What are Numbers

What are Numbers
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In the Wizard of Oz, Gilda, the Good Witch of the North told Dorothy to tap her ruby slippers 3 times to go back to Kansas.

Michael Phelps has a world record of 23 gold medals in the Olympic Games.

When Cathy bought a 25 cent candy, she gave the store manager a dollar. He returned 75 cents to her.

What do these three otherwise disconnected sentences have in common?

They have a number in it.

Numbers are an integral part of our everyday lives, right from the number of hours we sleep at night to the number of rounds we run around the racing track. Numbers define world records, sales, miles - you name it, and it has a number.

In this chapter, we'll give you an introduction to the different types of numbers there are.

Lesson Plan

What Is a number?

How do you define a number? You can define a number as a count, like in a race, when we say, “\(3\), \(2\). \(1\), GO!” or a measurement such as John Cena weighs \(275\) lbs.

There are also fractions such as \(\dfrac{22}{7}\) and decimals such as \(3.14\)

As you can see, the number universe is infinite and nothing in life can be managed without numbers. 

Even ingredients for your favorite double chocolate sundae requires numbers: right from measuring ingredients to the number of scoop and the cost per scoop.

Ingredients of chocolate sundae

In math, numbers can be 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.


Types of Numbers

Numbers are classified into several types like we mentioned above. These include:

  1. Prime and composite numbers
  2. Even and odd numbers
  3. Fractions and decimals
  4. Rational and irrational numbers
  5. Natural numbers
  6. Integers
  7. Real numbers
  8. Whole numbers
  9. Rational Numbers

Natural Numbers

All numbers from 1 to infinity (countless) are called natural numbers.

Whole Numbers

The numbers which start from zero (0) are called whole numbers.

In other words, the combination of zero and natural numbers are called whole numbers.

Integers

The positive and negative numbers along with zero are called integers.

Zero does not stand for positive or negative.

Number line and Integers

It is neutral in the center.

The positive numbers are called positive integers and negative numbers are called negative integers.

Rational Numbers

Any number which is defined in the form of a fraction or ratio is called a rational number.

This may consist of the numerator (p) and denominator (q).

A rational number can be a whole number or an integer.

Irrational Numbers

Irrational numbers are those in which the fraction or the ratio cannot be determined i.e. it cannot be expressed in the form of a fraction or a ratio.

Irrational numbers cannot be expressed in decimal form since the decimal numbers extend continuously and never repeat.

For instance, π (pi) is an irrational number because it does not have any decimal pattern and it go without an end.

Real numbers

Real numbers and its subsets

Real numbers in maths are numbers that include both rational and irrational numbers.

Therefore, it includes natural numbers, whole numbers, and integers.


Prime and Composite Numbers

In mathematics, composite numbers are the numbers that have more than two factors, unlike prime numbers, which have only two factors, i.e., one and the number itself. Such numbers with more than two factors also are called composites. 

Natural numbers greater than one that isn't prime are called composites because they're divisible by more than two numbers. The numbers 
\(0\) and \(1\) are neither prime nor composite, and therefore \(2\) is the first prime number, and \(4\), the first composite number

Prime and composite numbers

Meanwhile, prime numbers are any whole number greater than \(1\) that has exactly two factors, 1 and itself.

Another equivalent definition is: any whole number greater than \(1\) that is divisible only by \(1\) and itself, is defined as a prime number.

For example: \(13\) has just two factors: \(1\) and \(13\)

Hence it is a prime number.

There are around \(25\) prime numbers between \(1 \text{ and } 100\)

They are:

Prime Numbers between 1 and 10 2, 3, 5, 7
Prime Numbers between 11 and 20  11, 13, 17, 19
Prime Numbers between 21 and 30 23, 29
Prime Numbers between 31 and 40 31, 37
Prime Numbers between 41 and 50 41, 43, 47
Prime numbers between 51 and 100 53, 57, 61, 67, 71, 73, 79, 83, 89, 97

Even and Odd Numbers

A number that's divisible by \(2\) and generates a remainder of \(0\) when divided by \(2\) is known as an even number. An odd number is a number that isn't divisible by \(2\). The remainder within the case of an odd number is 1

An even number can be described as a number that may be divided into two equal groups. An odd number, on the other hand, can't be divided into two equal groups. 

Even numbers end in \(2, 4, 6, 8, \text{ and } 0\), no matter what preceding digits they have.

Odd numbers end in \(1,3,5,7,9\), no matter what preceding digits they have.

Even and Odd Numbers

Even numbers are often divided evenly into groups of two. The Number \(4\) is usually divided into two groups of Numbers of \(2(2+2)\).

Even numbers always end with numbers like \(0, 2, 4, 6, \text{ or } 8\)

All these numbers like 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 are even numbers.

Odd numbers can't be divided into two groups.

Number 5 is often divided into two groups of two and one group of \(1\)

Odd numbers always end with numbers like  \(1, 3, 5, 7, \text { or } 9\)

All these numbers like 1, 3, 5, 7, 9, 11, 13, 15, 17,19, 21, 23, 25, 27, 29, 31 are odd numbers.


Decimals and Fractions

A fraction is a part of a whole. The whole itself can be a number, an object, or a group of numbers (or objects). Fractions help us speak of numbers besides whole numbers. For instance, \(15\) and \(16\) are whole numbers — but a fraction is needed to write out a number that is between \(15\) and \(16\). Decimals help us like fractions do — but are separate entities.

This number line shows us a few whole numbers on the number line

Fractions and decimals both exist on the number line. In fact, every fraction has an equivalent decimal. The real difference between the two is how they are written out.

 
Challenge your math skills
Challenging Questions
  1. What are the prime numbers between 100 and 200?

Index Numbers

Index numbers tell you how many times you have to use the number in a multiplication.

Consider the following example:

\[8^2 = 8\times8=64\]

\[8^3=8\times8\times8=512\]

\[8^4=8\times8\times8\times8=4096\]

Using indices, you can also write large numbers using powers of \(10\). Scientists use this to calculate large values that go into billions and trillions.

For example, the speed of light is \(300,000,000\) m\s.

However, most science textbooks show it as \(3\times 10^8\)m\s.

This is the best way to use the power of \(10\)

Another example is the mass of the sun.

The sun has a mass of \(1.988\times 10^{30} \text{ kg}\).

It is easier to write this, than \(1,988,000,000,000,000,000,000,000,000,000 \text { kg}\). Chances are, you may make a mistake counting 30 zeroes!


Solved Examples 

Example 1

 

 

Which of the following are prime numbers and which are composite numbers?

\(1, 3, 4, 6, 12, 20, 29, 32\)

Solution

We have been given the following numbers

\[1, 3, 4, 6, 12, 20, 29, 32\]

\[\text{ The factor for 1 is 1}\]

\[\text{ The factors for 3 are 1 and 3}\]

\[\text{ The factors for 4 are 1, 2 and 4}\]

\[\text{ The factors for 6 are 1, 2, 3 and 6}\]

\[\text{ The factors for 12 are 1, 2, 3, 4, 6 and 12}\]

\[\text{ The factors for 20 are 1, 2, 4, 5, 10 and 20}\]

\[\text{ The factors for 29 are 1 and 29}\]

\[\text{ The factors for 32 are 1, 2, 4, 8, 16 and 32}\]

\(\therefore \) 1, 3 and 2 are prime numbers, while 4, 6, 12, 20, and 32 are composite numbers
Example 2

 

 

Can you write the following fractions as decimals?

\(\dfrac{5}{2}\), \(\dfrac{5}{7}\)

Solution

The first fraction is given as:

\[\begin{align}\dfrac{5}{2}\\\dfrac{5}{2}&=5\div2\\&= 2.5\end{align}\]

For the second fraction, we have

\[\begin{align}\dfrac{5}{7}\\\dfrac{5}{7}&=5\div7&\\&=0.71 \end{align}\]

\(\therefore \dfrac{5}{2}=2.5 \text{ and} \dfrac{5}{7} = 0.71\)
Example 3

 

 

Ron and Hermione are playing with number cards.

They want to classify the numbers based on their types.

Help Ron and Hermione complete their game.

Solution

classification of numbers

\(\therefore \text{the numbers have been classified}\)
Example 4

 

 

Can you help Stephen index this number using the power of 10?

\(50000\), \(100000\), \(1000000\)

Solution

\[50000 = 50\times 10^3\]

\[100000 = 100\times 10^3\]

\[1000000 = 100\times 10^4\]

\(\therefore \)Stephen can arrange the numbers using indices with the power of 10
Example 5

 

 

James has a bag of papers with the following numbers written on them. 

\(-1\), \(\sqrt{2}\), \(1\), \(4\), and \(-11\)

Can you help James pick out the natural numbers from this?

Solution

Since natural numbers are positive numbers, not fractions, and begin from 1, the only number James can choose are \(1\) and \(4\)

\(\therefore \text{ James should choose 1 and 4}\)

 
important notes to remember
Important Notes
  1. The numbers which start from 1 and go up to infinity are called natural numbers.
  2. The numbers which start from zero are called whole numbers.
  3. Integers consist of positive and negative numbers along with zero.
  4. Rational numbers consist of integers, fractions, and ratios.
  5. Irrational numbers are not expressed in the form of fractions or ratios.
  6. Real numbers consist of natural numbers, whole numbers, rational numbers, and irrational numbers.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of numbers. The math journey around numbers starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. What is the biggest number?

People may argue that the largest number is infinite, but we don't have a definite number of zeroes for that number. However, mathematicians have said that the largest number is called googol. It is 1 followed by one hundred zeroes and can be written as \(10^{100}\)

2. What are the basic numbers?

The basic numbers are natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, imaginary numbers, and complex numbers.

3. 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.

4. 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 \(1, 2, 3, 4, 5, 6, 7, 8, 9\text{ and } 10\)

  
More Important Topics
Numbers
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