# Number Systems Online

## Place-value systems

For a place-value system, the value of a digit results from the position of the digit within the number. The value of the number is determined by the sum of the digits, each digit being multiplied by the digits place value. The place value is the position-dependent power of the base value b of the place value system. The numbers are written as a sequence of digits from left to right, the left is started with the highest place value, and the place value is then reduced to the next position by one (b-adic representation) in the exponent. The transition to negative exponents of the base value is indicated by a ".". The number of different digit symbols required is equal to the base value b.

Structure of a number in the place-value system with the digits anan-1... a0.a-1... a-m. The value is calculated according to

$anbn+an-1bn-1+…+a1b1+a0b0+a-1b-1+…+a-mb-m = ∑ i = 0 m+n ai-mbi-m$

### Decimal system

Base value b = 10

Digits 0, 1, 2, ..., 9

Example of a decimal number:

$1265.42= ∑ i = 0 6 ai-210i-2 = 2⋅10-2+ 4⋅10-1+ 5⋅100+ 6⋅101+ 2⋅102+ 1⋅103 = 2100+ 410+ 5+ 6⋅10+ 2⋅100+ 1⋅1000$

### Binary system (dual system)

Base value b = 2

Digits 0, 1

Example of a binary number:

$1011012= ∑ i = 0 6 ai2i = 1⋅20+ 0⋅21+ 1⋅22+ 1⋅23+ 0⋅24+ 1⋅25 = 1+ 4+ 8+ 32 = 4510$

Base value b = 16

Digits 0, 1, 2, ...,9, A, B, C, D, E, F

$10FE1A16= ∑ i = 0 6 ai16i = A⋅160+ 1⋅161+ E⋅162+ F⋅163+ 0⋅164+ 1⋅165 = 10+ 16+ 3584+ 61440+ 1048576 = 111362610$

### More place-value systems

Octal system
Base 8
Duo-decimal system
Base 12
Sexagesimal system
Base 60
Base64
Base 64
Base 32
Vigesimal system
Base 20

### Conversion between different place-value systems

In the number system conversion calculator are digits greater then 9 to be set by letters A, B, C, ...

Number
with Base
Conversion to Base

#### Conversion of the decimal number to the new base

The starting point of the iteration is the decimal number. The number is divided by the base and the integer part of the division is used for the next iteration. The remainder of the division is the digit for the new base. The iteration is repeated until the remainder is 0.

The calculation of the decimal places is made by multiplication with the base. The number before the decimal point is the next digit. The decimal part is multiplied by the base until the remainder is 0 or the maximum number of desired digits is reached. Here after 10 positions is the calculation aborted.

## More Calculators

Here is a list of of further useful calculators:

Index Matrix Determinant Determinant 2x2 Determinant 3x3 Combinatorics