue dividing the quotient by 2 till you notice a quotient of zero. Then just draft out the remainders in the reverse order.