Binary Number System: Conversion, Formulas and Examples
Binary Number System uses only two digits, 0 and 1, to represent numbers. It is a base-2 positional system in which each position has a power-of-2 value. This page explains binary place values, decimal and binary conversions, binary arithmetic, fractions, formulas and quick methods with simple examples.
What Is the Binary Number System?
Starting from the right, the place values are 2⁰, 2¹, 2², 2³ and so on. For example, 1011₂ = 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰ = 8 + 0 + 2 + 1 = 11₁₀. The subscript indicates the base of the number.
Binary Number System Formula & Tricks
Important Formulas
Multiply every binary digit by the power of 2 assigned to its position and add the results.
Digits to the left of the binary point use non-negative powers of 2; digits to the right use negative powers.
Repeatedly divide the decimal integer by 2, record each remainder, and arrange the remainders in reverse order.
Use the positional expansion of the binary number and add all non-zero place values.
Quick Tricks
Write 1, 2, 4, 8, 16, 32 and so on from right to left. Select the powers whose sum equals the decimal number.
Divide by 2 until the quotient becomes zero. The binary answer is obtained by reading the remainders upward.
For a 4-bit group, use 8, 4, 2 and 1. For a 3-bit group, use 4, 2 and 1. Leading zeroes may be added to complete a group.
Binary Number System Concepts
Binary Place Value and Representation
In an integer binary number, positions increase from right to left as 2⁰, 2¹, 2², 2³ and so forth. A digit 1 includes that place value, while a digit 0 excludes it. Leading zeroes do not change the value.
Decimal to Binary Conversion
At each step, the remainder is either 0 or 1. Continue until the quotient is 0. For 19: 19÷2 gives remainder 1, 9÷2 gives 1, 4÷2 gives 0, 2÷2 gives 0 and 1÷2 gives 1. Reading upward gives 10011₂.
Binary to Decimal Conversion
The rightmost digit has weight 1, the next has weight 2, then 4, 8, 16 and so on. Only positions containing 1 contribute to the final value.
Binary Addition and Subtraction
When 1+1 occurs, write 0 and carry 1 to the next position. In subtraction, 0−1 requires borrowing from the next position; the borrowed 1 represents 2 in the current binary position.
Binary Fractions
The first digit after the point has value 1/2, the second has value 1/4, and the third has value 1/8. Multiply each fractional digit by its place value and add the results.
Binary Number System Video Lessons
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Prime Numbers: Test and Find Them Fast
Learn how to test whether a number is prime and identify prime numbers efficiently using clear divisibility checks and practical methods.
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Binary Number System Quick Revision
Remember these place-value rules and conversion methods for binary calculations.
- Binary is a base-2 system using only 0 and 1.
- From right to left, integer place values are 2⁰, 2¹, 2², 2³ and so on.
- Binary-to-decimal conversion uses the sum of digit × corresponding power of 2.
- Decimal-to-binary conversion uses repeated division by 2 and reverse-order remainders.
- In binary addition, 1+1 produces 0 with a carry of 1.
- Digits after the binary point use 2⁻¹, 2⁻², 2⁻³ and so on.
- Leading zeroes may be added or removed without changing an integer's value.
Binary Number System FAQs
What is the decimal value of 1111₂?
1111₂ = 1×8 + 1×4 + 1×2 + 1×1 = 15₁₀.
How do you convert 25₁₀ to binary?
25 = 16 + 8 + 1, so the digits for 16, 8, 4, 2 and 1 are 1, 1, 0, 0 and 1. Therefore, 25₁₀ = 11001₂.
What is 100000₂ in decimal?
The leftmost 1 has the value 2⁵, so 100000₂ = 32₁₀.
What is the result of 101₂ + 11₂?
Align the digits: 101₂ + 011₂ = 1000₂. In decimal terms, 5 + 3 = 8.
How is a binary fraction converted to decimal?
Multiply digits after the point by 2⁻¹, 2⁻² and so on. For example, 0.11₂ = 1/2 + 1/4 = 0.75₁₀.
What is the largest decimal number represented by n binary digits?
For n digits, the largest value is 2ⁿ − 1 because all n digits are 1. For 8 digits, the maximum is 2⁸ − 1 = 255.