Numbers: Definition, Classification, Properties, Examples

Numbers play a vital role in mathematics and in our life too. We use numbers to count things, money, etc., and we use numbers in date and time. Numbers can be real, rational, whole numbers, natural, composite, co-prime, even, odd, cardinal, ordinal, fractions, decimals, integers etc.,

There are 8 properties of numbers. The properties of the numbers are namely, commutative, associative, distributive, closure, identity, additive inverse, multiplicative inverse, and zero product properties. In this article, we will know more about numbers, types of numbers and more. Continue reading to learn more!

What are Numbers?

Definition: Numbers are the values that we use for representing the quantity and in making the calculations. We have the digits \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\) to make or form all the other numbers.

The Number Zero: Zero is a unique number. It is neither positive nor negative. The number \(0\) is both a number and the numerical digit used to represent the number in numerals. The number zero is denoted with the \(‘0’\) symbol.

Classification of Numbers 

The numbers can be classified in sets called the Number system. We have different types of numbers that are given below:

1. Natural Numbers \((N)\): Natural numbers are the real numbers that don’t have any decimal and are bigger than zero. They are the counting number from the number \(1\).
Example: \(1, 2, 3, 4, 5, 6, ….\) Natural numbers are the subsets of the real numbers.

2. Whole Numbers \((W)\): Whole numbers are positive real numbers, including the number zero. Natural numbers are also considered whole numbers. They are the subset of real numbers.
Example: \(0, 1,2, 3, 4, 5, 6,….\) are whole numbers.

3. Integers \((Z)\): The integers have no decimals, and they include both the positive and the negative numbers. Even whole numbers are also integers. 
Example: \(-3, -2, -1, 0, 1, 2, 3, 4,….\) They are known as integers. They are the subset of real numbers.

4. Rational Numbers \((Q)\): Rational numbers are the real numbers that can be written in the form of fractions  \(\frac{p}{q}\). Where \(p\) and \(q\) are co-prime integers and \(q≠0\). They are also a subset of real numbers.
Example: \(-\frac{2}{3}, 0,5, \frac{3}{4}\), etc. are rational numbers.

5. Irrational Numbers \((P)\): An Irrational number is expressed as the number that cannot be written in fraction fractions \(\frac{p}{q}\). Where \(p\) and \(q\) are co=prime integers and \(q≠0\). They are the subsets of the real numbers.
Example: \(\sqrt 2 ,\sqrt 5 ,\pi ,e\) etc. are irrational numbers.

6. Real Numbers \((R)\): The integration of rational numbers and irrational numbers is considered as real numbers. Real numbers can be either positive or negative, which can be denoted as \((R)\). Natural numbers, fractions, decimals all come under this category.

7. Complex Numbers or Imaginary Numbers: These numbers are the set \({a+bi}\), where, \(a\) and \(b\) are the real numbers, and \(‘i’\) is the imaginary unit, given by \(i = \sqrt { – 1} \).

8. Even Numbers: Any number (natural number, whole number, integer) divisible by \(2\) without leaving any remainder is called an even number. 
Example: Of even numbers are \(2, 72, 422, 38, …..\) etc. These are the numbers ending with the digits \(0, 2, 4, 6\) and \(8\).

9. Odd Numbers: Any number that can be (natural number, whole number, integer) that cannot be divisible by the number \(2\) is considered as the odd number. When you divide any odd number by \(2\), you get the remainder as the number \(1\) always. 
Example: Some odd numbers are \(11, 173, 107, 979, …..\) etc. These are the numbers ending with the digits \(1, 3, 5, 7\) and \(9\).

10. Prime Numbers: A number that has no factor other than the number \(1\) and the number itself is known as the prime number
Example: Numbers \(2, 3, 5, 7, 11, …..\) etc. are prime numbers. \(1\) is not a prime number.

11. Composite Numbers: A number is called a composite number; it has a least one factor other than the number \(1\) and the number itself.
Examples: The numbers \(4, 6, 8, 9, 10, 12, 14, 15, …..\) etc., are composite numbers.

In other words, a number other than \(1\) is a composite number if it not prime.
Note: It should be noted that the number \(1\) is neither prime nor composite. It is the only number with this property. Every number other than the number \(1\) is either the prime number or a composite number.

Below you can see the classification of the Numbers:

Numbers in Words 

Numbers are the beginning of the genesis of maths, and without them, there is no math. Number name is referred to the pattern of writing a number in words.

Example: The number \(1\) is written in words as one. Similarly, you can see a few more numbers and their number name below:

\(0\) is written as Zero, \(1\) is written as One, \(2\) is written as Two, \(3\) is written as Three, \(4\)  is written as Four, \(5\) is written as Five, \(6\) is written as Six, and so on.

In the same way, you can form the more significant numbers and then write their number names by understanding their place values.

Example: The number \(4783\) can be written in words as Four Thousand Seven Hundred Eighty-Three.

We have an Indian place value chart and also an International system of numeration chart as shown below:

Indian Place Value Chart

The Indian place value chart is as follows:

International System of Numeration Chart

The international system of numeration chart is as follows:

Special Numbers

The impressive numbers are mentioned below:

Cardinal Numbers: These are the numbers that tell us the total quantity or the total number of items in a group like one, three, four, six, ten etc.

Example: If there are \(40\) students in a class, we say that the cardinal number of the class is \(40\).

Ordinal Numbers: These are the numbers that tell us the position of any object, thing, person etc., 

Example: First: \({{\rm{1}}^{{\rm{st}}}}\) Second: \({{\rm{2}}^{{\rm{nd}}}}\), Third: \({{\rm{3}}^{{\rm{rd}}}}\), ….. etc. person in a row consisting of many people

Nominal Numbers: These numbers are used as only names. It is a number that has no information other than identification.

Example: The player with the number \(10\) written on his T-shirt, the area with pin code \(711410\).

Pi \(\left( \pi \right)\): This is an extraordinary number and is approximately equal to \(3.14159\). This is determined by dividing the ratio of the circumference of a circle by its diameter.

Euler’s number \((e)\): Euler’s numbers is one of the essential numbers in maths. It is approximately equal to the value \(e=2.718281\). It is an irrational number that is the base of the natural logarithm. 

Golden Ratio \(\left( \varphi \right)\): This golden ratio is an exceptional number in maths. It is equal to \(\varphi = 1.618034\)

Properties of Numbers

The properties related to the real numbers are given below:

1. Closure Property

When you add any two real numbers, the sum which you get is also a real number. If \(a\) and \(b\) are real numbers then, \(a+b\) is also a real number.

Example: \(9+7=16\), in which \(16\) is the sum and a real number similar to the numbers \(9\) and \(7\).

2. Commutative Property

The word commutative comes from “commute” or “move around”. Commutative properties mean that the numbers we operate can be changed or swapped from their position as the answer remains the same. This property is applicable only in the case of addition and multiplication but not for subtraction or division.

Addition: \(m+n=n+m\), example: \(6+3=3+6\) or \(2+8=8+2\)

Multiplication: \(m×n=n×m\), example: \(2×4=4×2\) or \(5×10=10×5\)

3. Associative Property

The associative property is the property in which you can add or multiply numbers regardless of how they are grouped. 

Addition: \(m + (n + r) = (m + n) + r,\) example: \(3 + (4 + 5) = (3 + 4) + 5\)

Multiplication: \((m \times n) \times r = m \times (n \times r)\) example: \((6 \times 2) \times 3 = 6 \times (2 \times 3)\)

4. Distributive Property

This property tells us how to solve the expression in the form of \(m(n + r) = mn + mr\) and \((m + n)r = mr + nr\)

An example of distributive property is \(4(3 + 2) = 4 \times 3 + 4 \times 2\)

5. Identity Property

Here are additive and multiplicative identities:

Addition: When any number is added with zero, the answer is the same number. \(m+0=m\). \((0\) is the additive identity).

Multiplication: When one is multiplied by any number, the answer is the same number. \(m×1=1×m=m. (1\) is the multiplicative identity).

Additive Inverse: Additive inverse of a number is a number, which, when added to the original number, results in zero \((0)\).

Examples: We see that \(7+(-7)=0\). So, we say \(-7\) is the additive inverse of \(7\).

Also, \(-15+(15)=0\). So, \(15\) is the additive inverse of \(-15\).

Multiplicative Inverse: 

When you multiply any number with the number one, you get the same number as the product.

Example: The number \(\frac{1}{x}\) is the multiplicative inverse of the number \(x\). In the same way, the number \(\frac{1}{{15}}\)  is the multiplicative inverse of the number \(15\). 

Zero product Property: When you multiply any number with zero, you get the answer as zero.

Example: \(x \times 0 = 0\), where \(x\) is any number. In the same way \(8 \times 0 = 0,67 \times 0 = 0\) or \(0 \times 23 = 0,0 \times 90 = 0\)

Solved Examples on Numbers

Q.1. Find the value of: \([32 + 2 \times 17 + ( – 6)] \div 15\)
Ans: We have, 
\([32 + 2 \times 17 + ( – 6)] \div 15\)
\( = [32 + 34 + ( – 6)] \div 15 = (66 – 6) \div 15 = 60 \div 15 = \frac{{60}}{{15}} = 4\)
Hence, the required answer is \(4\).

Q.2. Write the expanded form of the numbers \(375\) and \(921\).
Ans: Given the numbers \(375\) and \(921\)
You need to write the given numbers in the expanded form.
Now, \(375=300+70+5\)
In the same way, the number \(921\) in which the digit \(9\) is in the hundreds place, \(2\) is in tens place, \(1\) is in the ones place.
So, \(921=900+20+1\)
Hence, the required answer is \(375\) is \(300+70+5\) and \(921\) is \(900+20+1\).

Q.3. Find the sum of the greatest three-digit number and the smallest three-digit number.
Ans: We know that the greatest three-digit number is \(999\).
The smallest three-digit number is \(100\).
So, we need to add these two numbers.
Now,\(999+100\)
\(=1099\) 
Hence, the required answer is \(1099\).

Q.4. Write the given numbers in words: \(830, 154, 942, 789\).
Ans: Given to write the given numbers in words \(830, 154, 942, 789\)
Here, the number name of \(830\) is Eight Hundred Thirty.
The number name of \(154\) is One Hundred Fifty-Four.
The number name of \(942\) is Nine Hundred Forty-Two.
The number name of \(789\) is Seven Hundred Eighty-Nine.
Hence, the required answer is given above.

Q.5. Find \(x\) such that \(\frac{{ – 3}}{8}\) and \(\frac{x}{{ – 24}}\) are equivalent rational numbers.
Ans: It is given that \(\frac{{ – 3}}{8} = \frac{x}{{ – 24}}\)
Therefore, \(\frac{{ – 3}}{8} = \frac{x}{{ – 24}} \to 8 \times x = ( – 3) \times ( – 24)\)
\( \to 8 \times x = 72\)
\( \to x = \frac{{72}}{8} = 9\)
Hence, \(x=9\)

Summary

In the given article, the topics covered are about the various types of numbers, including the number zero. Numbers are defined as the mathematical symbols that we use in our day-to-day life. Numbers can be divided into various types namely, Natural, real, whole, integers, rational, irrational, complex, even, odd, prime, and composite numbers. We have the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, to form all or other numbers. Furthermore, the special numbers are classified into cardinal, ordinal, nominal, Euler’s numbers, and the golden ratio.

Learn All About Real Numbers

FAQs on Numbers

Q.1. What are numbers? 
Ans: Numbers are the mathematical symbols that we use in our daily life for counting the quantity or in making calculations. 

Q.2. What are the properties of numbers?
Ans: The properties of the number are given below:
1. Commutative property
2. Associative property
3. Distributive property
4. Closure property
5. Identity property
6. Additive Inverse property
7. Multiplicative Inverse property
8. Zero Product property

Q.3. Write the difference between rational numbers and irrational numbers. 
Ans: The difference between rational numbers and the irrational number has been shown below:

Q.4. What are the associative properties of numbers? 
Ans: The associative property of the number both in addition and multiplication is given below:
Associative Property: The associative property is a property in which you can add or multiply numbers regardless of how they are grouped. 
Addition: \(m + (n + r) = (m + n) + r\), example: \(5 + (7 + 9) = (5 + 7) + 9\)
Multiplication: \((m \times n) \times r = m \times (n \times r)\), example: \((4 \times 7) \times 2 = 4 \times (7 \times 2)\)

Q.5. What are real and complex numbers?
Ans: Both the numbers are explained below:
Real Numbers: The combination of rational numbers and irrational numbers is known as real numbers. Real numbers can either be positive or negative or both, which can be denoted as \(‘R’\). Natural numbers, fractions, decimals all come under this category.
Complex Numbers: These numbers are the set \({ a + bi} ,\), where, \(a\) and \(b\) are the real numbers, and \(‘i’\) is the imaginary unit, given by \(i = \sqrt { – 1} \).

We hope this detailed article on numbers helped you in your studies. If you have any doubts or queries regarding this topic, feel to ask us in the comment section and we will be more than happy to assist you.