Category: Maths Notes PPT

In our daily routine, we come across many shapes that are round, such as a wall clock, a wheel, the sun, the moon, the earth, a coin, bangles, rings, etc. All these shapes are in the form of a circle. A circle is a 2-dimensional shape defined as a collection of all the points in a plane that are equidistant from a given fixed point.

An interesting way to imagine a circle is thinking of it as a single line segment that is bent in a circular shape.

In this article, we will study what a circle is, how to draw a circle, related terms, properties of a circle, and examples of circles.

Circle Definition

A circle is a collection of points in a plane that are equidistant from a given fixed point.

The fixed point is called the centre of the circle. And the fixed distance from the centre is called the radius of a circle. From the figure O is the centre of the circle, r is the radius of a circle and the blue lines indicate the circumference of the circle that is the perimeter of the circle.

The circle formula is given as:

(x – h)² + (y – k)² = r²

where (x, y) are the coordinate points of the x and y axes, (h, k) are the coordinates of the centre of the circle, and r is the radius of the circle.

How to Draw a Circle?

The following steps are required to draw a perfect circle:

• Take plain paper.

• Plot a point on it.

• Take a compass to measure some length on a ruler.

• Now, place the sharp end of the compass on that point.

• Keeping the pointed leg on the point, rotate another leg one complete round drawing a line.

• You will see that a circle is drawn. And thus, we get a collection of points on the line that is equidistant from a given fixed point, and that is what defines a circle.

Terms Related to a Circle

Some important terms related to circles are explained below. It will prove helpful to you in an example of a circle.

Centre: The fixed point of a circle is called the centre of the circle,

Radius: The fixed distance from the centre to any point on the circle is called the radius of a circle.

Chord: a line segment joining any two points on the circle is called a chord. In the figure below the green line indicates the chord. 

Diameter: A chord that passes through the centre of the circle is called the diameter of a circle. The orange line indicates the diameter. Also, diameter is the longest chord and can be expressed as 2r where r is the radius of a circle. There are infinite numbers of diameter in a circle.

Secant: A chord that intersects the circle in two points is called a secant. In the figure below, the blue line indicates a secant.

Tangent: A line that touches a circle at one point is called tangent to the circle, and that point is called the point of contact. In the figure below, the red line indicates the tangent.

Arc: A piece of the circle between two points is called an arc. The red curve line is called the arc of a circle.

Circumference: The length of the complete circle is called the circumference of a circle. It is also said to be the perimeter of the circle. The circumference of a circle is equal to 2πr, where r is the radius of the circle.

Segment: The region between the chord and the arc is called the segment of the circle. In the below figure, the purple region indicates the segment. The segment containing the minor arc is called the minor segment and the segment containing the major arc is called the major segment.

Sector: The region between the arc at the two radii is called the sector. The blue region indicates the sector.

Interior and Exterior of a Circle: The circle divides the plane into three parts:

• The ‘inside’ of the circle is also called the interior of the circle, as B lies in the interior of the circle.

• The ‘outside’ of the circle is also called the exterior of the circle, as point A lies outside the circle.

• ‘On’ the circle is also called the circumference of the circle, as point C lies on the circle.

Properties of a Circle

Some of the properties of a circle are as follows:

• Circles with the same radii are said to be congruent.

• The longest chord of a circle is the diameter of a circle.

• The diameter of a circle is double the radius.

• The diameter divides the circle into two equal semicircles.

• The radius that is drawn perpendicular to the chord bisects the chord.

• A circle can be inscribed in a square, triangle or kite.

• The chords that are equidistant from the centre are equal in length.

• The distance from the centre of the circle to the longest chord (diameter) is zero, i.e., the longest chord is the diameter.

• The perpendicular distance from the centre of the circle decreases when the length of the chord increases.

• An isosceles triangle is formed when the radii join the ends of a chord to the centre of a circle.

Formulas Related to a Circle

Some of the important circle formulas that are widely used in solving examples of circles are as follows:

Diameter d: The diameter of a circle is twice the radius of a circle. The formula for the diameter is given as follows:

Circumference: The outer boundary of a circle is the circumference of the circle.
The formula for circumference is given by-as follows :

Area: Area of a circle is defined as the space occupied by the circle. The formula for area is given as follows:

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Solved Problems Related to Circles

Example 1: Find the area and the circumference of a circle whose radius is 7 cm. (Consider the value of π to be 22/7).

Solution:

Given radius r  = 7 cm.

Area A = π r² 

= 22/7  × 7 x 7

Area  = 154cm²

Circumference C = 2πr

     = 2  x 22/7 x 7 

Circumference =  44 cm

Example 2 : If the diameter of a circle is 12 cm, then find its area.

Solution:

It is given that the diameter d is equal to 12 cm.

So, radius r = d /2 

= 12/2 

= 6cm

Hence, area A = πr²

          A = π x (6)²

  = 3. 14 x 36

          A = 113.04 cm²

Did you Know?

Circles are special shapes that have the largest areas for any given length of their perimeters. Circles are shapes with high symmetry. Each line through a circle’s centre forms a line of reflection symmetry. It also has rotational symmetry around its centre for each and every angle.

Quiz Time

It is time for you to test your knowledge of what you have learnt in this article. Solve the following questions to check if you are ready to take up challenges in problems related to circles.

1. A circular garden has a radius of 21 m. The owner wants to put a plastic edge around the garden, and so the owner wants to know what the circumference of the garden is. Find the garden’s circumference. (Consider the value of π to be 22/7).

2. A circle with a radius of 21 cm is given. Determine its area and its circumference.

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Conclusion

Circles are an important component in Geometry and in Mathematics as a whole. Several theorems related to circles are constantly referenced in order to solve problems in exams or even in real-life situations like construction and architecture. Experts always rely on the terms and concepts, which have already been discussed in this comprehensive article on the topic, to put them into practical use. Circles are sure to constitute a significant portion of your Maths syllabus, and skipping this topic will cause you to lose precious marks that can otherwise be easily scored. Hence, it is advised to read through and internalise as many formulas and properties related to circles as possible, to be able to score good marks in your exams.   

Counting has been an ancient and important part of our lives. Likewise, it is also an important field when it comes to Mathematics. Now, counting objects that are less in number is an easier task but when it comes to counting large numbers of objects, it becomes a little difficult. Especially for students, the task of counting large volumes of units can be boring to say the least. So how do we speed up this process? Combinatorics is one such concept that can make the counting process easier. Mathematical calculations can be easily carried out by using this concept. 

What is Combinatorics? Combinatorial Meaning

As per the scientific definitions and concepts, combinatorial meaning can be perfectly explained as the process of counting objects in a dataset by making use of the enumeration, permutation, and combination techniques so that accurate results are obtained. Therefore, the study of combinatorics, as the name suggests, is a combination of various branches of mathematics. This is an essential application which will help students to gain a lot of knowledge from the diverse segments of mathematics. Thus, this is the overall combinatorial meaning and its concept.

 

The range that this concept has in its applications includes logic, statistical physics, computer science, evolutionary biology, etc. Therefore, students planning to build a career in research, development, and IT industries must have a deep rooted knowledge of Combinatorics in general.

Combinatorics Formula

The Combinatorics Formula is a union of both the Permutation and Combination concepts. The formula for finding out the Permutation for a set of objects is as given below.

P(n,r) = [frac{n!}{(n-r)!}]

Where n is the total number of objects 

And r is the conditions considered at a given time.

A combination is an unordered arrangement of objects in a collection. The objects can be randomly arranged. They are further classified into two types; repetition and without repetition of objects. The formula for finding out the Combination for a set of objects is as given below.

C(n,r) = [frac{n!}{(n-r)!r!}]

Where n is the total number of objects 

And r is the conditions considered at a given time  

The Combinatorics Formula helps to carry out mathematical operations on large datasets easily.

What are the Combinatorics Applications? 

Combinatorics deals with the counting of things in a dataset by following a particular pattern. It has its main applications when studying discrete objects.  

It has its major applications in the field of Computer Science as it deals with the study and application of several programming related real-time applications that are based on the concepts of Mathematics.  

Some of the Other Combinatorics Applications are as Follows:

• Discrete Mathematics 

• Additive Number Theory 

• Discrete Harmonic Analysis 

• Computer Architecture 

• Scientific Research and Development 

• Data Mining and Pattern Analysis 

• Communications Networks and Security

• Probabilistic Methods 

• Combinatorial Geometry 

• Extremal Problems for Graphs and Set Systems

• Ramsey Theory

What are Permutation and Combination?

In Mathematics, Permutation and Combination are used to describe and define a collection of individual objects. Both of them are related to the collection of objects but are different.  

A permutation is an ordered arrangement of the objects in a collection. They are further classified into two types; repetition and without repetition of objects.

A combination is an unordered arrangement of objects in a collection. The objects can be randomly arranged. They are further classified into two types; repetition and without repetition of objects.

Counting is an integral part of Mathematics and is easily carried out when the number of objects to be counted is less. The process is much more difficult when large datasets have to be handled. To solve this issue and to gain accurate output, the methods of Permutation and Combination are used.

Combinatorics Problems – Solved Example

There are various types of Combinatorics problems that students can work on. One of the most significant format of combinatorics problems has been highlighted below:

1.  Calculate the number of groups of 4 students that can be elected from the class of 28 students. 

A) Given that 4 students have to be selected in a team. Since no specifications have been provided, their order should be considered as random. Therefore, the formula for Combination will be used. 

C(28,4) = [frac{28!}{(28-4)!4!}] = 20475

There are 20475 ways in which a team of 4 students can be made out of 28 students, considering no proper order is followed.

A Guide to the Complex Numbers and Quadratic Equations

Mathematics includes a lot of topics that give an edge to your problem-solving abilities and critical thinking. Complex numbers and quadratic equations is a segment of maths that deals with crucial theorems and concepts along with various formulae. It comprises of linear and quadratic equations along with roots related to the complex number’s set (known as complex roots).

Although maths is a scoring subject, yet we find problems tricky because of insufficient knowledge of different topics. If you get stuck with cumbersome mathematical problems, try seeking complex numbers and quadratic equations NCERT solutions. 

Let us take a tour for a better understanding.

Define Complex Numbers

A mathematical equation having a complex number comprises of the real and imaginary sections. Complex numbers are nothing but a combination of two numbers (real, imaginary). Real ones mostly comprise of 1, 1998, 12.38, whereas imaginary numbers generate a negative result when they get squared.

For instance, consider an equation in the form (a + ib). Here, both a and b constitutes a complex number having a, as the real portion of the complex number and b acts as an imaginary one.

What are Quadratic Equations?

A quadratic equation is a mathematical equation in algebra that comprises of squares of a variable. It derives the name from the word ‘quad’ which implies square. It is also known as ‘equation of a degree 2’ (because of x2).

In complex numbers and quadratic equations, the standard form of a quadratic equation appears as:

  ax2 + bx + c = 0

Where x is a variable or an unknown factor, and a, b and c are known values.

The below chart show a few examples of a quadratic equation: 

Exercises on Complex Nos. and Quadratic Equations

Below there is a complex numbers and quadratic equations miscellaneous exercise. Go through it carefully!

1. Will be the Equation of the Following if they have Real Coefficients with One Root? 

1. 1 -2i

2. -2 – i√3

3. 1/(2 + i√2)

Solution: Assume, (a + b) and (a – b) are roots for all the problems.

1. Sum of the value of roots is (1 + 2i) + (1 – 2i) = 2 

Then, products of these roots will be (1 + 2i) * (1 – 2i) = 1 + 4 = 5

Therefore, the equation is equal to x2 – 2x + 5 = 0

2. Here, sum of the value of roots is -2 – i√3 – 2 + i√3 = – 4

Multiplication of these roots is (-2 – i√3) * (- 2 + i√3) = – 3 – 4 = 7

Hence, the quadratic equation will be x2 + 4x – 7 = 0

3. Sum of the roots will be (2 + i√2)/2 + (2 + i√2)/2 = 2 (If, 1/(2 + i√2) = (2 + i√2)/2)

Multiplication of them will result in (4 + 2)/4 = 3/2

Hence, the equation is equal to 2x2 – 4x + 3 = 0

Rack Your Brains

Read the following question on complex numbers and quadratic equations thoroughly to excel in your studies.

1. Suppose z = (√3/2 + i/2)5 + (√3/2 – i/2)5. If the real portion is R(z) and the imaginary section constitutes I(z) of the value z, then what will be the answer?

1. R(z) > 0 and I(z) > 0

2. R(z) < 0 and I(z) > 0

3. I(z) = 0 and R(z) < 0

4. R(z) = 3

5. I(z) = 0

6. None of them

Some chapters apart from quadratic equations can appear problematic to solve, such as integrals, permutation and combination, etc. However, you find solutions to these problems quickly if you understand the theorems well. Also, you can try searching for complex numbers and quadratic equations solutions to tackle cumbersome topics.

Download our app to learn amazing mathematics concepts with ease.

In mathematics, we define statement as a declarative statement which may either be true or may be false. Often sentences that are mathematical in nature may not be a statement because we might not know what the variable represents. For example, 2x + 2 = 5. Now here we do not know what x represents thus if we substitute the value of x (let us consider that x = 3) i.e., 2 × 3 = 6. Therefore, it is a false statement. So, what is a conditional statement? In simple words, when through a statement we put a condition on something in return of something, we call it a conditional statement. For example, Mohan tells his friend that “if you do my homework, then I will pay you 50 dollars”. So what is happening here? Mohan is paying his friend 50 dollars but places a condition that if only he’s work will be completed by his friend. A conditional statement is made up of two parts. First, there is a hypothesis that is placed after “if” and before the comma and second is a conclusion that is placed after “then”. Here, the hypothesis will be “you do my homework” and the conclusion will be “I will pay you 50 dollars”. Now, this statement can either be true or may be false. We don’t know. 

Hypothesis

A hypothesis is a part that is used after the ‘if’ and before the comma. This composes the first part of a conditional statement. For example, the statement, ‘I help you get an A+ in math,’ is a hypothesis because this phrase is coming in between the ‘if’ and the comma. So, now I hope you can spot the hypothesis in other examples of a conditional statement. Of course, you can. Here is a statement: ‘If Miley gets a car, then Allie’s dog will be trained,’ the hypothesis here is, ‘Miley gets a car.’ For the statement, ‘If Tom eats chocolate ice cream, then Luke eats double chocolate ice cream,’ the hypothesis here is, ‘Tom eats chocolate ice cream. Now it is time for you to try and locate the hypothesis for the statement, ‘If the square is a rectangle, then the rectangle is a quadrilateral’?

Conclusion

A conclusion is a part that is used after “then”. This composes the second part of a conditional statement. For example, for the statement, “I help you get an A+ in math”, the conclusion will be “you will give me 50 dollars”. The next statement was “If Miley gets a car, then Allie’s dog will be trained”, the conclusion here is Allie’s dog will be trained. It is the same with the next statement and for every other conditional statement.   

How Do We Know If A Statement Is True or False? 

In mathematics, the best way we can know if a statement is true or false is by writing a mathematical proof. Before writing a proof, the mathematician must find if the statement is true or false that can be done with the help of exploration and then by finding the counterexample. Once the proof is discovered, the mathematician must communicate this discovery to those who speak the language of maths. 

Converse, Inverse, contrapositive, And Bi-conditional Statement

We usually use the term “converse” as a verb for talking and chatting and as a noun we use it to represent a brand of footwear. But in mathematics, we use it differently. Converse and inverse are the two terms that are a connected concept in the making of a conditional statement.

If we want to create the converse of a conditional statement, we just have to switch the hypothesis and the conclusion. To create the inverse of a conditional statement, we have to turn both the hypothesis and the conclusion to the negative. A contrapositive statement can be made if we first interchange the hypothesis and conclusion then make them both negative. In a bi-conditional statement, we use “if and only if” which means that the hypothesis is true only if the condition is true. For example, 

If you eat junk food, then you will gain weight is a conditional statement.

If you gained weight, then you ate junk food is a converse of a conditional statement.

If you do not eat junk food, then you will not gain weight is an inverse of a conditional statement.

If yesterday was not Monday, then today is not Tuesday is a contrapositive statement. 

Today is Tuesday if and only if yesterday was Monday is a bi-conventional statement.   

A Conditional Statement Truth Table

In the table above, p→q will be false only if the hypothesis(p) will be true and the conclusion(q) will be false, or else p→q will be true. 

Conditional Statement Examples

Below, you can see some of the conditional statement examples.

Example 1) Given, P = I do my work; Q = I get the allowance

What does p→q represent?

Solution 1) In the sentence above, the hypothesis is “I do my work” and the conclusion is “I get the allowance”. Therefore, the condition p→q represents the conditional statement, “If I do my work, then I get the allowance”. 

Example 2) Given, a = The sun is a ball of gas; b = 5 is a prime number. Write a→b in a sentence. 

Solution 2) The conditional statement a→b here is “if the sun is a ball of gas, then 5 is a prime number”.

The way of constructing a triangle with a compass depends on the information given in the question. The construction of triangles is very important while applying the Pythagorean theorem and trigonometry.

Here, we will learn how to construct a triangle if we have the following data.

• All the three sides of a triangle are given.

• The measure of the hypotenuse and one side is given in the right triangle.

• Two sides of a triangle and included angle are given.

• Two angles of a triangle and included sides are given.

How to Construct Triangles?

Triangle is a two-dimensional polygon shape with three sides and three angles, which can be formed by joining the points in a plane.

But, the question arises how to construct triangles?

A unique triangle can be easily constructed using the concept of Geometry.

Geometry is a branch of Mathematics that deals with lines, angles, shapes, size, and dimension of different things we observe in everyday life. In Euclidean Geometry, there are different two dimensional and three -dimensional shapes. Flat shapes such as square, triangle, and circle are known as two -dimensional shapes. These shapes have only length and width.

Solid shapes such as cube, cuboid, sphere, cone, etc are three-dimensional shapes. These shapes have length, width, or height.

These geometric shapes can be easily constructed using compass, ruler, and protractor. Let us learn the steps of constructing triangles with compass, ruler, and protector below.

Constructing Triangle When Hypotenuse and One Side is Given

To construct a triangle when hypotenuse and one side is given, we need the following geometric tools:

Let us learn to construct a triangle when hypotenuse and one side is given through examples:

Construct a right-angled triangle ABC with the length of the hypotenuse AB = 3 cm and side BC = 5 cm. The steps of construction are:

Step 1:

Draw a line of any length and mark a point C on it.

Step 2:

Set the width of the compass to 3 cm.

Step 3:

Place the pointer of the compass at C and draw an arc on both sides of C.

Step 4:

Mark the point as P and A where both the arc crosses the line.

Step 5:

Taking P as the centre, draw an arc above the point C.

Step 6:

Taking A as the centre, draw an arc that cuts the previous arc.

Step 7:

Mark the point B, where two arcs intersect each other.

Step 8:

Join the points B and A along with B and C with the help of the ruler.

Thus, ΔABC is the required right-angled triangle.

Constructing Triangle When Two Sides and Included Angle are Given

To construct a triangle when two sides and angle, we need the following geometric tools:

• A Ruler

• A Protractor

• A Compass

Let us learn to construct a triangle when the length of two sides and included angle are given through an example.

Example:

Construct a triangle PQR with PQ = 4 cm, QR = 6.5 cm , and ∠PQR = 60°.

The steps of construction are:

Step 1:

Draw a line QR = 6.5 cm using a ruler.

Step 2:

Using protractor at Q, draw a line QX making an angle of 60° with QR

Step 3:

Taking Q as the centre, draw an arc of radius 4 cm to cut the line QX at P.

Step 4:

Join PR.

Therefore, PQR is the required triangle.

Constructing Triangle When Two Sides and Included Angle are Given

To construct a triangle when two sides and angle, we need the following geometric tools:

Let us learn to construct a triangle when the length of one side and included angle are given through examples.

Construct a triangle XYZ with XY = 4 cm, ∠ZXY = 100° and ∠ZXY = 30°.

The Steps of Construction are:

Step 1:

Draw a line segment XY = 4 cm using a ruler

Step 2:

Using protractor at X, draw a ray XP forming an angle of 30° with XY 

Step 3:

Using protractor at Y, draw another ray YQ making an angle of 100° with XY

Step 4:

Let the rays XP and QY intersect at Z.

Step 5:

Using the property, sum of all the angles of a triangle is equal to 180°, we can easily find the third angle of the triangle which is 50°. Hence, ∠Z = 50°.

Step 6:

Hence, XYZ is a required triangle.

Constructing Triangle Given Three Sides

To construct a triangle when all the three sides are given, we need the following geometric tools:

Before knowing how to construct a triangle with given sides, we should check the following property of triangles is met by the length of all the three sides.

“ The sum of all the three sides of a triangle should always be greater than its third side”.

We will not be able to construct a triangle with the given three sides if the above-mentioned property is not met by the given three sides.

Let us learn to construct a triangle given three sides through an example.

Example:

Construct a triangle ABC with side AB = 4 cm, BC = 6 cm and AC = 5 cm.

The steps of construction are:

Step 1:

Draw a line BC = 6 cm ( the longest side).

Step 2:

Taking B as centre, draw an arc of radius 4 cm above the line segment BC.

Step 3:

Taking C as centre, draw an arc of radius 5 cm that intersects the previous arc at ‘A’.

Step 4:

Join line segments AB and AC

Hence, ABC is the required triangle.

Drawing Triangle With Protractor

Construct an isosceles triangle PQR with PQ = 6 cm, QR = 6 cm and ∠PQR = 50°.

Steps of drawing a triangle with protractor for the given sides and angles are as follows:

1. Draw a line QR 6 cm long.

2. Taking Q as the centre, draw an angle of 50° using the protractor.

3. Taking R as the centre, draw an angle of 50° using the protractor (angles opposite to the equal sides of an isosceles triangle are equal).

4. Mark the point P where two lines intersect.

Therefore, PQR is the required isosceles triangle.

Solved Examples:

1. Construct an equilateral triangle with a side 5 cm long using a protractor?

Ans: An equilateral triangle is a triangle whose all the three sides are equal in length. Another property of the equilateral triangle is that three angles of the triangle are equal, and each angle of a triangle is equal to 60 degrees.

Following are steps to construct an equilateral triangle with each side 5 cm long.

Step 1:

Draw a line AB of 5 cm long.

Step 2:

Taking A as centre, draw an angle of 60° using a protractor.

Step 3:

Taking B as centre, draw another angle of 60° using a protractor.

Step 4:

Mark the point C where both the lines meet.

Hence, ABC is a required equilateral triangle of length 5 cm.

2. Write down the steps in constructing a triangle ABC with sides AB = 3.5 units, BC = 6 units and AC = 4.5 units.

Solution

Step 1:

Draw a line segment BC measuring 6 units.

Step 2:

With B as center, and draw an arc of radius 3.5 units

Step 3:

With C as center, draw an arc of radius 4.5 units to intersect the previous arc at A

Step 4:

Join the line segment AB and AC.

Hence, the triangle ABC is drawn.

Fun Facts

• Triangle is a polygon with the minimum possible number of sides (three).

• Hatch marks, also known as tick marks are used in triangles to identify the sides of equal length.

• Two triangles are considered similar if each angle of one triangle has the same measure as the corresponding angle in the other triangle.

Firstly, know what a variable is to have a better understanding of the continuous variable. Thus, a variable is a numerical expression whose value varies. There exist only 2 kinds of variables i.e. Continuous variable and Discrete variable. Of which, the continuous variable refers to the numerical variable whose value is attained by measuring. The variables can take almost any type of numeric value and can be further divided into meaningful smaller increments which include fractional and decimal values. This, in particular, is a kind of quantitative variable often used in machine learning and statistical modeling to describe data that is measurable in some way. Continuous variables are generally measured on scales such as height, weight, temperature, etc. With the help of continuous variables, one can measure mean, median, variance, or standard deviation. Continuous variables are the ones which in between any two numeric values have an infinite number of values. Different techniques of calculus are used in continuous optimization problems in which the variables are also continuous. 

Types of Continuous Variables

The continuous variables are of two types:

1. Instant variable: The variables that define the level or distance between each category which is equal and static are known as instant variables.

2. Ratio variable: The variables having only one variation from the interval variable are known as ratio variables. The ratio between the score gives the information between the responses regarding the relationship. 
   

Grab Discrete Variable to Grasp Continuous Variable

As we said above continuous variable meaning is all quantifying the data in some way. To learn about the mechanism, you would need to know more than a little about discrete variables.  So, what does a discrete variable mean? Data that deals with counting is considered to be discrete.  And for all you know, when we count things, we take into account whole numbers like 0, 1, 2, and 3. To better understand the discrete variables, let’s take an example.

Examples of Continuous Variables

Tell us how many eggs a hen lays? A chicken may or may not lay egg/eggs each day, but there are two things that certainly can never happen. There can never be eggs in a negative number, and there can never be a section or a fraction of an egg. 

 

Now that you know the two variables are distinctive of each other. Then surely, there must be some key differences between the two that set them apart for better description of data.

 

An example of a continuous variable is temperature as we can have decimals while measuring temperature and it can take on any value in an interval. In nature, almost all the variables present are continuous until the size reaches a quantum level. Therefore, at a macroscopic level, the mass, temperature, energy, speed, length, and so on are all examples of continuous variables. 

Another example of a continuous variable is height. Suppose you want to take an accurate measurement of your height and you are having the most advanced device in the world. You already know that you are between 150 and 152 cm but you want to know the accurate number. At first measurement, the device reads 151.2, then you measure it again and now the device reads 152.21. This means that even in between 150 and 152 there is an infinite number of possibilities.

Continuous Variables vs. Discrete Variables:

A variable holding any value between its maximum value and its minimum value is what we call a continuous variable; otherwise, it is called a discrete variable.

Step 1: First thing to do is to discover how long it would take you to count out the possible values of your variable. For instance, if your variable is “Temperature in North India”. See to it as how long would it take you to find every possible temperature reading? It would literally take you a lifetime.

45°, 47.11° 48.4°, 49.11°, 49.111°,…

 

For variables like these where you begin counting now and end up never (i.e. the numbers go on and until eternity), you have what’s known as a continuous variable.

 

If your variable is “Number of palm trees in a nursery,” then you can literally count all of the numbers (there can’t be innumerable numbers of palm trees). And with that, you have what’s called a discrete variable.

 

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Difference Between Discrete and Continuous Variables

 

Solved Examples

For the below given cases, identify whether a continuous random or discrete variable is involved?

Case 1

The length of a polar bear

Answer: The length is essentially treated as a continuous variable, since a polar bear will precisely not measure 3m. Even, the length of an adult male may measure between 2.4-3m, while reaching above 3m (more than 10 feet). That said, the length will vary by some foot or a fraction and thus is a continuous variable.

 

Case 2

The age of a polar bear

Answer: Age can every so often be considered as continuous or discrete variable. For example, we usually depict age as only a number of years, but occasionally we discuss a polar bear being to live beyond 18-20years old. Technically, since age can be regarded as a continuous random variable, then that is what it is reviewed, unless we have logic to deal with it as a discrete variable.

 

Now let’s take a fun quiz to know how far you have discrete and continuous variables.

 

Fun Quiz

For 1-10, find out whether each condition is a continuous or a discrete random variable, or if it is none

 

 

Did You Know

A variable in algebra is not quite alike to a variable in statistics.

Continuous Variables – Some more Solved Examples

1. Is bodyweight an example of a continuous variable?

Solution: Yes, bodyweight is an example of a continuous variable.

2. Is gender an example of a continuous variable?

Solution: No, gender is not an example of a continuous variable; however, it is a discrete variable.

3. How many gallons of milk does a cow give and is this an example of a continuous variable?

Solution: Cow’s milk is another example of a continuous variable. Although the amount of milk it gives daily varies, one day the cow can give 2.89 gallons, and the other day it can give 4 gallons. Therefore, there is an infinite number of possibilities which includes decimals and fractions as well.