# 8 Number systems

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Definition

A *number* is a quantity.

A* numeral* is a symbol of representation.

A *system of numeration*/*number system* is a set of numerals and a rule for combining them to make numbers.

There have been multiple number systems used throughout history and in various cultures.

Example \(\PageIndex{1}\):

**Ciphered Number System**

Names are given to 1, the powers of the base, as well as the multiples of the powers.

The first known Ciphered Number System was the Egyptian Hieratic Numerals. The Egyptians had no concept of a place value system, such as the decimal system. Therefore it required exact finite notation. It was based on multiples of 10, often being rounded off to the higher power.

Other ciphered number systems include Greek, Coptic, Hindu Brahmin, Hebrew, Syrian, and early Arabic.

Example \(\PageIndex{2}\):

**Roman Number System**

Numbers in the Roman number system are represented by combinations of letters from the Latin Alphabet.

Symbol |
I | V | X | L | C | D | M |

Base 10 |
1 | 5 | 10 | 50 | 100 | 500 | 1,000 |

So for example, 6 is VI, one more than 5 and 4 is IV, one less than 5. Similarly, 11 is XI, and 9 is IX. Thirty-three would be written as XXXIII.

What is this number represented in Roman Numerals MMMMDCCXXVII?

Example \(\PageIndex{3}\):

**Addition in Roman Numerals**:

Let's say we want to add 27 and 39 in Roman Numerals: XXVII + XXXVIV

Simply line up XXVII and XXXVIV, group, add and convert, yielding LXVI = 66.

**Multiplication in Roman Numerals:**

Suppose we wish to multiply 38 and 3. First, convert thirty-eight to roman numerals - XXXIII. Then, they would have simply written it out 3 times and added as follows:

XXXVIII

XXXVIII

XXXVIII

=LXXXXXXIV

=LLXIV

=CXIV

**Subtraction in Roman Numerals:**

Suppose we wish to subtract 29 from 63. First, expand or do in parts

=LXIII - XXVIV

=XXXXXXIII - XXVIV

=XXXIV.

Example \(\PageIndex{4}\):

**Hindu-Arabic Decimal System**

This system came to western civilization from India in 1000 AD. In 1200 AD it was translated from Latin, 300 years later (15^{th} and 16^{th} century) we began using Arabic numerals commonly. This is the system that we use today, it uses only 10 digits, with 0 as a placeholder. It is a position based system.

10 Symbols - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

We need not create any other symbols as we have the positioning system which determines the quantity of the symbol.

For Example: 509 - can be expanded to 500 + 0 + 9, or 5X10^{2} + 0X10^{1} + 9X10^{0}.

Base 10 columns explained using 24,962

10^{4} |
10^{3} |
10^{2} |
10^{1} |
10^{0} |

2 |
4 |
9 |
6 |
2 |

2(10,000) | 4(1000) | 9(100) | 6(10) | 2(1) |

20000 | 4000 | 900 | 60 | 2 |

The Babylonians began the idea for this positional system, however, they used base 60. They, however, had 2 oversights that were corrected with our current decimal system:

- They did not create a 0 or such a symbol to represent a placeholder in the position that had no value such as in our 10's position in the above example, and
- They did not create a decimal point, which made reading numbers very difficult. The decimal point in our current system is extremely important when reading number systems.

For example, we know that $21.45, tells us the product cost 21 dollars and 45 cents, so there is a 2 in the 10's position, a 1 in the 1's position, and 4 in the tenths position, and 5 in the hundredths position. What if we were to simply write 2145, how much would the product cost?

Note

**FOUR components that make up a good number system:**

- positional notation

- multiplicative base

- abstract symbolization

- 0: very important to distinguish number position. For example 12 5 or 1205.

**Interesting number system fact: **

The Magical World of Harry Potter’s currency system uses a non-decimal system:

1 Galleon = 17 sickles = 493 Knuts