Numbers

Numbers can be represented in two ways: numerically and in words. For example, 45 in words is "forty-five", while in numbers it is "45". A number system organizes numbers logically using symbol or digits, and a numeral system defines a collection of numbers within an arithmetic or algebraic structure. By combining the digits 0 to 9, we can create an infinite number of numbers.

In math, there are different types of numbers like:

• Composite numbers
   
• Prime numbers
   
• Rational numbers
   
• Irrational numbers
   
• Odd numbers
   
• Even numbers
   
• Integers
   
• Complex numbers, and more.

Numbers have not always looked the way they do today. Long ago, early humans used simple marks and symbols to keep count of things around them. As time passed, people created new ways to count and calculate more easily.

• Early humans used symbols and tally marks (////) to count things, but it was hard for large numbers.

• As civilization grew, people created different symbols to represent bigger quantities. Romans used I, V, X, L, C, D, M but had no zero.

• Around the 7th century, India perfected the decimal system (base ten). 

• This system used only ten digits (0-9) to show any number. Aryabhata explained the concept of zero and place value. For example, even Pythagoras once said, "Everything in this world is made of numbers, even music".

• Arab merchants and scholars spread it to Europe, and today we call it the Hindu-Arabic numeral system, which is still used everywhere.

• Scientists believe counting systems first emerged in ancient societies like Egypt, Mesopotamia, and India.

• Different numeral systems were developed across the world, and as people traveled, these systems spread across continents. The Indian numeral system spread to the Middle East and Europe, becoming the base of the modern decimal system.

• The Brahmi script (around 3rd century BCE, during Emperor Ashoka's time) is the earliest known numeral system of India. Brahmi numerals had symbols for 1-9 and special symbols for 10, 100, and 1000. They are considered the ancestors of modern numerals (0-9) used worldwide.

• Later, numbers were written in Devanagari script, which is used for Sanskrit.

Before kids can learn numbers, they need to learn how to do math without numbers. These early skills, such matching, sorting, categorizing, ordering, and comparing, are the building blocks for acquiring a good understanding of numbers.

Similarly, they need to grasp these pre-number ideas before they can understand math in a meaningful way. Kids usually learn these skills in preschool, when they look for patterns, similarities, and contrasts in the things around them.

Numbers follow specific special rules are called properties that help us solve problems easily and understand math better. These properties demonstrate how numbers behave when we add, multiply, or combine them in various ways.

1. Commutative Property:  If you change the order of two numbers and the answer stays the same, that operation is commutative.

Example:
Addition: 2 + 3 = 3 + 2 = 5
Multiplication: 4 × 5 = 5 × 4 = 20

2. Associative Property: If you group numbers differently but still get the same result, the operation is associative.

Example
Associative property of addition and multiplication: 
Addition: (1 + 2) + 3 = 1 + (2 + 3) = 6
Multiplication: (2 × 3) × 4 = 2 × (3 × 4) = 24

3. Distributive Property: When one number is multiplied by a group of numbers inside brackets, you can "distribute" it to each number inside.

Example
Distributive property of addition: 
2 × (3 + 4) = (2 × 3) + (2 × 4) = 14.

4. Identity Property: An identity is a special number that doesn't change the value when used in an operation.

Example 
Addition: 5 + 0 = 5 and Multiplication: 8 × 1 = 8

5. Inverse Property: An inverse is a number that cancels out another number to bring you back to the identity.

Additive inverse: Adding a number to its additive inverse results in 0.
Example: 6 + (-6) = 0.

Multiplicative inverse: Multiplying a number by its reciprocal gives one.
Example: 5 × (1/5) = 1

6. Closure Property: When you operate (like addition or multiplication) on two numbers from a set, the answer always stays in the same set. That operation is closure.
For instance, we get a whole number if we add two whole numbers together. However, subtracting two natural numbers may not give a natural number. 

Example: 
Integers: 5 + 2 = 7 (an integer)
Whole numbers: 4 × 3 = 12 (a whole number)

Numbers can be classified in different ways based on how we use them and their unique features. Let's explore the different kinds of numbers.

1. Natural Numbers - These are the basic counting numbers that start from 1 and go on without end. They are represented by the letter 'N'. For example: N = {1, 2, 3, 4, 5,...}

2. Whole Numbers - Whole numbers include all natural numbers along with 0. They are represented by the letter 'W'. For example, W = {0, 1, 2, 3, 4, 5,...}


3. Integers - Any positive, negative, or zero whole numbers are called integers. For example: -2, -1, 0, 1, 2,...

4. Rational Numbers - If a number is written as p/q, it is called a rational number, where q is not zero and both p and q are integers. For example: 1/2, -3, 57/100.

5. Irrational Numbers - These numbers can never be expressed as fractions. For example: π, √2.

6. Real Numbers - Rational and irrational numbers together make real numbers. For example, 2, -5, √3, 0.5.

7. Complex Numbers - Complex numbers consist of a real part and an imaginary part and are written in the form a + bi, where i = √(-1).

Numbers are significant for students because they play a crucial role in our daily lives. A good understanding of numbers makes learning math easier and more useful in daily life.

• A number is one of the building blocks of mathematics. Understanding numbers and how they relate to each other helps children learn more advanced math concepts later.
  
   
• Learning numbers helps children think better. It teaches them to follow sequences and solve problems, which improves their thinking capacity and helps them make connections.
  
   
• Learning numbers is also part of language development. When children learn numbers, they practice saying them. This helps improve their vocabulary and makes them better communicators.
  
   
• Knowing numbers well helps children do better in school. Children who grasp numbers early can learn not only math, but also other subjects more easily.
  
   
• Numbers play a crucial role in learning about money. Being good with numbers helps people understand and manage their finances, such as budgeting, saving, and making informed investments.
  
   
• Counting helps children understand that numbers represent real objects when they link numbers to the things around them.
  
   
• Children need to understand that numbers represent quantities and can be used to compare different amounts.

Begin your journey into Numbers by exploring key concepts. Understand important number topics in detail by selecting from the list below:

Numbers can be grouped based on their types and usage. Let us see what are all the types of numbers.

1. Cardinal and Ordinal Numbers - The numbers that indicate quantity are called cardinal numbers. For example, 1, 2, 3, 4... The numbers that show position or order are called ordinal numbers. For example, 1st, 2nd, 3rd…

2. Even and Odd Numbers - Even numbers are divisible by 2. For example, 2, 4, 6. While odd numbers are not divisible by 2. For example, 1, 3, 5.  


3. Consecutive Numbers - Numbers that follow one another consecutively. For example, 1, 2, 3 or 5, 6, 7.


4. Prime and Composite Numbers - Prime numbers have only two divisors. For example, 2, 3, 5. Composite numbers have more than two divisors. For example, 4, 6, 8.


5. Co-Prime Numbers - Co-prime numbers are pairs of numbers with no common factors except 1. For example, 8 and 15 are co-prime because the common factor between them is 1. However, 15 and 9 are not co-prime numbers, as they share 3 as a common factor.


6. Perfect Numbers - Numbers equal to the sum of their proper divisors. For example, 6 has the divisors 1, 2, and 3, and by adding them we get 6.
   

7. Fractions and Decimals - Fraction show parts of a whole. For example, 12. While decimals express numbers in base 10. For example, 0.5.


8. Factors and Multiples - Factors divide a number exactly. For example, factors of 12 are 1, 2, 3, 4, 6, and 12. Multiples are products of a number. For example, multiples of 3 are 3, 6, 9, 12 etc. 


9. GCF and LCM - GCF is the greatest common divisor. For example, the GCF of 12 and 18 is 6, because it divides both evenly. While LCM is the smallest common multiple. For example, LCM of 5 and 10 is 10. So, 10 can be divided by both 5 and 10 evenly. 


10. Prime Factorization - Expressing a number as a product of its prime factors. For example, prime factorization of 18 is 2 × 3 × 3.


11. Algebraic and Transcendental Number - Algebraic numbers satisfy polynomial equations, while transcendental numbers do not. For example, π.

The term PEMDAS stands for P – Parentheses, E – Exponents (powers and roots), M – Multiplication, D – Division, A – Addition, S – Subtraction. PEMDAS represents the order of performing mathematical operations.

This can be understood better by using an example.
 
Solve :

\(8 + \left( \frac{6}{3} \right) \times 2^{2} \) 

Step 1: Parentheses → \(6 \div 3 = 2 \)

Step 2: Exponents → \(2^{2} = 4 \)

Step 3: Multiplication →  \(2 \times 4 = 8 \)

Step 4: Addition → \(8 + 8 = 16 \)

In this example, the correct order of operations that are performed is shown.

Understanding numbers can be made simpler using a few tricks. Like a game, the more we practice, the easier it gets. Here are a few tips and tricks that can make understanding numbers easier.

• Think about how you use numbers every day. For example, if your alarm is set for 6:48 am, you can break it down to see how many minutes you have left to sleep.
   
• Use the distributive property: Big numbers may sometimes seem intimidating, so break them down into smaller numbers. Why not try dividing them into parts?

For example: 48 + 36

Group the numbers according to their place values

40 + 8 = 48;

36 = 30 + 6

Now add the numbers in the tens place and ones place,

40 + 30 = 70;

8 + 6 = 14;

70 + 14 = 84

• Rounding numbers for estimations makes calculations easier. For example, to add 487 + 293, round off to 490 + 290 = 780.
   
• Use the rule of divisibility: When the last digit is even, it is divisible by 2; when the last digit is 0 or 5, it is divisible by 5. The sum of digits is divisible by 3; then the number is a multiple of 3.

Numbers are an important part of our daily lives. They are used for simple and complex tasks. Here are a few real-world applications of numbers:
 

From Numbers to Geometry and beyond, you can explore all the important Math topics by selecting from the list below: