Positional Notation Calculator

Positional notation is a system for representing numbers using digits and their positions. This calculator helps you convert numbers between different bases (binary, decimal, hexadecimal, octal) and understand how positional notation works.

What is Positional Notation?

Positional notation is a method of representing numbers where the value of each digit depends on its position in the number. The most common example is the decimal system (base 10), where each digit represents a power of 10.

General formula for positional notation:

Number = d_n × bⁿ + d_n-1 × b^n-1 + ... + d₁ × b¹ + d₀ × b⁰

Where: d_i = digit at position i, b = base

Other common bases include:

Binary (base 2) - uses digits 0 and 1
Octal (base 8) - uses digits 0-7
Hexadecimal (base 16) - uses digits 0-9 and letters A-F

Understanding positional notation is fundamental in computer science, mathematics, and digital systems where different bases are used to represent information.

How to Use This Calculator

Enter the number you want to convert in the "Number to convert" field.
Select the base of the input number from the "From base" dropdown.
Select the target base from the "To base" dropdown.
Click the "Convert" button to see the result.
Use the "Reset" button to clear all fields.

Note: The calculator supports bases from 2 to 36. For bases higher than 10, letters A-Z represent values 10-35.

Conversion Formulas

The calculator uses these fundamental conversion methods:

Decimal to Any Base

Divide the decimal number by the target base.
Record the remainder (this is the least significant digit).
Repeat the process with the quotient until the quotient is 0.
The digits are read in reverse order.

Any Base to Decimal

Multiply each digit by the base raised to the power of its position (starting from 0 on the right).
Sum all the results to get the decimal equivalent.

Between Non-Decimal Bases

First convert the number to decimal.
Then convert the decimal number to the target base.

Example conversion from binary (base 2) to decimal:

1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13₁₀

Examples

Here are some example conversions using our calculator:

Number	From Base	To Base	Result
1010	2	10	10
1A	16	10	26
1000	10	2	1111101000
FF	16	8	377

These examples demonstrate how numbers can be represented in different bases and how the calculator handles the conversions.

FAQ

What is the difference between binary, octal, decimal, and hexadecimal?

These are different numeral systems with different bases:

Binary (base 2) uses digits 0-1
Octal (base 8) uses digits 0-7
Decimal (base 10) uses digits 0-9
Hexadecimal (base 16) uses digits 0-9 and A-F

Can I convert numbers between any two bases?

Yes, the calculator supports conversions between any two bases from 2 to 36. For bases higher than 10, letters A-Z represent values 10-35.

What happens if I enter an invalid number for the selected base?

The calculator will display an error message. For example, entering "2" in a binary field is invalid because binary only uses 0 and 1.

Is there a limit to how large a number I can convert?

The calculator can handle numbers up to a certain size, but very large numbers may cause performance issues or display inaccuracies due to JavaScript's number precision limits.