Numbers
Numbers are an important part of our daily lives, from how long we sleep at night to how many times we run around a track, and much more. In math, there are even and odd numbers, prime and composite numbers, decimals, fractions, rational and irrational numbers, natural numbers, integers, real numbers, rational numbers, irrational numbers, and whole numbers. In this chapter, we’ll learn about the different kinds of numbers and the ideas that go along with them.
What are Numbers
A number is one of the most basic parts of math. Numbers are used to keeping things in order, count, measure, index, etc. We have natural numbers, whole numbers, rational and irrational numbers, integers, real numbers, complex numbers, even and odd numbers, and so on. We can use the most basic arithmetic operations to figure out the number that comes out. At first, count marks were used instead of numbers. Now let’s talk about what numbers are and what their different kinds and properties are.
Introduction to Numbers
Numbers are the building blocks of math. We should get to know numbers if we want to learn math. There are different kinds of numbers. We have a long list of ordinal numbers, consecutive numbers, odd numbers, even numbers, natural numbers, whole numbers, integers, real numbers, rational numbers, irrational numbers, and complex numbers.
We meet the interesting world of factors and multiples when we learn about numbers. There are prime numbers, composite numbers, co-prime numbers, and even perfect numbers (yes, numbers can be perfect!) in this world. HCF, LCM, and factorization by primes.
Let’s start our trip through the world of numbers. You can learn more about all of the important topics in Numbers by choosing from the list below:
| Classification | Number Names | PEMDAS |
| Number Systems | Ordinal Numbers | Consecutive Numbers |
| Integers | Natural Numbers | Whole Numbers |
| Even Numbers | Odd Numbers | Prime Numbers |
| Composite Numbers | Co-prime Numbers | Perfect Numbers |
| Fractions | Decimals | Rational Numbers |
| Irrational Numbers | Real Numbers | Complex Numbers |
| Factors | Multiples | Prime Factorization |
| Least Common Multiple (LCM) | Highest Common Factor (HCF) | Number Line |
Number System
Number systems are mathematical systems that let you write numbers in different ways and that computers can understand. A number is a mathematical value that can be used to count things, measure them, and do math calculations. There are many different kinds of numbers, such as natural numbers, whole numbers, rational numbers, and irrational numbers. In the same way, there are different kinds of number systems, such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system, and each has their own properties.
In this article, we’ll talk about the binary number system, the octal number system, the decimal number system, and the hexadecimal number system, which are all ways to write numbers. We will learn how to convert between these two number systems and work through some examples to help us understand the idea better.
Most numbers are written using the decimal system. Numbers are written using the digits 0 through 9. Each number’s digits have a place value. The decimal number system is the most common way to talk about integers and numbers that are not integers. We use the decimal number system to show 2 Digits Number, 3 Digits Number, 4 Digits Number, 5 Digits Number, 6 Digits Number, 7 Digits Number, 8 Digits Number, 9 Digits Number, and Numbers up to 10 Digits.
- Binary number system (Base – 2)
- Octal number system (Base – 8)
- Decimal number system (Base – 10)
- Hexadecimal number system (Base – 16)
Math Before Numbers
Building math skills before numbers is a must if you want to understand numbers. Matching, sorting, classifying, putting in order, and comparing are all pre-number skills that set the stage for building a strong number sense. Math skills before numbers are learned in preschool. Before they take their first steps, kids learn how to stand. In the same way, they need to know about numbers before they can start to understand math. In this section, we’ll talk about the different pre-number skills, such as matching and sorting, comparing and ordering, classifying, and making patterns with shapes.
Check out the picture below, which shows two columns. In the left column, the numbers 1 through 4 can be seen. Rows of items are shown in the right column. The numbers are put with the amounts they stand for. This is a very important skill for kids ages 3 to 4.
Number Name
People use number names to show numbers in alphabetical order. Each number is called by a different word. To write a number in English, we need to know what each digit in the number stands for.
In the number names form, 36 is written as “thirty-six.” Look at the picture below to see how this is done.
BODMAS and PEMDAS
The rules of PEMDAS show how the operations should be done and give structure to operations that are inside other operations. In math, PEMDAS is an acronym that stands for P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction.
One of the most basic things to do in math is to evaluate a number expression. The “BODMAS rule” tells you which operation to simplify first when an expression has addition (+), subtraction (-), multiplication (x), division (), a bracket (), and the word “of.” First, we need to figure out what the BODMAS rule is. What does the BODMAS formula mean?
In math, the BODMAS rule is a rule or order that is used to make expressions with more operators easier to understand. When there are more than two operators in an equation, we have to figure out in what order to solve it. The BODMAS rule helps us figure out how to solve this problem.
BODMAS is an acronym for:
- Brackets (Parts of a calculation inside the bracket always come first).
- The order of the brackets is: [{(bar)}]
- Orders (powers and square roots)
- Division
- Multiplication
- Addition
- Subtraction
Number Operation
- Addition
- Multiplication
- Subtraction
- Division
Type of Numbers
The different kinds of numbers are based on the things they can do. For example, natural numbers are those that start with 1, while whole numbers are those that start with 0. Prime numbers are those that can only be divided by 1 and the number itself. Let’s learn more about the different kinds of numbers.
Even Numbers
Even numbers can be divided into two same groups or pairs and are exactly divisible by two. Example: 2, 4, 6, 8, 10, etc. In other words, these are whole numbers exactly divided by 2.
Look at the following diagram, which shows that even numbers are completely divisible by 2.
Odd Numbers
Odd numbers are whole numbers that can’t be completely divided by 2. There is no way to pair up these numbers. All whole numbers except those that are multiples of 2 are odd numbers, which is a bit strange.
If you look at the figure below, you can see that odd numbers are not completely divisible by 2, and when they are divided by 2, they leave a remainder of 1.
Composite Number
A composite number is one that can be divided by 1, itself, and at least one other integer. You can also say that a composite number is any number greater than 1 that is not a prime number. There are always more than 2 factors in a composite number. For example, 6, 8, 9, 12, and so on are all composite numbers because they have more than 2 parts.
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)
Ordinal Numbers
Ordinal numbers, like first, second, third, and so on, show the position or order of something in relation to other numbers. This order or sequence could be based on size, importance, or any kind of time order. Let’s look at an example of an ordinal number to see how it works. Ten students took part in a competition. The top three winners got medals and were given the rankings of 1st, 2nd, and 3rd. First, second, and third are all ordinal numbers in this case.
Cardinal Numbers
A cardinal number is one that tells how many of something there are. A cardinal number is a number that comes from nature, like 1, 2, 3, etc. An ordinal number is a number that shows the position or place of something. For instance, first, second, third, fourth, fifth, etc. Now, let’s say, “There are three ants and five bears.” This is what a cardinal number looks like. But if we say, “The runners in the running event are in order of first, second, third, and so on,” we are using ordinal numbers. Look at the table below to see how cardinal numbers are different from ordinal numbers.
Consecutive Number
Before we can understand consecutive numbers, we need to know what comes before and what comes after. The number that comes right before another number is called that number’s predecessor. The number that comes right after another number is called that number’s successor. Think about the list of natural numbers: 1, 2, 3, 4, and 5. The number 1 comes before 2, and the number 3 comes after 2. Numbers that come right after each other, from the smallest to the biggest, are said to be consecutive. Every pair of numbers has a difference of 1 most of the time. Note that the difference between any two things is always the same. Let’s look at some examples of numbers that come one after the other.
Natural Number
The set of natural numbers is made up of all whole numbers except 0. We use and talk about these numbers a lot in our everyday lives and conversations. Numbers are used to count things, represent or exchange money, measure temperature, tell time, etc., and they are all around us. These numbers are called “natural numbers” because they are used to count things. For example, when counting things, we might say “5 cups, 6 books, 1 bottle, etc.”
Whole Number
Whole numbers include both natural numbers and zero (0). We know that natural numbers are a set of numbers that start with 1 and go up to 9. Whole numbers are just a group of numbers that don’t have any fractions, decimals, or even negative numbers. It is a group of positive numbers and the number zero. Or, we can say that the set of non-negative integers is the set of whole numbers. The main difference between natural numbers and whole numbers is that whole numbers have a zero.
Prime Number
Prime numbers are those that can only be broken down into two parts: 1 and the number itself. Take the number 5, for example, which has only two parts: 1 and 5. It’s a prime number because of this. Let’s look at the number 8, which has more than two factors 1, 2, 4, and 8. Because of this, 8 is not a prime number. Taking number 1 as an example, we know that it only has one factor. So, it can’t be a prime number, since a prime number should only have two factors. This means that 1 is neither a prime number nor a composite number. It is a unique number.
Co-Prime Number
Co-prime numbers are two numbers that have nothing in common except 1. To make a set of co-prime numbers, there must be at least two numbers. Co-prime numbers, like 4 and 7, or 5, 7, and 13, have only 1 as their highest common factor. It’s important to remember that co-prime numbers don’t always have to be prime numbers. Co-primes are also made up of two numbers that are not prime, like 4 and 13.
Perfect Number
A perfect number is a positive integer that is equal to the sum of all of its factors, except for itself. In other words, perfect numbers are the sums of the numbers that divide them. The smallest perfect number is 6, which is the sum of its factors, which are 1, 2, and 3. Note that this total does not include the number itself, since the number is also a factor.
Fraction Number
In math, a fraction is written as a number, which shows how much of a whole it is. A fraction is a part or section of a whole number, value, or thing. Let’s use an example to understand this idea. The picture below shows a pizza that has been cut into 8 equal pieces. Now, if we want to talk about one part of the pizza in particular, we can write it as 1/8, which means that out of 8 equal parts, we are talking about 1 part.
It means one of eight parts that are equal. It can also be written as “one-eighth” or “one by eight.”
The number at the top is called the numerator, and the number at the bottom is called the denominator.
Now that we know what fractions are and how they are written, we can learn more about them on pages like Equivalent Fractions, Improper Fractions and Mixed Fractions, Adding and Subtracting Fractions, Multiplying and Dividing Fractions.
Decimal Number
On a number line, decimals are the numbers between integers. They are just another way for mathematicians to show fractions. With decimals, we can write more precise values for things like length, weight, distance, money, etc. that can be measured. The numbers to the left of the decimal point are whole numbers, and the numbers to the right of the decimal point are decimal fractions. If we go right from one place, the next place will be (1/10) times smaller, which is (1/10)th or tenth place value. For example, look at the decimal place value chart below for the number 12.45
Decimals are a lot of fun to think about. They have a part that is a whole number, but they can also be shown as fractions. You can learn more about decimals on pages like Adding and Subtracting Decimals, Multiplying and Dividing Decimals.
Rational number
Rational numbers look like p/q, where p and q can be any whole numbers and q is ≠ 0. This means that natural numbers, whole numbers, integers, fractions of integers, and decimals are all types of rational numbers (terminating decimals and recurring decimals). In this lesson, we will learn more about rational numbers, how to recognize them, and what they look like.
On pages like Decimal Representation of Rational Numbers and Operations on Rational Numbers, you can read about other things related to rational numbers that will help you understand rational numbers better.
Irrational Number
Irrational numbers are real numbers that can’t be written as a fraction, p/q, where both p and q are whole numbers. The denominator q is not equal to zero (q ≠ 0). Also, the decimal expansion of an irrational number doesn’t end or repeat.
Irrational Numbers: Irrational numbers are real numbers that can’t be shown as a simple fraction. These can’t be written as a ratio, like p/q, where p and q are both whole numbers and q is greater than 0. It is opposite of rational numbers.
Real Number
A real number is any number that exists in the real world. Everywhere we look, we see numbers. Natural numbers are used to count things, rational numbers are used to show fractions, irrational numbers are used to find the square root of a number, integers are used to measure temperature, and so on. Real numbers are made up of all of these different kinds of numbers. We will learn all about real numbers and the important things about them in this lesson.
Complex Number
A complex number is a number that can be written as (a + bi), where a and b are real numbers and I is a solution to the equation x2 = 1. Since this equation can’t be solved with a real number, I is called an imaginary number. There is a real part and an imaginary part to a complex number. Wait, do you really think that Complex numbers are hard to understand? So, let’s take a close look at them to find out. In this part, we talk about different things, such as Complex Numbers – Points in the Plane, A point on the plane is what a complex number is. 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 the Meaning of |z1-z2|.
Integers
All whole numbers and negative numbers are also integers. This means that a set of integers can be made up of both whole numbers and negative numbers.
All integers are written with the letter Z, and they have neither a decimal nor a fractional part. There are integers all along a number line. On the left, there are negative integers, and on the right, there are positive ones. Don’t forget the zero in between!
Z = { …., -4, -3, -2, -1, 0 , 1, 2, 3, 4,….}
Factors and Multiples
HCF
The HCF (Highest Common Factor) of two or more numbers is the number that is the highest among all of the numbers that they have in common. The HCF (Highest Common Factor) of two natural numbers x and y is the biggest number that can be used to divide both x and y. Let’s use the numbers 18 and 27 to figure out what this means. 18 and 27 have the same factors: 1, 3, and 9. The number 9 is the largest of these numbers. The HCF of 18 and 27 is therefore 9. It looks like this: HCF (18, 27) = 9. Look at the figure below to understand this idea.
LCM
In math, the LCM or lowest common multiple is another name for the least common multiple. The least common multiple of two or more numbers is the smallest multiple of those numbers that is also a multiple. Take the numbers 2 and 5, for example. Each one will have its own set of multiples.
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
Multiples of 5 are 5, 10, 15, 20, …
Prime Factorization
Prime factorization is the process of writing a number as the sum of its prime factors. The only things that make up a prime number are 1 and the number itself. Prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, and so on. Any number can be written as a product of prime numbers, which is called “prime factorization.” For example, 40 can be broken down into its prime factors in the following way:
Factors
In math, a factor is a number that can be used to divide another number without leaving any extra numbers. We use factors and multiples all the time. For example, they are used to put things in a box in the right order, to handle money, to find patterns in numbers, to solve ratio problems, and to work with increasing or decreasing fractions.
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
Multiples
When you multiply one whole number by another whole number, you get a multiple. When you multiply a number, you get the number of times that number. Do you still remember how to multiply? We will use them to find groups of things. Let’s see how listing the first five (non-zero) multiples of the number 6 helps us understand what a multiple is. The first five multiples of 6 that don’t start with 0 are 6, 12, 18, 24, and 30. In the table of 6, we can see that the 6 times are listed.
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, ….
Properties of Numbers
- Commutative Property
- Associative Property
- Distributive Property
- Identity Property
- Reflexive Property
- Symmetric Property
- Transitive Property
- Inverse Property
- Commutative Property: This property of numbers is applicable to addition and multiplication. It is expressed as, a + b = b + a and a × b = b × a.
- Associative Property: This property of numbers is applicable to addition and multiplication. It is expressed as, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
- Distributive Property: The product of the sum of two numbers and a third number is equal to the sum of the product of each addend and the third number. It is expressed as a × (b + c) = a × b + a × c.
- Identity Property: We have an additive identity equal to 0 and a multiplicative identity equal to 1. It is written as, a + 0 = a and a × 1 = a.
- Reflexive Property: This property implies that every number is equal to itself. It is written as, for all a, a = a.
- Symmetric Property: If a number x is equal to y, then y is equal to x. It can be written as, x = y ⇒ y = x.
- Transitive Property: If x is equal to y and y is equal to z, then we can say that x = z. It is expressed as, x = y and y = z ⇒ x = z.
- Inverse Property: When an arithmetic operation is applied between a number and its inverse, we can get the identity. It is expressed as, a + (-a) = 0 and a × (1/a) = 1.
