Category: Maths Notes PPT

What is the Angle of Elevation? 

The term ‘angle of elevation’ is defined as the angle formed between the horizontal line and the oblique line from some object above the observer’s eye.

In other words, whenever you see an object above you, the angle formed between the horizontal line and the oblique line joining your eye and the object is called the angle of elevation.

In the above figure, an observer is looking at an object which is above the horizontal line forming an angle ‘θ’ between the line of sight and the horizontal line. Here, if we join an imaginary perpendicular line from the object to the horizontal line, a right-angled triangle will be formed. Thus, we can now use the concepts of trigonometry to find the distance between observer and object or, the height at which the object is from the horizontal line.

Terms Related to Angle of Elevation

There are three terms related to angle of elevation and are given below: 

1. Line of sight

2. Horizontal line

3. Angle of elevation

Line of Sight

The line of sight is the oblique line drawn from the observer’s eye to the point being viewed on the object. 

In the above figure, the oblique line AC is called the line of sight.

Horizontal Line

The imaginary line assumed for the horizontal eye level of the observer is here termed as horizontal line. 

In the above figure, the line AB is the horizontal line.

Angle of Elevation

When we raise our head to look at the object, the angle so formed by the line of sight with the horizontal line is called angle of elevation. It is measured in degrees and denoted by ‘θ’.

In the above figure, angle BAC is called the angle of elevation.

Formula for Angle of Elevation

The formula for finding the angle of elevation depends on the information provided to us like, height of the object from horizontal level, horizontal distance of object from the observer and the oblique distance of the object from the observer. Based on this provided information, we can use the concept of trigonometric ratios to find the angle of elevation.

Let ‘θ’ be the angle of elevation then, the formula for angle of elevation if the height of object and its horizontal distance from the observer is given by:

Tan θ = (Height of the object from horizontal level)/(Horizontal distance of object form the observer)

Comparison Between Angle of Elevation and Angle of Depression

Solved Examples:

Q.1. The angle of elevation of the top of a tower is 30° from a point on the ground, which is 15 m away from the foot of the tower. Find the height of the tower.

Solution: 

Let AB be the tower of height ‘h’ meters, and the angle of top of tower from a point ‘C’ on the ground is 30°.

In right angled triangle ABC,

Tan 30⁰ = (Height of the object from horizontal level)/(Horizontal distance of object form the observer)

Or,  Tan 30⁰= [frac{h}{15 m}]                      (∵ Tan 30° =[frac{1}{sqrt 3}])

Or,  [frac{1}{sqrt 3}] = [frac{h}{15 m}]

Or,  h = [frac{15 m}{sqrt 3}]

Or,  h = 5 [sqrt 3] m = 5 × 1.732 = 8.66 m 

Therefore, the height of the tower is 8.66 m.

Lowest Common Multiple (LCM):
The smallest number (other than zero) that is the common multiple of any two or more given natural numbers are termed as LCM. For example, LCM of 6, 8, and 12 is 24.

Highest Common Factor (HCF):
The greatest factor which is  common to any two or more given natural numbers is termed as HCF of given numbers. Also known as GCD (Greatest Common Divisor). For example, HCF of 8,and 40 is 8.

Both HCF and LCM of given numbers can be found by using two methods, they are division methods and prime factorization.

HCF and LCM have many applications in our daily life. Let us understand the applications of LCM and HCF, also we will understand the relationship between these two, which will make the concept more clear.

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Relationship Between LCM and HCF 

LCM and HCF have an interesting correlation between them. Some of LCM and HCF relations are as follows:

Relation 1: The product of LCM and HCF of any two given numbers is equivalent to the product of the given numbers.

LCM × HCF = Product of the Numbers

Suppose P and Q are two numbers, then.

LCM (P & Q) × HCF (P & Q) = P × Q

Relation 2: As HCF of co-prime numbers is 1. Therefore we get that the LCM of given co-prime numbers is equal to the product of the numbers.

LCM of Co-prime Numbers = Product of The co-prime Numbers.

Relation 3: H.C.F. and L.C.M. of Fractions

= LCM of fractions= LCM of numerators/  (GCD/HCF of denominators)

= HCF of numerators =HCF of fractions / LCM of denominators

What are the Applications of LCM and HCF?

We use H.C.F. method in the following fields:

Real Life Example: ​Priyanka has two pieces of cloth. One piece is 45 inches wide and the other piece is 90 inches wide. She wants to cut both the strips of equal width. How wide should she cut the strips?

Answer: ​

This problem can be solved using H.C.F. because we are cutting or “dividing” the strips of cloth into smaller pieces (Factor) of 45 and 90 (Common) and we are looking for the widest possible strips (Highest)

So,

H.C.F. of 45 and 90

45 = 3 x 3 x 5

90 = 2 x 3 x 3 x 5

HCF is 3 x 3 x 5 = 45 

So we can say that

Priyanka should cut each piece to be 45 inches wide.

We use L.C.M. method in  the following fields :

Real Life Example: Ram exercises every 8 days and Deepika every 4 days. Ram and Deepika both exercised today. After how many days do they exercise together again?

This problem can be solved using Least Common Multiple because we are trying to find out the time they will exercise, time that it will occur at the same time (Common).

Answer:

L.C.M. of 8 and 4  is

8 = 2 x 2 x 2

4 = 2 x 2

LCM is 2 x 2 x 2 = 8

SO, they will exercise together again in 8 days.

Solved Examples on LCM and HCF

1. Find the HCF of the following numbers:

36

48

60

Solution:

36 = 2 x 2 x 3 x 3

48 = 2 x 2 x 2 x 2 x 3

60 = 2 x 2 x 3 x 5

HCF(36, 48, 60)= 2 x 2 x 3 = 12

Therefore HCF of 36, 48, 60 is 12

2. Find the LCM of the following numbers:

25

40

Solution:

25 = 5 x 5

40 = 2 x 2 x 2 x 5

LCM = 5 x 2 x 2 x 2 x 5 = 200

Therefore LCM of 25 and 40 is 200

Quiz Time

Mr Patil has three classes. Each class has 28, 42 and 56 students respectively. Mr Patil wants to divide each class into groups so that every group in every class has the same number of students and there are no students left over. What is the maximum number of students Mr. Patil can put into each group?

(answer: 14 students).

Find the Highest Common Factor of 18, 24 and 42.

(answer: 6).

Fun Facts

Tricks for finding the HCF and LCM easily :

Trick 1 –

• Observe the numbers and list them out.

• Express each of them as a product of prime factors using prime factorization. For example, if we take the number 18 , it can be broken as 2 × 3². Similarly, factorise each of the numbers. 

• Now, the product of all these highest powers of prime factors will be equal to the LCM. 

Trick 2 –

• Note the numbers in a horizontal line and separate them with the help of commas.

• Take the smallest prime number and divide the given number from it. 

• Note the undivided numbers and quotients in a line as well.

• Repeat this process until there is no prime factor common between two numbers.

• Then, multiply all the divisors and the numbers left in the last row. The final product of these numbers will be equal to the LCM of those numbers.

Trick 3 – How to Find LCM of Co-Prime Numbers

Hint: This question is using the concept of LCM as well as co-prime numbers. Co-prime numbers are the numbers having 1 as a common factor. Use this property to get find their LCM and hence the solution to this question:

Complete step-by-step answer:

As we know, the two numbers which have only 1 as their common factor are known as co-primes.

For example, Factors of 5 are 1 and 5

Factors of 3 are 1 and 3.

Here, the common factor is 1.

Thus, 5 and 3 are the co-primes.

Now, to find out the LCM of two or more numbers using the factorization method, we have to find their factors. Then common will be taken as LCM.

We have to choose each prime number with the greatest power and then we have to multiply them to get their LCM. So, in the above example case,

LCM of 3 and 5 will be,

3 × 5 = 15

Hence, we c
an say that the LCM of two co-prime numbers is nothing but their product.

∴ L.C.M. of two or more co-prime numbers is their product.

Note: Here two terms must be clear for getting a solution. One is LCM and the other is Co-primes. Least Common Multiple i.e. LCM is a method to find the minimum common multiple between any two or more numbers. LCM denotes the least value of common factor or multiple of any two integers. Co-prime number is a set of numbers that have only 1 as their common factor, which means their HCF will be 1.Then after this question it is very easy to get a solution.

Thank you, This was all about Applications of LCM and HCF. provides courses which include lectures, PDF notes, question and MCQ series etc to provide better and easy learning for students. Students can study more math topics further with the help of this course.

A circle is characterized as the locus of a considerable number of focuses that are equidistant from the inside. However, a quadrant is one-fourth segment of a circle which is achieved when a circle is partitioned equally into four segments or rather 4 quadrants by two lines that are perpendicular to each other in nature. In this article, let us talk about what a quadrant is, how to compute the area of a quadrant and the area of quadrant formulas with some examples for a better understanding of the concepts of quadrants.  At the end of this section, you will be able to state what a quadrant is and use the area of the quadrant formula to solve your problems.

 

In this article, students will learn the concept of quadrants, other relevant terms, and examples of accurately identifying and plotting points using quadrants. Students are required to study the concept of quadrants thoroughly in order to solve a wide range of math problems. Knowing how to find the area of quarter circles will serve students well as they advance in math classes and as they take mathematics tests. 

What is Quadrant?

The coordinated frame system has four quarters or segments in it and these quarters or segments are known as the quadrants. These quadrants are all equal in size and area. With respect to a circle, the quarter of a circle is known as a quadrant, which is a segment of 90 degrees. Let us consider four such quadrants attached together. What does it make? It forms a circle. Let’s understand this better with the help of the image given below. You can see that the circle has been divided into four equal parts. Now, these parts are known as quadrants. Each of these quadrants is equal in size and at the midpoint or the center O, they all make a 90-degree right angle. 

 

How to Calculate the Area of a Quadrant of Circle?

Before we go ahead and learn how to calculate the area of a quadrant of a circle, there are a few things you must know. Stated below are the key factors you must keep in handy before you solve the problem.

 

In order to find the area of a quadrant of a circle, you first need to know the area of the circle. Here’s a list of things you need to know.

1. The Center of a Circle: All the points of the circle are at an equal distance from the center of the circle. Hence, this is known as the center. 

2. The Radius of a Circle: The distance from the center of the circle to any point on the circle, is known as the radius of the circle. It is denoted by the letter R.

3. The Diameter of a Circle: The diameter is twice the radius of the circle. It is denoted by the letter D.

4. The Circumference of a Circle: The distance around the edge of the circle is called the circumference of the circle.

5. The formula of the circumference of Circle: Circumference = 2πr

6. The Area of a Circle: The total amount of sq units occupied by a circle is known as the area of the circle.

7. The formula of Area of Circle: Area = π x radius x radius or Area of circle = πr2

Now that we know the area of the circle, let’s calculate the area of the quadrant of the circle. 

 

We know that the circle can be divided into four segments or portions and these portions can be called quadrants. Since there are four quadrants in a circle, you can just divide the area of the circle by 4. 

 

Therefore, 

 

Area of a Quadrant = [frac{pi r^{2}}{4}].

 

Methods to Calculate the Area of a Quadrant

The area of a quadrant of a circle can be calculated by two methods.

 

Method 1: By dividing the area of a circle by 4 to proportionate it to the area of the one-fourth part of the circle, we can obtain the area of a quadrant

Area of circle =  [pi times r^{2} ].

One-fourth area of circle = [frac{1}{4}]. 

Area of quadrant = [frac{1}{4} times pi times r^{2} ]

Also, Area of quadrant = [frac{1}{4} times pi times (frac{d}{2})^{2} ]

 

Method 2: By using the area of the sector of a circle, we can obtain the area of a quadrant.

Area of a sector of a circle = [(frac{θ}{360º}). pi. r^{2}]

where θ is equal to 90° because the quadrant of a circle is a type of sector having a right angle.

Area of a quadrant = area of a sector of a circle of θ as 90°

Area of a quadrant = [(frac{90°}{360°}). pi. r^{2}].

Area of a quadrant = [frac{1}{4}. pi. r^{2}]

 

The Formula for The Perimeter of a Quadrant

Pulling both parts together, the formula for the perimeter (​p​) of a quadrant is:

p = 0.5πr + 2rp = 0.5πr + 2r

This is really easy to use. For example, if you have a quadrant with ​r​ = 10, this is:

p​=( 0.5 x π x 10 ) + ( 2 x 10 ) = 5π + 20 = 15.7 + 20 = 35.7​

 

Solved Examples

Question 1: The radius of a circle is 6 cm. Find the area of the circle, the perimeter of the circle, and the area of the quadrant of the circle. 

Solution: 

Given, 

The radius = 6 cm 

We know that,

Area of circle = πr2

Circumference = 2πr

Area of a Quadrant = [frac{pi r^{2}}{4}]

 

(i) Area of circle

Area of Circle = πr2

Area of Circle = 3.14 * 62

Area of Circle = 3.14 * 6 * 6

Area of Circle = 3.14 * 36

Area of Circle = 113.04 cm2

 

(ii) Circumference of the Circle

Circumference = 2πr

Circumference = 2 * 3.14 * 6

Circumference = 12 * 3.14

Circumference = 37.68 cm.

 

(iii) Area of a Quadrant of the Circle

Area of a Quadrant = [frac{pi r^{2}}{4} ]

Area of a Quadrant =  [frac{pi 6^{2}}{4} ]

Area of a Quadrant =  [frac{pi 6 times 6}{4} ]

Area of a Quadrant = 3.14 * [frac{36}{4}]

Area of a Quadrant = [frac{113.04}{4}]

Area of a Quadrant = 28.26 cm2

 

Question 2: The radius of a circle is 2m. Find the area of the circle, the perimeter of the circle, and the area of a quadrant of the circle. 

Solution: 

Given, 

The radius of a circle is = 2m 

We know that,

Area of circle = πr2

Circumference = 2πr

Area of a Quadrant = [frac{pi r^{2}}{4}]

 

(i) Area of circle

Area of Circle = πr2

Area of Circle = 3.14 * 22

Area of Circle = 3.14 * 4

Area of Circle = 12.56 m2

 

(ii) Circumference of the Circle

Circumference = 2πr

Circumference = 2 * 3.14 * 2

Circumference = 4 * 3.14

Circumference = 12.56 m.

 

(iii) Area of a Quadrant of the Circle

Area of a Quadrant = [frac{pi r^{2}}{4}]

Area of a Quadrant = [frac{pi 2^{2}}{4}]

Area of a Quadrant = [frac{pi times 2 times 2}{4}]

Area of a Quadrant = [frac{3.14 times 4}{4}]

Area of a Quadrant = 3.14 m2.

A quadrilateral is a two-dimensional closed shape that has four sides, four corners, and four vertices. According to Euclidean Geometry, a quadrilateral is a polygon having 4 sides, 4 vertices. There are seven quadrilaterals and they are:

Square:

It has all sides equal and makes 90° at the edges. Diagonals intersect each other. The sum of all the interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Rectangle: 

The pair of sides are equal and all the edges are at 90°. Diagonals intersect each other. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Rhombus:

It is similar to a square. All sides are equal. It is in the shape of a diamond. Opposite angles are equal. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Trapezium:

It has a pair of parallel sides. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Kite: 

Two pairs of equal-length sides that are adjacent to each other. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Parallelogram: 

Opposite sides are parallel and equal in length. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Isosceles Trapezoid:

It is also the same as trapezium. The only difference is non-parallel sides are of equal length. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

This article focuses on Trapezium, its basic concept, its properties, area, application, formulae, and derivation of the area of trapezium. 

Trapezium Definition

The trapezium is a two-dimensional closed figure having a pair of parallel sides. It has 4 sides and 4 vertices. Parallel sides of the trapezium are bases and non-parallel sides are called legs. 

Different Types of a Trapezium

Scalene Trapezium:

All the sides and angles are of different measures. Just trapezoid is scalene trapezium as shown in the figure below.

  

(Images will be uploaded soon).

Isosceles Trapezium: 

If in trapezium any two pairs of sides are equal, that is, bases or legs then the trapezium is Isosceles. 

Right Trapezium: 

At least two of the angles are right angles i.e.,90°.

The basic difference between the trapezium and trapezoid is shown below. This difference is only due to British and American versions. 

Difference Between Trapezium and Trapezoid

British Trapezoid:

A quadrilateral with no sides parallel

USA Trapezium:

A quadrilateral with no sides parallel. 

Basic Concepts of Trapezium

1. Bases are the parallel sides of the trapezium and non-parallel sides are the legs.

2. A line drawn from the middle of non-parallel sides is the midpoint.

3. The arrows and equal marks shown in the figure denotes that the lines are parallel and the length of the sides are equal respectively.

4. The trapezium will get divided into two unequal parts if one cuts it into two sides from the middle of non-parallel sides.

5. In Isosceles trapezium, the two non-parallel sides are equal and form equal angles on the bases. 

Properties of Trapezium

Some of the properties of Trapezium are as follows:

Exactly one pair of opposite sides are parallel.

• Diagonals intersect each other. 

• The sum of the internal angles of the trapezium is 360° i.e., ∠A + ∠B + ∠C + ∠D = 360°.

• Except for isosceles trapezium, trapezium has non-parallel sides unequal.

• The line that joins the mid-point of the non-parallel sides is always parallel to the bases of the trapezium. Mid-segment = (AB + CD)/2 

• The legs are congruent in Isosceles Trapezium.

• The two angles of a trapezium are supplementary to each other. Their sum is equal to 180°.

Area of Trapezium

The area of trapezium is calculated as it is half of the sum of parallel sides and height. The formula of area of Trapezium is written as ½ × sum of parallel sides × times distance between them = ½ × (b₁ × b₂) × h

The Perimeter of Trapezium 

The perimeter of the trapezium is the sum of all four sides. Mathematically it is given as, Perimeter = AB + BC + CD + DA. 

Derivation of the Area of Trapezium

The derivation of the area of trapezium is given below. To Derive: Area of trapezium

Derivation: Here, let one side be ‘b1’ and another side be ‘b2’. The distance between the parallel sides is ‘h’

From the figure, it can be seen that there are two triangles and one rectangle. Hence, the area of the trapezium is

Area = area of triangle 1 + area of rectangle + area of triangle

Area = ½ × AE × DE + DE × EF + ½ × FB × CF

          = [frac{ah}{2} + b_{1} h + frac{ch}{2}]

          = [frac{ah+2b_{1} h+ch}{2}]

On simplifying the equation, we get, 

          = [frac{h}{2}] (a + 2b[_{1}] + c)

From the figure, b[_{2}] = a + b[_{1}] + c

Substituting the value from the above equation we get,

          = [frac{h}{2}] [[b_{1}+b_{2}]]

          = [frac{1}{2}] [[b_{1}+b_{2}]] × h

is the required equation. Hence, the area of the trapezium has been derived where b₁ and b₂ are bases and h is the altitude.

Hence, the area of the trapezium has been derived where b₁ and b₂ are bases and h is the altitude. 

How To Find The Angles Of A Trapezium

We now know that there are several types of trapeziums. Keeping a regular or isosceles trapezium in mind, students should remember that the sets of angles adjoined by parallel lines are equal. Moreover, we also know the mathematical fact that the sum of all the interior angles is equal to 360 degrees in any given quadrilateral. This will help us to calculate the angle of any trapezium.

For example, let us assume that an angle named x is given between the two parallel sides and non-parallel sides of the trapezium provided. Now, subtract twice of this angle from 360. This will give you the sum of two angles on the formed opposite side of the angle we named x. Once you have successfully calculated the sum of these two angles, you can proceed to find the fourth angle by dividing this sum by 2. 

Application of Trapezium

The concept of trapezium has a wide range of applications. It is used in physics for solving various questions based on trapezium whereas in mathematics it is used in a variety of applications, i.e, solving various questions based on surface area or for finding the complex figure area or perimeter. Trapezium formula can be used in construction also like the shape of the roof is trapezoidal. It has multiple applications in daily life. 

Example:

The length of the parallel sides of a trapezium are in the ratio 5:2 and the distance between them is 20 cm. If the area of the trapezium is 325 cm², find the length of the parallel sides.

Solution:

Let x = common ratio

The parallel sides are = 5x and 2x

Altitude = 20cm

Area of trapezium = 325cm²

Then from the formula of area of trapezium

Area = [frac{1}{2}] [[b_{1}+b_{2}]] × h

325  = [frac{1}{2}] [5x + 2x] × 20

         = x = 4.64 cm

Hence, the parallel sides are 5x = 23.2 cm and 2x = 9.28 cm

Therefore, the length of the non – parallel sides is 23.2 cm and 9.28 cm. 

Example:

Two parallel sides of a trapezium are of lengths 15 cm and 10 cm respectively, and the distance between them is 22 cm. Find the area of the trapezium. 

Solution:

Given: Parallel sides of the trapezium = 15 cm and 10 cm

Distance between the parallel sides = 22 cm

Area of trapezium is given by = [frac{1}{2}] [[b_{1}+b_{2}]] × h

Hence,

Area = [frac{1}{2}] (15 + 10) × 22 = 275 cm²

Therefore, the area of the trapezium is 275 cm². 

Example:

Find the perimeter and area of the trapezium whose parallel sides are 10 cm and 25 cm. The distance between the bases is 30 cm and the non-parallel side length is 20 cm each.

Solution:

 

Given: Parallel sides length = 10 cm and 25 cm

Altitude = 30 cm

Area = [frac{1}{2}] [[b_{1}+b_{2}]] × h

         = [frac{1}{2}] (10 + 25) × 30

         = 1050 cm²

Perimeter = Sum of all the sides= 10 + 25 + 20 + 20= 75 cm

Hence, the area and perimeter is 1050 cm² and 75 cm respectively.

Statistics, by its simplest understanding, is the analysis that involves collection, review, and the inference to be drawn from data. While, there is usually a large volume of data involved in this academic discipline, the concept of central tendency deviates from it. 

Central tendency focuses on a solitary value for the description of a given set of data. Such function is undertaken with the identification of central position located in the provided data set. There are three ways to measure central tendency – Mean, Median and Mode. It is an arithmetic mean statistics that are being elaborated further.  

What is Understood by Arithmetic Mean Statistics? 

Definition of arithmetic mean in Statistics simply covers the measurement of average. It involves the addition of a collective of numbers. The resulting sum is further divided with the count of numbers that are present in a given series. 

Simple arithmetic mean formula can be understood from the following example –

Say, within a series the numbers are – 36, 46, 58, and 80. The sum is 220. To arrive at arithmetic mean, the sum has to be divided by the count of numbers within the series. Hence, 220 is divided by 4, and the mean comes out to be 55. 

Arithmetic mean statistics includes the formula –

[bar{X}] = [frac{(x_{1}+x_{2}+…..+x{n})}{n}] = [frac{sum_{i=1}^{n}xi}{n}]

In the above equation,

X̄ = arithmetic mean symbol ___________________ (a)

X1,…,Xn =  mean of ‘n’ number of observations _____ (b)

∑ = summation ______________________________ ©

Concept of Arithmetic Mean Median Mode 

Even though arithmetic mean statistics has been elaborated, it can be better understood in the context of median and mode as well. 

Within a given data set –

Mean of a data set can comprise of several different series – (1) Individual, (2) Discrete, (3) Continuous, (4) Direct. On the other hand, for calculating the median, the data set has to be arranged in descending or ascending order. Mode covers such data which occurs the most number of times within a given series. The mode formula may be applicable in case of discrete, individual and continuous series. 

Finding Arithmetic Mean

Following example illustrates the application of arithmetic mean formula.

The Arithmetic Mean Formula in Statistics is – 

[bar{X}] = [frac{(x_{1}+x_{2}+…..+x{n})}{n}] = [frac{sum_{i=1}^{n}xi}{n}]

In the first two steps, midpoints of values (f) and aggregate of such values (fi xi) have to be found out. 

Midpoint = (upper value) + (lower value) / 2

From the above table, it can be derived – 

∑ fi = 30  ………………………………… (i)

∑ fixi = 1020 …………………………… (ii)  

Therefore, the arithmetic means of given data amounts to –

X̄ = ∑ fixi / ∑ fi

= 1020/30

= 34

Arithmetic mean statistics can be a complex concept to grasp. You can have all your doubts clarified in ’s online classes. To find out more about this, download the app today!

A number base (also known as base) for short is a numeral system that tells us about the unique or different symbols and notations that can be used to represent a value.

For example, the base 2 number system tells that there are only 2 unique notations 0 and 1 to represent the value.

The most commonly used number base is base 10, also known as the decimal number system. The decimal number system uses ten different notations which are the digit 0-9 to represent a value Bases can be either positive, negative, 0, complex, or non-integer. The most frequently used bases are base 2 and base 16. They are also used for calculating and are known as binary, and hexadecimal respectively.

What is a Base Number?

A base number is a number raised to the power that represents the number of units of a number system. For example, the base number of the binary number system is 2.

For Example,

y^x

Here, y is a base number.

Base 2 Number System

In Mathematics, the base 2 number system, also known as the binary number system uses 2 as the base and therefore requires only two digits i.e. 0 and 1 to represent any value, rather than 10 different symbols required in the decimal number system. The numbers from 0 to 10 in the binary number system are represented as “.” .The base 2 number system is widely used in Mathematics and Computer Science as bits are easy to create using physically logic gates (the logic gates are either open or closed meaning 0 or 1).

Counting in Different Bases

Counting in different bases substitutes the base 10 with a different bases. We often use Base 10. It is our decimal number system. It has 10 digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

We count numbers with base 10 as shown below:

Let us Understand How to do Counting in Different Base

(Base 2) Binary Number System Has Only 2 Digits: 0 and 1

We count the base 2 like shown below:

(Base 3) Ternary Number System Has 3 Digits: 0,1, and 2

We count numbers with base 3 as shown below:

(Base 4) Quaternary Number System Has 4 Digits: 0, 1, 2, and 3

We count numbers with base 4 as shown below:

(Base 5) Quinary Number System Has 5 Digits: 0, 1, 2, 3, and 4

We count numbers with base 5 as shown below:

(Base 6) Senary Number System Has 6 Digits: 0, 1, 2, 3, 4, and 5

We count numbers with base 6 as shown below:

(Base 7) Septenary Number System Has 7 Digits: 0, 1, 2, 3, 4, 5, and 6

We count numbers with base 7 as shown below:

(Base 8) Octal Number System Has 8 Digits: 0, 1, 2, 3, 4, 5, 6, and 7

We count numbers with base 8 as shown below:

Nonary (Base 9) Number System Has 9 Digits: 0, 1, 2, 3, 4, 5, 6, 7, and 8

We count numbers with base 9 as shown below:

(Base 10) Decimal Number System Has 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 9, and 10

We count numbers with base 10 as shown below:

Facts to Remember

In the number system, base, also known as radix, is the number of different digits or combinations of digits and letters that the number system uses to represent numbers.