Bfh-sts: Created page with "= Numeral Systems: Overview & Positional Notation = This page introduces numeral systems, explains why different bases exist, and outlines the principle of positional notation. == Decimal system == Humans use the decimal system with base 10. * Digits are 0–9. * A number is a sequence of digits, where each digit has a positional value. * Example: 123 = 1 × 100 + 2 × 10 + 3 × 1. Other bases are also used in daily life: * Base 12 for hours on a clock. * Base 60 for m..."

2025-10-20T13:35:14Z

Created page with "= Numeral Systems: Overview & Positional Notation = This page introduces numeral systems, explains why different bases exist, and outlines the principle of positional notation. == Decimal system == Humans use the decimal system with base 10. * Digits are 0–9. * A number is a sequence of digits, where each digit has a positional value. * Example: 123 = 1 × 100 + 2 × 10 + 3 × 1. Other bases are also used in daily life: * Base 12 for hours on a clock. * Base 60 for m..."

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= Numeral Systems: Overview & Positional Notation =
This page introduces numeral systems, explains why different bases exist, and outlines the principle of positional notation.

== Decimal system ==
Humans use the decimal system with base 10.
* Digits are 0–9.
* A number is a sequence of digits, where each digit has a positional value.
* Example: 123 = 1 × 100 + 2 × 10 + 3 × 1.

Other bases are also used in daily life:
* Base 12 for hours on a clock.
* Base 60 for minutes and seconds.
* Example: 50 minutes and 33 seconds = 50 × 60 + 33 = 3033 seconds.

== Binary system and bits ==
Computers work electronically, using two states: voltage present or absent.
This gives the binary system with digits 0 and 1.

The smallest unit of information is the bit.
Each additional bit doubles the number of possible states:
* 1 bit = 2 states
* 4 bits = 16 states (nibble)
* 8 bits = 256 states (byte)
* 16 bits = 65 536 states (word)
* 32 bits = over 4 billion states (double word)
* 64 bits = over 18 quintillion states (quad word)

== The issue with binary notation ==
Even moderate numbers require many digits in binary.
A 64-bit value is too long to be readable, even when grouped into bytes.
For this reason, shorter notations are used by grouping bits:
* 3 bits → octal (base 8)
* 4 bits → hexadecimal (base 16)

== Essence of a number base ==
Roman numerals were not column-based, but modern systems use positional notation.
In positional systems, the value of a digit depends on its position:
* In any base, the number written as 10 represents the base itself.
* Column 0 (rightmost) is multiplied by base^0 = 1.
* Column 1 is multiplied by base^1.
* Column 2 is multiplied by base^2, and so on.

Example in decimal:
* 532 = 5 × 10^2 + 3 × 10^1 + 2 × 10^0.

This principle applies to all bases: binary, octal, hexadecimal, and beyond.

[[Category:Numeral Systems]]

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