Binary To Decimal: How To Convert Binary And Decimal Numbers

In mathematics, there are multiple number systems, each used for its own specific reasons; one of which is the binary numeral system, through which modern computers and computer-based devices work. Other than binary, there is the decimal number system, which is used widely by us, humans, in our daily lives. While learning programming, networking, digital devices, computer science, or when dealing with numeral systems, one needs to convert between different number systems, and we are going to focus on learning especially binary and decimal numbers and their conversions between each other. 

It can be done manually, and with the help of available online tools that take only a few seconds to make the conversion between different number systems. One such fine example is Toolflora’s number base converter tool, which is designed to produce accurate results. We must first understand how these two widely used number systems work and their conversions before using any converter. 

What Is Binary? 

Binary’s recorded history stretches back to the time of the Mauryan Empire, in the accounts of the ancient Indian scholar Pingala. The binary system consists of only two digits: 0 & 1, with powers of 2 for each digit, on which the binary language has been created to let computers understand the input. 

What Is Decimal? 

Decimals also have an established history in different cultures as non-positional decimal numbers until, with the help of Chinese rod calculations, the decimal system became positional. Decimals contain 10 digits ranging from 0 to 9, with 10 as its base. The number system we use daily is considered the decimal system because of its biological root and precise representation as per the needs. 

Binary Vs Decimal: What’s The Difference? 

Both of the numeral systems are used for their specific purposes in the world right now, and the difference between them is only in the symbols and the use of place values. 

Number system	Base	Digits used	Example
Binary	2	0 & 1	101101
Decimal	10	0-9	542
Octa	8	0-7	25
Hexadecimal	16	0-9 & A-F	FF

How to Convert Binary to Decimal Step by Step 

STEP-1:

Write down the given binary number and assign it the powers of two starting from 0, from right to left. 

STEP-2:

Write down the equation with the product of the binary digits and their corresponding powers of 2. 

STEP-3:

Multiply and add them to get the decimal form of the binary number. 

EXAMPLE: 

Consider any binary number like this, 

10101₂

Assign powers of 2 to each binary digit,

2⁴, 2³, 2², 2¹, 2⁰

Write them together in the sum as the product,

=1 × 2⁴ + 0 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰

=16 + 4 + 1 

=21₁₀ ( converted binary number into decimal number) 

Binary To Denary: Is It The Same As Binary To Decimal? 

Denary is the same as decimal, with the same base of 10; their names may sound a bit different but have the same meaning, which is ‘the tenth’ for decimal and denary, which comes from the word denarius, meaning ‘containing tenth’. So whenever you see or read the denary instead of decimal, you don’t need to worry about the result; the result will be the same as with decimals. We can conclude that binary to denary carries the same meaning as binary to decimal. 

How To Convert Decimal To Binary? 

Converting decimals to binary might sound hard to you, but it’s actually pretty simple and based on the maths of primary class. Let’s understand this with an example:- 

Division	Quotient	Remainder
45 ÷ 2	22	1
22 ÷ 2	11	0
11 ÷ 2	5	1
5 ÷ 2	2	1
2 ÷ 2	1	0
1 ÷ 2	0	1

After the whole process, have a look at only the remainders:-

1, 0, 1, 1, 0, 1; reverse the order to get the converted binary number: 101101₁₀ 

But what is the need for reversing the order? This question must arise in your mind, and the explanation is quite simple: the first remainder is the least significant bit of the binary number, and each division gives you the next bit moving toward the left. 

So, you have learned both methods of conversion, whether it’s from binary to decimal or from decimal to binary, and you may have realized how easy these conversions are, but what if you need to convert several values in a limited time? Let’s discuss further. 

Convert Binary To Decimal Online

The question arises when learning conversions between number systems: ‘What if you need to convert several numbers in a limited time?’ Learning conversions for manual use is good unless you have several values waiting to be converted into another number base. 

At that time, you need a tool that does the work for you within seconds, and for that you can find many tools that convert the number base, but in search of one you may encounter numerous kinds of tools, some of which run ads, some of which are paid, and some of which may not be beginner-friendly or may not have a simple user interface, but you will ultimately find the one that suits you. 

Allow me to save you time by directing your attention to ToolFlora’s number base converter, which is designed in such a way that it will help anyone with its beginner-friendly use and interface and provides direct results with no ads, no interruption of any kind. You don’t need to download it or pay it for its use because it is free and can be used directly through your browser.

How Toolflora’s Number Base Converter Works  

step-1: Open the number base converter in your browser 

step-2: Write the number and choose the suitable base system 

Step-3: Copy the result and use it wherever you like 

Common Mistakes When Converting Binary To Decimal 

Starting the powers of 2 from the wrong side

The rightmost binary digit represents:

2⁰ = 1

The next position is:

2¹ = 2

Then:

2² = 4

and so on.

Always start from the right.

Adding positions containing zero

A zero contributes nothing to the decimal value.

For 10101, you only need: 16 + 4 + 1

You do not add the positions containing zero.

Reading decimal-to-binary remainders in the wrong direction

When using repeated division by 2, read the remainders from bottom to top, not top to bottom.

Mixing number bases

A number such as 1010 could be interpreted differently depending on its base.

1010₂ = 10₁₀

The subscript helps make the base clear.

When Converting Binary To Decimal Be Useful? 

It can be useful in the following fields:-

• Computer science 
• Programming 
• Networking 
• Digital electronics 
• Education 
• Debugging & development 

Frequently Asked Questions