Numbers

Introduction to Numbers

Numbers form the basis of mathematics. We should befriend numbers in order to understand maths. Numbers are of various kinds. We have a long list that includes ordinal numbers, consecutive numbers, odd numbers, even numbers, natural numbers, whole numbers, integers, real numbers, rational numbers, irrational numbers, and complex numbers.

Building pre-number math skills is a prerequisite to understanding numbers. Pre-number skills like matching, sorting, classifying, ordering, and comparing will set the stage to build a strong number sense. Pre-number Math skills are built-in preschool years. Kids learn how to stand before they start taking small steps. In the same way, the pre number concept is very important for them to start understanding Mathematics. In this section, we will cover the different pre-number concepts like Matching and Sorting, Comparing and Ordering, Classification, and Shapes and patterns.

Number names are used to represent numbers in an alphabetical format. A specific word is used to refer to each number. To write a number in words in English, you should know the place value of each digit in the number.

The rules of PEMDAS outline the order of the operations and give structure to nested operations. In mathematics, PEMDAS is an acronym that stands for P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction. 

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

What about Decimals? A decimal number has a whole number part and a fractional part. These parts are separated by a decimal point.

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.

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'.

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' .

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 as 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|.

Factors and Multiples

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.

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

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

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.

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.

What is the Smallest Whole Number?

0 is the smallest whole number.

What are the First 10 Numbers?

If someone asks you this question, it means that they want to know the first ten natural numbers. These are 1,2,3,4,5,6,7,8,9 and 10.

What is Not a Real Number?

If you look at the diagram above, all basic numbers, including irrational numbers fall under the subset of real numbers. There is NO number that isn't a real number unless it's an imaginary number.

How Numbers are Formed?

A number is an arranged group of digits. Numbers can be formed with or without the repetition of digits. For example, the largest number which can be formed using 8 and 9 is 98.

Why Numbers are Important in our Life?

Numbers are a part of our everyday life. These are used in an unlimited range of ways, from mathematical calculations, mobile numbers, and phone calls, identification of bank accounts, exchange of money to cooking, etc.