# Numerals

A numeral is a symbol used to represent a number, while a number is a mathematical object used to count, measure, and label.

## What is a numeral

Numerals are a way of representing numbers. The most commonly used today is the Arabic numeral system. Below are a few examples of numerals representing the number 8:

Examples

Hindu-Arabic numerals | : | 8 |

Roman numerals | : | VIII |

Tally marks | : | llll lll |

Word form | : | Eight |

These are just a few ways that numerals are used to represent the concept of the quantity 8. Throughout the course of history, there have been many others, and even today, different regions of the world represent numbers differently to some degree.

## Roman numerals

Roman numerals are the numeral system that originated in ancient Rome. They are one of the most commonly known numeral systems after the decimal numeral system, which is an extension of the Hindu-Arabic numeral system. Roman numerals are still used today on things like clocks and watches, book chapters, numbering events, and more.

### How to read Roman numerals

Roman numerals work by assigning values to specific letters as follows:

- I - 1
- V - 5
- X - 10
- L - 50
- C - 100
- D - 500
- M - 1000

To determine the value of Roman numerals, use the rules below:

- When consecutive numerals are equal, or the left-most numeral is greater than that of the following numerals (e.g. XX, VIII), their values are added.
- When a numeral of lesser value is followed by a numeral of greater value (e.g. IV), the lesser value is subtracted from the greater one.
- A bar written over a numeral (called a vinculum) indicates multiplication by 1000. For example, M = 1000 × 1000 = 1,000,000.
- No more than 3 of the same numeral can be written consecutively. For example, IIII does not exist. Instead, the subtractive principle (described below) is used.
- Numerals V, L, and D are never repeated; they also are never written before a larger symbol (so are never subtracted).
- The numeral I is only ever written before V or X, meaning that these are the only numerals it can be subtracted from. Similarly, X is only ever written before L, C, or M.

To expand on rule 4, presumably this rule exists because the more consecutive letters there are, the more difficult it is to determine the value of the whole. For example, it would be tedious to have to count the number of "I" in IIIIIIII to determine the value is 8. Using VIII = 5 + 3 is much clearer and easier to read. To account for this, Roman numerals use the subtractive principle, where writing a letter with a smaller value before one with a larger value indicates that the smaller value is subtracted from the larger one. Using 4 as an example, from the list above we know that the value of V is 5, and the value of I is 1, so writing IV means 5 - 1 = 4. Similarly,

- IX = 9
- XIV = 14
- XXIV = 24
- XL = 40
- XC = 90
- CD = 400
- CM = 900

The above example show a number of cases in which the subtractive principle is used. These are not all of the cases but provides an idea of how the principle works with Roman numerals. Let's look at some more complicated Roman numeral examples:

Examples

Convert between Roman numerals and numerals in the decimal system.

1: 5437

5437 = VCDXXXVII

2: CMXXIV

CMXXIV = 900 (CM) + 20 (XX) + 4 (IV) = 924

3: 98

98 = XCVIII

4: IVCDXLIV

IVCDXLIV = 4000 (IV) + 400 (CD) + 40 (XL) + 4 (IV) = 4,444

5: 1,536,424

1,536,424 = MDXXXVICDXXIV

To break this down, MDXXXVI = 1536 × 1000 = 1,536,000. CDXXIV = 424. 1,536,000 + 424 = 1,536,424.

## Numeral vs number

The terms number and numeral are often used interchangeably, but they are not actually the same thing.

A number is a mathematical object used for counting and measurement. Numerals are the symbols used to represent numbers. For example, if John has 5 cookies, he can indicate this by writing the number "5," the word "five," the roman numeral "V," or he can hold up 5 of his fingers. Regardless which numeral we use to express the number of cookies that John has, John still has the same quantity of cookies.

### Numeral system

A numeral system is the collection of symbols and rules used to represent all numbers. The Roman numeral system and decimal system are just two examples of numeral systems. There are many more used today such as the hexadecimal system and binary system. There are even more that have been used in the past such as Egyptian numerals, which were written in hieroglyphs.

### Digit

A numerical digit is the symbol used to represent a number in a positional numeral system. It is another term that can cause confusion along with numerals and numbers, but it is important to understand the distinction in order to have a better understanding of the number system that we use. In the decimal system, the digits used are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. All numbers can be represented using just these 10 digits.

The decimal system (described below) is a positional numeral system; each digit has a given value based on its position in the numeral. For example, each digit in 5,241 has a different value than they would individually. In 5,241, the 5 represents 5,000, the 2 represents 200, the 4 represents 40, and the 1 represents 1. This differs from other number systems such as Roman numerals. Each letter used in Roman numerals has a different value, and representing values follows a certain set of rules (described above). This works well for smaller values, but gets more and more complicated for larger values. Positional numeral systems significantly simplify our representation of numbers, allowing us to perform calculations and communicate numbers efficiently and effectively.

## Decimal system

A numeral system is a set of rules and symbols used to represent numbers. The most commonly used numeral system is the Hindu-Arabic numeral system, a positional numeral system that uses the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The decimal numeral system is a positional numeral system that is an extension of the Hindu-Arabic numeral system. Unlike the Hindu-Arabic system, the decimal system includes non-integer numbers. In the decimal numeral system, each digit in a numeral has a place value.

For example, the numeral "2" represents the number two. On the other hand, the numeral "222" represents the number two hundred twenty-two. Each of the digits is the same, but in the numeral 222, each of the 2s has a different value. The left-most 2 is in the hundreds place, the middle 2 is in the tens place, and the right-most 2 is in the ones place. Their value is determined by what position they are in, which is referred to as their place value. The numeral 222 can be expanded as follows:

222 = 2 × 10^{2} + 2 × 10^{1} + 2 × 10^{0}

222 = 2 × 100 + 2 × 10 + 2× 1

222 = 200 + 20 + 2

In the decimal numeral system, negative numbers are indicated using a negative sign (-), and a decimal point (.) is used to separate the integer portions of a numeral from the decimal portions. For example, negative two hundred twenty-two and two tenths is written as: -222.2.