Decimal To Binary Conversion

1 dec = 1 bin

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How to convert decimal to binary (dec to bin)

Converting from decimal to binary involves transforming a decimal (base-10) number into its binary (base-2) representation, which is composed of only two digits: 0 and 1. The process of converting from decimal to binary is relatively straightforward. You start by dividing the decimal number by 2 repeatedly, keeping track of the remainders at each step. You continue this division until the quotient becomes zero. To extract the binary representation, you read the remainders in reverse order, from the last remainder obtained to the first. These remainders will form the binary equivalent of the decimal number. The process of converting from decimal to binary can be expressed by the following process:Step 1: decimal ÷ 2 = x₀ remainder r₀Step 2: x₀ ÷ 2 = x₁ remainder r₁Step 3: x₁ ÷ 2 = x₂ remainder r₂...Step n: x_n ÷ 2 = 0 remainder r_n+1The converted binary number is each of the remainder values i.e. r_n+1...r₂r₁r₀. This may seem confusing but will become clear with the example below.

Example Conversion

For example, to convert 13 decimal to binary, you would do the following:

13 ÷ 2 = 6 remainder 1

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1

The remainders in reverse order are: 1, 1, 0, and 1. Therefore, the decimal 13 is 1101 in binary.

Converting binary to decimal

Here's a step-by-step guide on how to convert binary to decimal.1. Binary is a base-2 numbering system that uses only two digits, 0 and 1, to represent values. While decimal is a base-10 numbering system that employs ten digits, 0 through 9. Converting from binary to decimal involves understanding the positional value of each binary digit and summing them up according to their respective positions. The formula for converting from binary to decimal is: dec = b_n-1 × 2^n-1 + b_n-2 × 2^n-2 + ... + b₀ × 2⁰. See below for an example on how this works.2. Positional Value: In a binary number, each digit represents a power of 2, starting from the rightmost digit. The rightmost digit has a positional value of 2^0 (which is 1), the next digit to the left has a positional value of 2^1 (which is 2), the next has a positional value of 2^2 (which is 4), and so on.3. Binary Digits: Take the binary number you want to convert and assign positional values to each digit, starting from the rightmost digit.4. Calculate Decimal Equivalent: To convert the binary number to decimal, multiply each digit by its positional value and sum up the results. For "10110," the calculation would be: (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (0 × 2⁰) = 16 + 0 + 4 + 2 + 0 = 22.

Conversion Unit Definitions

What is a Decimal?

The decimal number system, often called base-10, is the numerical system most commonly used by humans for everyday counting and arithmetic. It utilizes ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The concept behind the decimal system is that each digit's value is determined by its position within the number, with each position representing a power of 10.

In the decimal system, the rightmost digit holds the smallest place value and represents 10^0, which is 1. The next digit to the left represents 10^1, which is 10, and the pattern continues as you move to the left. For example, in the number 456, the digit '6' is in the ones place (10^0), '5' is in the tens place (10^1), and '4' is in the hundreds place (10^2). To convert a decimal number to its numerical value, you sum up the product of each digit with its corresponding power of 10. In the case of 456, it's 410^2 + 510^1 + 6*10^0, which equals 400 + 50 + 6, resulting in the decimal value 456.

The decimal system is widely used in daily life because it's well-suited for human cognition, particularly for counting and representing quantities. It's also the standard system for representing fractional values, where digits to the right of the decimal point represent negative powers of 10 (e.g., 0.1, 0.01, 0.001). While the decimal system is prevalent in everyday arithmetic and commerce, the binary system (base-2) is favored in computing because electronic devices, such as computers and calculators, operate naturally with binary switches (0s and 1s). However, conversions between decimal and binary systems are essential for translating human-readable information into machine-readable formats, making these two number systems fundamental in the realm of mathematics and technology.

What is a Binary?

The binary number system, often referred to as base-2, is a fundamental numerical representation used in digital computing and information technology. Unlike the decimal system, which is based on powers of 10, the binary system relies on powers of 2, making it particularly well-suited for electronic devices and digital data storage. In the binary system, there are only two possible digits, 0 and 1, which are analogous to the on/off states of electronic switches. This simplicity is essential in the context of computing because it aligns perfectly with the binary nature of electronic circuits.

In binary, each digit represents a power of 2, starting from the right and increasing by one for each position to the left. For example, the rightmost digit represents 2^0 (which is 1), the next digit to the left represents 2^1 (which is 2), the next one represents 2^2 (which is 4), and so on. To convert a binary number to its decimal equivalent, you sum up the values of all the positions where a '1' appears. For instance, the binary number 1101 is equal to 12^3 + 12^2 + 02^1 + 12^0, which simplifies to 8 + 4 + 0 + 1, resulting in the decimal value 13.

The binary system is foundational in the field of computer science and digital electronics because it forms the basis for representing data, executing calculations, and transmitting information within electronic devices. It enables computers to process and store information in the form of binary code, where each piece of data, from text and numbers to images and videos, is ultimately represented as sequences of 0s and 1s. This universal language of computing facilitates the development of complex software, the operation of hardware components, and the exchange of data across the digital landscape, making it a cornerstone of modern technology.