Unit, Number and Numeral. The Number Systems and Their Rules

What is Unit?

Everywhere in our daily life we come across hearing or visualising the count of things or number of items something like 6 pens, 12 bananas, 2 bikes etc. Let’s take “6 pens” to understand what a unit is. “6” refers to count of pens which is a number, but the “pens” is a quantity which can be thought of type a “pen”, which is also known as unit.

Let’s understand with one more real life scenario. There are 5 pears in a basket. So, in there, we have a thing that to be counted is pear and the number of items of type pear are 5. One pear or a pear is a single thing, which is also called a unit.

By definition, a unit is something that denotes a single thing.

What is Number?

Again from the above example of pears, there were 5 pears in the basket which tells that a pear is taken five times. Or in other words, a unit (which is pear here) has been taken 5 times, so 5 is called a number. So, a number is how many times a unit is taken.

By definition, a number denotes how many times a unit is taken.

If you visit a shop and observe, the customers buy items in numbers and pay the price for the items to the shopkeeper again in numbers.
So, what do you observe in the shop while buying items and paying for the items? It is the numbers which play their role in counts of things e.g. 6 pens or 12 chocolates or 2 sandwiches etc. or while paying the price of items e.g. 4$ for pens, 10$ for chocolates or 5$ for sandwiches.

What is Numeral?

From the above examples of numbers, nine is the number used in “Joseph is nine years old” and seven in “There are seven days in a week”. So, is there any way to represent the numbers “nine” and “seven” in mathematics? The answer is yes, there is. Such symbols which we can use to represent the “nine” and “seven”, are called as numerals and in this case “nine” is represented with symbol 9 and seven with symbol 7.

By definition, a numeral is a symbol which is used to represent a number.

What are the Numeral Systems?

A numeral system is a system used to write numbers using a set of symbols. If we explore the history of mathematics, we could find mathematicians have developed many numeral systems to write numbers. The two systems are used majorly in the world. One is Roman Numerals and another is Hindu Arabic System. The most widely accepted system to write numbers is Hindu Arabic Number system.

The above “Table of numerals from zero to nine” uses symbols from the Hindu Arabic Numeral system.

1. Hindu Arabic Numeral System

Hindu Arabic Numeral System is the most adapted system in the world. This system was invented by Indian mathematicians between the Ist and 4th centuries. Arabs also started using this system in their Arabic mathematics but after 500 years.
Europeans started calling it as Arabic numerals also when Arabs introduced them to the Hindu numerals.

In this system, the numerals are represented by symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 as shown below in figure.

Hindu Arabic Numerals

2. Roman Numeral System

Roman numerals were invented by Romans. They used 7 letters of Latin alphabets to represent numbers. Each of the letters has a number value. So, different combinations of Roman letters can be used to represent any mathematical number.
Following is the list of Roman numerals, each of which has a value in numbers also.

Roman numerals with their number value

Convert Romans into Hindu Arabic

There are two rules which are kept in mind while converting a Roman numeral to Hindu Arabic number.

1. Additive notation
2. Subtractive notation

1. Additive Notation

In this notation, the Roman numerals next to each other are added up. But there arises two cases where these numerals are added up.

First case, when Roman numerals are repeated and are next to each other.
For example, II, here I is repeated two times. So, the number it makes is 1 + 1 = 2.

In additive notation, the numerals next to each other are added up when:
1. they are repetitive
2. the first numeral on the left is greater than the next second numeral on the right

Second case, when a greater numeral is followed up by a smaller numeral.
For example, VI, here V has value of 5 and I has a value of 1. So, V is greater than I and V is followed by I. Therefore, we can add them up as 5 + 1, which is 6. So, VI is number 6.

2. Subtractive Notation

With Subtractive notation, Roman numerals are subtracted. When a smaller numeral is followed up by a greater numeral, a smaller numeral is subtracted from a greater numeral. For example, IV, here V has value of 5 and I has a value of 1. So, I is followed by greater numeral V. Therefore, we subtract them as 5 – 1, which is 4. So, IV is a number 4.

As said above, symbols V, L and D are never repeated, that is why 10 is written as X, not VV. Similarly, 100 is written as C, not LL.

In subtractive notation, the numerals next to each other are subtracted, when:
the first numeral is smaller than the next numeral on the right of the smaller numeral

Important rules to form roman numbers

There are some rules that must be followed while forming any roman number. Let’s see below what is a right or wrong for a roman number.

1. Only I, X, C and M can be repeated in a number.
   XX = 10 + 10 = 20 ✅
   II = 1 + 1 = 2 ✅
   VIIII = 5 + 1 + 1 + 1 + 1 = 9 ❌
2. I and X can be repeated maximum three times only.
   XXI = 10 + 10 + 1 = 21 ✅
   VII = 5 + 1 + 1 = 7 ✅
   XIIII = 10 + 1 + 1 + 1 + 1 = 14 ❌
3. Symbols V, L and D are never repeated.
   VV = 5 + 5 = 10 ❌
   CC = 100 + 100 = 200 ✅
4. V, L and D are never subtracted.
   XC = 100 – 10 = 90 ✅
   VX = 10 – 5 = 5 ❌
5. The symbol X can be subtracted from L, M and C only.
   XL = 50 – 10 = 40 ✅
   XD = 500 – 10 = 490 ❌
6. The symbol I can be subtracted from V and X only.
   IX = 10 – 1 = 9 ✅
   IL = 50 – 1 = 49 ❌
7. The symbol C can be subtracted from D and M only once.
   CD = 500 – 100 = 400 ✅
   CM = 1000 – 100 = 900 ✅
   CCD = 500 – 100 – 100 = 300 ❌
8. If a bar is placed over a numeral, that shows that the numeral is multiplied by 1000.
   V = 5 × 1000 = 5000 ✅
   X = 10 × 1000 = 10000 ✅

List of Hindu arabic numerals

List of number values of Roman numerals