The Ultimate Guide to Binary and Decimal Conversion

The world of technology runs primarily on the binary system. This is the first and most crucial language of computers, and consists of only two numbers, 0 and 1. People, on the other hand, use the decimal system, which is made up of ten digits ranging from 0 to 9. To study and practice computing, programming, and electronics, one must be able to convert numbers from one system to another. Our binary to decimal converter is designed to show you how to convert step by step in order to help you learn how to convert manually.

The Evolution of Number Systems

The decimal system is the first and oldest number system in existence, dating from the 1st to 4th century in India. This system is thought to be based on the ten digits of the Arabic numeral system, which corresponds to the ten fingers of a human hand. This system of numerals spread to the Arab world and to Europe, where it became the dominant system of calculation and commerce worldwide. It is also the system of numerals that introduced the concepts of zero and positional notation, which is a cornerstone of all computing.

It is funny that we can still study a binary code system today that has been around since the 3rd Century BC. Pingala, an Indian scholar, was the first to mention this. Then, some 1,500 years later, German mathematician and philosopher Gottfried Wilhelm Leibniz thoroughly described it in the 17th Century. He thought it was spiritual, with ‘1’ standing for god and ‘0’ the void. The 20th century was finally able to apply it with the first electronic computers.

To answer the core concept, let’s work with positional notation.

Realizing how number systems function delays the valuation systems notation. This involves calculating how much a digit represents based on its position in a number. Each digit corresponds to a value based on a power of ten. Therefore we consider the number 345:    (3 x 10^2) + (4 x 10^1)+(5 x 10^0) = 3100 + 410 + 51 = 345.    The same principle works in the binary system. However, each position represents a power of two. In the binary number 1101, the calculation is:    (1 2^3) + (1* 2^2) + (0* 2^1) + (1* 2^0) = 12^0 + 12^1 + 0 + 1*2^1 =1 + 0 = 4 + 8 = 13.    The value base 2 to base 10 converter performs this exact calculation for you.    For a manual conversion, take binary number 101101.    As customary, we note the binary number to be converted and the positional notation below it, starting with the rightmost digit in position 0.

Binary Number: 101101

As mentioned before, the first column starting from the right (Position 0) denotes 2^0, the next column (Position 1) denotes 2^1, and so on.

Thus, using Position powers of 2, we have:

Position 5 = 2^5 = 32

Position 4 = 2^4 = 16

Position 3 = 2^3 = 8

Position 2 = 2^2 = 4

Position 1 = 2^1 = 2

Position 0 = 2^0 = 1

Now, we have to multiply: (1 * 32) + (0 * 16) + (1 * 8) + (1 * 4) + (0 * 2) + (1 * 1)

To get total: 32 + 0 + 8 + 4 + 0 + 1 = 45

Therefore, the above-stated binary of 101101 is 45 when expressed in the decimal number system.

How to Convert Decimal to Binary?

When changing from decimal to binary, we use the repeated division by two technique. For instance, converting 45 to binary involves. First, we take 45 and divide it by 2. Then, we take the quotient and divide it by 2. This process continues until the quotient is equal to 0. This results in: 45 – 2 = 22 remainder 1; 22 – 2 = 11 remainder 0; 11 – 2 = 5 remainder 1; 5 – 2 = 2 remainder 1; 2 – 2 = 1 remainder 0; 1 – 2 = 0 remainder 1. Once the quotient is 0, we then read the remainders as binary numbers from the last quotient to the first, which gives 101101. Thus, we have completed the conversion.

Additional Number Systems, while decimal and binary numbers are predominantly used, engineers and computer scientists also rely on other systems, the octal number system and the hexadecimal number system. These systems provide a shorthand way to use binary numbers. The octal number system is a base-8 system. Like all other base-8 systems, it also uses 8 digits, which are made with numbers only from 0 to 7. It is also very easy to use, as a single octal digit can represent 3 binary digits. For example, the binary number 111 is equal to 7 in octal, and the octal form of 101101 can be separated into (101) (101), which is 55. Hexadecimal (Base-16)

The hexadecimal system is very common. It consists of 16 characters (0-9 and A-F; A is 10, B is 11, C is 12, D is 13, E is 14, and F is 15). Each digit of a hexadecimal number corresponds to four binary digits (16 is 2 to the 4th). As a base 16 number system, it is very efficient to represent large binary numbers. For example, the byte 11101011 (EB in hexadecimal) can compactly represent 8 bits.

Data Measurement and Its Different Types

First, let’s learn about binary.

Bit: The smallest digit of binary data (0 or 1) and also the smallest amount of data measurable.

Byte: A Group of 8 bits used to represent a single character of data (A, B, C, etc).

Kilobyte (KB): There are 1,024 bytes in a kilobyte.

Megabyte (MB): 1.0244 kilobytes rounded to about 1.024 kilobytes.

Gigabyte (GB): 1,024 megabytes.

Terabyte (TB): 1,024 gigabytes.

Frequently Asked Questions (FAQ’s)

Is this binary to decimal converter free?

Yes. We do not charge you anything, and this converter runs totally in your browser. Your data is never sent to any server.

How big a number can I convert?

You can work with rather large numbers. You can work with a number bigger than 9 quadrillion (2^53 – 1), and this is fine for lots of use cases.

What do base-2 and base-10 mean?

When we talk about ‘base,’ we mean how many unique characters there are in a number system. For base-2 (binary), it has two characters: 0 and 1. For base-10 (decimal), there are 10 characters: 0-9. The base also determines how the value of a place in a number system is calculated. Binary has powers of 2 for each place, and decimal has powers of 10.

How Are Letters and Symbols Converted to Binary?

When converting letters and symbols to binary, the complete set of characters is referenced, and each character’s position is mapped to a code, either using ASCII or UNICODE. Each character is assigned a unique code. The code is represented in decimal format before being converted to binary. The decimal value representation of the capital letter ASCII ‘A’ is 65; in binary format, 65 is 01000001.

Conclusion

When someone is digitally literate, they have the skills to differentiate between characters represented in decimal format and how the underlying data structure is composed of binary. The tool we created to convert value representations in binary and decimal is designed for enhanced clarity and accessibility. It is an educational tool as much as it is a utility. Real-time, two-way conversion is accompanied by a breakdown of each value. It seamlessly integrates with programming tools without the need for document conversion. It is also useful for students and advanced programmers, as well as developers. There are many ways to help support the structure of your number bases. This tool helps you achieve your desired standard. Given the time and structure of your number bases, it is easy to support other tools in your programming workflow. There are many ways to help support the structure of your number bases. This tool helps you achieve your desired standard.

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