Let’s learn in detail the different concepts of PEMDAS such as Addition, Subtraction, Multiplication, and Division.

The decimal number system is the most commonly used number system. The digits 0 to 9 used to represent numbers. A digit in any given number has a place value. The decimal number system is the standard system for denoting integers and non-integers. We will be using the decimal number system for representation of Numbers up to 2-Digits, Numbers up to 3-Digits, Numbers up to 4-Digits, Numbers up to 5-Digits, Numbers up to 6-Digits, Numbers up to 7-Digits, Numbers up to 8-Digits, Numbers up to 9-Digits and Numbers up to 10-Digits.

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A cardinal number is a number that denotes the count of any object. Any natural number such as 1, 2, 3, etc., is referred to as a cardinal number, whereas, an ordinal number is a number that denotes the position or place of an object. For example, 1^st, 2^nd, 3^rd, 4^th, 5^th, etc. It indicates the order of things or objects, such as first, second, third, fourth, and so on.

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In the example given above, ordinal numbers will help define the position of the children. Such as, Jim is the fourth child from the left.

Consecutive numbers are numbers that follow each other in order from the smallest number to the largest number. They usually have a difference of 1 between every two numbers.

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Integers are numbers that are whole numbers and negative numbers. All integers are represented by the alphabet Z. A number line is full of integers. On the left side, you can find negative integers while on the right side you have the positive ones. Don’t forget the zero in between! Integers do sound interesting right? We will now learn about the Addition and Subtraction of Integers, Multiplication, and Division of Integers, Euclid's Division Lemma, and Euclid's Division Algorithm.

Z = { ...., -4, -3, -2, -1, 0 , 1, 2, 3, 4,....}

A natural number is a non-negative integer and is always greater than zero. It is represented by the symbol N. We will learn about the various properties of natural numbers as part of understanding this concept. Now, the whole number does not contain any decimal or fractional part. It means that it represents the whole thing without pieces. It is represented by the symbol W.

We will learn about the various properties of whole numbers as part of understanding this concept.

Even numbers are those numbers that can be divided into two equal groups or pairs and are exactly divisible by 2. For example, 2, 4, 6, 8, 10, and so on. In other words, these are whole numbers that are exactly divisible by 2.

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Odd numbers are whole numbers that cannot be exactly divided by 2. These numbers cannot be arranged in pairs. Interestingly, all the whole numbers except the multiples of 2 are odd numbers.

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A prime number is a number that has exactly two factors, 1 and the number itself whereas a composite number is a number that has more than two factors, which means it can be divided by number 1 and itself, and at least one more integer. Any whole number greater than 1 that has exactly two factors, 1 and itself is defined to be a prime number.

Examples: 2, 5, 7, 11 , etc

Well, now we know that a prime number has just two factors. One and the number itself. But how do we know that a number is prime? You will learn How to find out if a Number is a Prime Number in this section. along with the concepts of Prime Numbers and Euclid’s Proof 3.

What about Composite Numbers? Any number greater than 1 that is not a Prime number, is defined to be a composite number. Thus composite numbers will always have more than 2 factors.

Examples: 6, 8, 9, 12, etc

• Factors of 6 = 1, 2, 3, 6 (factors other than 1 and 6)
• Factors of 8 = 1, 2, 4, 8 (factors other than 1 and 8)
• Factors of 9 = 1, 3, 9 (factors other than 1 and 9)
• Factors of 12 = 1, 2, 3, 4, 6, 12 (factors other than 1 and 12)

Composite numbers are positive integers and you already know that they have more than one factor. Let us now learn about the Fundamental Theorem of Arithmetic.

If a pair of numbers has no common factor apart from 1, then they are called co-prime numbers. In other words, a set of numbers or integers which have only 1 as their common factor, which means their highest common factor (HCF) will be 1, are co-primes. These are also known as mutually prime numbers or relatively prime numbers. Also, there should be two numbers in order so to form co-primes.

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Perfect numbers are the positive integers that are equal to the sum of its factors except for the number itself. In other words, perfect numbers are the positive integers which are the sum of their proper divisors. The smallest perfect number is 6, which is the sum of its proper divisors: 1, 2 and 3

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Fractions are a part of a whole. They are represented by numbers that have two parts to them. There is a number at the top, which is called the numerator, and the number at the bottom is called the denominator.

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Now that we already know about fractions and how it is represented, let us explore some more fraction related topics like Equivalent Fractions, Improper Fractions and Mixed Fractions, Addition and Subtraction of Fractions, Multiplication of Fractions and Division of Fractions.

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Decimals are really interesting. They have a whole number part and they can also be represented as fractions. Let’s dive deeper and find out how! In this section, we will cover decimals related concepts such as the Relationship between Fractions and Decimals, Addition and Subtraction of Decimals, Multiplication of Decimals, and Division of Decimals.

A rational number, denoted by Q, is represented in the form p/q, where q is not equal to zero. Integers, Fractions, Decimals, Whole numbers, and Natural numbers are all Rational numbers.

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In order to get a better understanding of Rational numbers, we will cover topics like Decimal Representation of Rational Numbers, and Operations on Rational Numbers.

Now, Irrational Numbers are the numbers that cannot be represented using integers in the p/q form. The set of irrational numbers is denoted by Q'.

Example: √5, √2, etc

Irrational numbers cannot be represented as a simple fraction. Their decimal expansion neither terminates nor becomes periodic. You must be wondering how! We will find out once we study some more topics related to irrational numbers such as Square Root of Two is Irrational, Decimal Representation of Irrational Numbers, The exactness of Decimal Representation, Rationalize the Denominator, Surds, and Conjugates and Rationalization.

Any number that can be found in the real world is a real number. Any number that we can think of, except complex numbers, is a real number. The set of real numbers is the union of the set of Rationals (Q) and Irrationals (Q'). It is denoted by R. The set of real numbers, R = Q ∪ Q'.

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A complex number is a number that can be expressed in the form (a + bi) where a and b are real numbers, and i is a solution of the equation x² = −1. Because no real number satisfies this equation, i is called an imaginary number. Complex numbers have a real part and an imaginary part. Wait, do you think Complex numbers are really complex? Well, let us study them in detail to find out. In this section, we cover different topics like Complex Numbers – Points in the Plane, A Complex Number is a Point in the Plane, What is i? Magnitude and Argument, Powers of iota, Addition, and Subtraction of Complex Numbers, Multiplication of Complex Numbers, Conjugate of a Complex Number, Division of Complex Numbers, Addition, Subtraction, and Interpretation of |z1-z2|.

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The factors and multiples are the two key concepts studied together. Factors are the numbers that divide the given number completely without leaving any remainder, whereas the multiples are the numbers that are multiplied by the other number to get specific numbers.

Factors of a given number are numbers that can perfectly divide that given number.

Examples:

• Factors of 6: 1, 2, 3, 6
• Factors of 8: 1, 2, 4, 8
• Factors of 14: 1, 2, 7, 14
• Factors of 36: 1, 2, 3, 4, 6, 9, 18, 36

A multiple of a number is a number obtained by multiplying the given number by another whole number.

Examples:

• Multiples of 3: 3, 6, 9, 12, 15, ......
• Multiples of 5: 5, 10, 15, 20, 25, .....
• Multiples of 10: 10, 20, 30, 40, 50,...
• Multiples of 12: 12, 24, 36, 48, 60, ....

The highest common factor (HCF) of the two numbers is the largest whole number which is a factor of both. It is also called the Greatest Common Factor(GCF).

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When we consider two numbers, each will have its own set of multiples. Some multiples will be common to both numbers. The smallest of these common multiples is called the least common multiple (LCM) of the two numbers.

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Prime factorization allows us to write any number as a product of prime factors. It is a way of expressing a number as a product of its prime factors. To do prime factorization, you need to break a number down to its prime factors. In this section, we will learn about concepts such as Divisibility, GCD, and LCM. We will also have a look at the various applications of prime factorization.

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Important Notes

• The numbers which start from 1 and go up to infinity are called natural numbers.
• The numbers which start from zero are called whole numbers.
• Integers consist of positive and negative numbers along with zero.
• Rational numbers are of the form p/q and consist of integers, fractions, and ratios.
• Irrational numbers are not expressed in the form of fractions or ratios.
• Real numbers consist of natural numbers, whole numbers, rational numbers, and irrational numbers.