Category: Maths Notes PPT

In Mathematics, most of the researchers used logarithms to transform multiplication and division problems into addition and subtraction problems before the process of calculus has been found out. Logarithms are continuously used in Mathematics and Science as both subjects contend with large numbers. Here we will discuss the log 0 value (log 0 is equal to not defined) and the method to derive the log 0 value through common logarithm functions and natural logarithm functions.

Logarithm Functions

Before deriving the Log 0 value, let us discuss logarithm functions and their classifications. A logarithm function is an inverse function to an exponential. Mathematically logarithm function is defined as:

If Logab = x, then ax =b

Where, a

Note= The variable “a” should always be a positive integer and not equal to 1.

Classification of Logarithm Function

Common logarithm functions – Common logarithm function is the logarithm function with base 10 and is denoted by log10 or log.

F(x) = log10 x

Natural logarithm functions – Natural logarithm functions are the logarithm functions with base e and is denoted by loge

F(x) = loge x

Log functions are used to find the value of a variable and eliminate the exponential functions. Tabular data will be updated soon.

What is the Value of Log 0? How Can it be Derived?

Here, we will discuss the procedure to derive the Log 0 value.

The log functions of 0 to the base 10 is expressed as Log10 0

On the basis of the logarithm function,

Base = 10 and 10x = b

As we know,

The logarithm function logab can only be defined if b > 0, and it is quite impossible to find the value of x if ax = 0.

Therefore, log0 10 or log of 0 is not defined.

The natural log function of 0 is expressed as loge 0. It is also known as log function 0 to the base e. The representation of natural log of 0 is Ln

If ex = 0

No number can agree with the equation when x equals to any value.

Hence, log 0 is equal to not defined.

Loge 0 = In (0) = Not defined

Value of Log of 0, and its Calculation to the Base 10

The inverse function to the exponentiation is generally regarded as the Logarithm, in Mathematics. Logarithm shows how much the base of the b must be raised to meet the exponent of the number x. In simple terms, the logarithm counts how many times the same factor occurs in the repeated multiplication.

Let us take an example of the number 1000. It can be formed by multiplying the number 10 with itself three times. 1000 = 10 × 10 × 10 = 1000, that is to say, 103. It means for 1000 the logarithm base is 3. It can be denoted as log10(1000) = 3. 1000 is the base here and the exponent 3 is the log.

logb(x) shows the logarithm for the x to the base b, it can also be shown without the use of brackets or parenthesis logbx. or sometimes even without the base log x. Logarithms are of great use in mathematics, science, and technology, and they are used for various reasons and purposes. 

Logarithm Value Table from 1 to 10

Logarithm Values to the Base 10 are:

Ln Values table from 1 to 10

Logarithm Values to the Base e are:

Solved Example

1. Solve for y in log₂ y =6

Solution: The logarithm function of the above function can be written as 26 = y

                  Hence, 25 =2 x 2 x 2 x 2 x 2 x 2 =64 or Y =64

2. Find the value of x such that log x  81 =2

Solution:

Given that, log x  81=2

On the basis of Logarithm definition

If logx b=x

ax  = b – (1)

a=x, b= 81, x =2

Substituting the value in equation (1), we get

x2 =81

Taking square root on both sides we get,

x = 9

Therefore, the value of x = 9

Fun Facts

• The logarithm with base 10 is known as common or Briggsian, logarithms and can be written as log n. They are usually written as without base.

• Concept of Logarithm was introduced by John Napier in the 17th century

• The logarithm is the inverse process of exponentiation.

• The first man to use Logarithm in modern times was the German   Mathematician, Michael Stifel (around 1487 -1567).

• According to Napier, logarithms express ratios.

• Henry Briggs proposed to make use of 10 as a base for logarithms.

Quiz Time

1. Which of the following is incorrect?

a. Log10 = 1

b. Log( 2+3) = Log( 2×3)

c. Log10 1 = 0

d. Log ( 1+2+3) = log 1 + log 2+ log 3

2. If log [ frac{a}{b} ] + log [ frac{b}{a} ] = log( a+b), then:

a. a + b=1

b. a – b = 1

c. a = b

d. a² – b² = 1

A vector is an object that contains  both a magnitude and a direction. Force and velocity are the two examples of vector quantities. Understanding the magnitude of the vector would indicate the strength of the force and similarly, the speed of any object is associated with the velocity. In the following article, the magnitude and direction of vectors are explained.

The picture below shows a vector:

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A vector has magnitude (that is the size) and direction:

The length of the line or the arrow given above shows its magnitude and the arrowhead points in the direction.

Now, we can add two vectors by simply joining them head-to-tail, refer the diagram given below for better understanding:

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And it doesn’t matter in which order the two vectors are added, we get the same result anyway:

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Notation

A vector can often be written in bold, like a or b.

Subtraction of Vectors:

We can also subtract one vector from another, keeping the two points given below in our mind:

• Firstly, we need to reverse the direction of the vectors we want to subtract, this changes the sign of the vector from positive to negative.

• Secondly we need to add them as usual:

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What is the Magnitude of a Vector?

As we know, that vector can be defined as an object which has both magnitudes as well as it has a direction. Now if we have to find the magnitude of a vector formula and we need to calculate the length of any given vector. Quantities such as velocity, displacement, force, momentum, etc are the vector quantities. But the quantities like speed, mass, distance, volume, temperature, etc. are known to be scalar quantities. The scalar quantities are the ones that have the only magnitude whereas vectors generally have both magnitude and direction.

Magnitude of a Vector Formula:

The magnitude of a vector formula can be used to calculate the length for any given vector and it can be denoted as |v|, where v denotes a vector. So basically, this quantity is used to define the length between the initial point and the end point of the vector.

[mid vec{v} mid] = [sqrt{x^{2}+y^{2}}]

[mid vec{v} mid] =  [sqrt{left ( x_{2}-x_{1} right )^{2}+left ( y_{2}-y_{1} right )^{2}}]

Formula for the magnitude of a vector

Note: The magnitude of a vector can never be negative this is because | | converts all the negatives to positive. Thus, we can say that the magnitude of a vector is always positive.

Direction of A Vector

The direction of a vector is nothing but it can be defined as the measurement of the angle which is made using the horizontal line. One of the methods to find the direction of any given vector AB is :

Tan α is equal to y/x; endpoint at 0.

Where the variable x denotes the change in horizontal line and the variable y denotes a  change in a vertical line.

Or we can write that : tan[alpha] = [frac{y_{1}-y_{0}}{x_{1}-x_{0}}]

Where, the variable (x₀, y₀) is known to be the initial point and  (x₁, y₁)is known to be the end point.

We may know a vector’s direction and magnitude, but want its x and y lengths (or we can say vice versa):

Magnitude and Direction

Important points to remember, these points given below will be helpful to solve problems:

The magnitude of a vector is always defined as the length of the vector. The magnitude of a vector is always denoted as ∥a∥.

For a two-dimensional vector a,  where a = (a₁, a₂ ), ||a|| = √a¹₁+a²₂

For a three-dimensional vector a, where a = (a₁, a₂, a₃), ||a|| = √a²₁+a²₂+a²₃

The formula for the magnitude of a vector is always  generalized to dimensions that are arbitrary, Now let’s see for example, if we have a four-dimensional vector namely a, where a =a = (a₁, a₂, a₃, a₄), ||a|| = √a²₁+a²₂+a²₃+a²₄

Solved Questions

Q1) What is the magnitude of the vector b = (2, 3) ?

Ans: We know the Magnitude of a vector formula,

 |b| = (√3²+4²) = √9+16 = √25= 5

Q2) What is the magnitude of the vector a = (6, 8) ?

Ans: We know the Magnitude of a vector formula,

|a| =  (√6²+8²) = √36+64 = √100 = 10

Q3) Find the magnitude of a 3d vector 2i + 3j + 4k.

Ans) We know, the magnitude of a 3d vector xi + yj + zk = √x²+y²+z²

Therefore, the magnitude of a 3d vector , that is 2i + 3j + 4k is equal to 

√x²+y²+z² = √(2)²+(3)²+(4)² = 5.38

Hence, the magnitude of a 3d vector given, 2i + 3j + 4k ≈ 5.38.

Note: The symbol ≈ denotes approximation.

What Exactly is a Vector and How is it Different from a Scalar?

A vector is any mathematical quantity that includes both magnitude and direction. This might be a bit confusing to understand, however, it is relatively simple once you get the hang of it. 

There are certain quantities in the universe that express different things. Mathematicians over the years have broadly classified mathematical quantities into two categories: scalar and vector quantities.

Scalar quantities are those that have only magnitude. Most of the numbers you would have dealt with in school would have been scalar quantities. 

Vector quantities, meanwhile, express both magnitude and direction, so they have two aspects to them. Since they cannot generally be used in the same mathematical equations as scalar quantities, there is a whole different branch of mathematics focused on the algebra of vectors. 

Let’s look at some common scalar quantities:

• Time

• Mass

• Volume

• Density

• Energy

• Speed

• Temperature

Now let’s look at some common vector quantities:

• Gravity

• Acceleration

• Force

• Displacement

• Thrust

• Velocity

• Angular Momentum

• Linear Momentum

To find out more about vectors, you can click here.

Studying Magnitude of Vectors with

On this page, you will find a detailed explanation and examples of vector quantities and using and finding the magnitude of a Vector. 

To study more about vectors, you can use the search bar or navigation menu at the top of the page to look for specific study materials that will aid you in learning about vectors. These resources have been created and compiled by experts in mathematics who have worked carefully with to provide excellent quality study materials for free. All of the resources available in PDF format on the website and app are downloadable for free if you have a account.

For more in-depth information, you can also book a one-on-one session with a maths teacher who will help you learn more about vectors.

Do you know what subtraction is? What if I give you 10 apples and your brother took 2 apples from you. How many apples will you be left with? Well, you will be left with 8 apples. How did I know that? Voila! Here is the magic. Math provides us with a lot of interesting concepts and one of them is called “subtraction”. Do you know what this is all about? Let’s read Riya’s story and understand how it works.

Story of Riya and Her Parrot Friend

Riya lives with her parrot friend “Kuku” and one day she offered her 7 green chilies out of which the parrot ate only 1. So, how many chilies are left with the parrot?

Here, we will determine the number of chilies left by using a very interesting concept.

Please remember that when we are asked “How many are left or How many remain?” is where we need to perform the subtraction operation. So, now we will explain Subtraction for Class 2?

Now, let us understand how many chilies are remaining for the above story.

So, we get the answer to our question as 6. Therefore, 6 chilies were left after the parrot ate only 1. 

Now, let us see another story on subtraction sums for Class 2 to understand subtraction better. Moving forward, we will also understand how subtraction is actually performed. Now, we need to see how the subtraction is done for one-digit numbers, for a two-digit number and a one-digit number, for 2 two-digit numbers, and so on.

How do we perform subtraction on numbers?

One day, my best friend Sonali was invited to an award ceremony. 55 guests were invited to the ceremony, out of which 23 turned up. How many guests could not turn up at the ceremony? Well, to find the number of guests who could not turn up at the ceremony, we need to understand one of the very interesting concepts called “place value”. 

Place Value Concept

Here, the sequence of placing each digit in numbers 55 and 23 is as follows:

Here, Ones place means 5 * 1, and 5 at tens place means 5 * 10 = 50

So, 50 + 5, we get 55.

Simialty, 3 at Ones place means 3 * 1 = 3 and 2 at tens place means 2 * 10 = 20, we get 20 + 3 = 23.

Now, let us learn how the subtraction is performed for the guests who didn’t turn up at the ceremony.

After we have got the result, we write as per the place value, which is as follows:

Below is the visual presentation of this table.

Subtraction of Three-Digit Numbers

Yesterday, my friends went to a very big library, and you know what, they came to know that 267 people visit this library every day, however, on average, only 143 people have a membership with this library. 

My friends told me this story, and I was very shocked to hear such a big number, but we wanted to find how many people didn’t have a membership. Well, for this, we will have to again use the “Place Value Concept”.

Placing the value 267:

2 hundreds = 2 *  100  =200

6 tens = 6 * 10 = 6

7 ones =7 * 1 = 7

Similarly, for 143:

1 hundred = 1 *  100  = 100

4 tens = 4 * 10 = 40

3 ones = 3 * 1 = 3

Now, let us find the number of non-paid membership people by subtraction.

So, subtract hundreds from hundreds, tens from tens, and so on. We can see that 124 persons were non-paid members of the library.

Isn’t it so simple? Yes, it is. But what if we have to arrange a few more players for our cricket team? Let’s take a look. Assume that in an 11-member team, seven of your teammates couldn’t play cricket, so how many more members would you need to make it to 11? Here, we will understand subtraction with borrowing for class 2.

Subtraction with Borrowing for Class 2

In the above example of 11 and 7, we have the following arrangement:

Now that we subtract 7 ones from 1 ones, we notice that 1 < 7, so what we do is take another 1 from 1 tens and make a single digit in the Ones Place to a two-digit.

Now, 11-7 gives us 4. So, 4 teammates are required to complete the cricket team.

A similar example for subtraction sums for class 2 with borrowing for three-digit numbers is as follows.

Story 2 on the Subtraction of Numbers with Borrowing

Some students were asked to participate in a drawing competition and they had to participate in a team. Here, if you look at the image below, some kids agreed to participate in a team.

Similarly, there were some more teams that participated in the same competition.

Now, we have to see what difference do we find in the number of participants in both the teams, i.e., Team 1 and Team 2. This we can do by performing subtraction between the number of participants shown below.

Now, we see that 11 =  1 tens and 1 ones

5 =  0 tens and 5 ones

So, let us borrow 1 from 11 and we get another form:

You can see that 11 participants were there in team 1 and 5 in team 2. So, the difference in the members was 6.

So, this was all about Class 2 Subtraction. We started with subtraction of two, and three-digit numbers and proceeded with the concept of borrowing.

Let us first discuss what is mean value theorem?The mean value theorem defines that for any given curve between two ending points, there should be a point at which the slope of the tangent to the curve is similar to the slope of the secant through its ending points.

If f(x) is a function, so that f(x) is continuous on the closed interval [p,q] and also differentiable on the open interval (p, q), then there is point r in (p, q) that is, p < r < q such that

f’(r) = f(q) -f(p)/ q-p

Lagrange’s Mean ValueTheorem or first mean value theorem is another name for the mean value theorem. This article discuss about Mean Value Theorem for Integrals, Mean ValueTheorem for Integrals problems and Cauchy Mean Value Theorem

Geometrical Representation of Mean Value Theorem 

The above mean value theorem graph represents the graph of the function f(x).

• Let us consider the point A as (a,f(a)) and point B as = (b,f(b))]

• The point C in the graph where the tangent passes through the curve is (cf(c)).

• The slope of the tangent line is similar to the secant line i.e.both the tangent line and the secant line are parallel to each other.

What is The Mean Value Theorem for Integrals?

For defining what is mean value theorem for integrals, let us consider

 f (θ) be continuous on [p, q]. 

F(θ) = [int_{P}^{Q}] f(z) dz

The Fundamental Theorem of Calculus indicates (θ) = f (θ). The Mean Value Theorem indicates  the inclusion of r ϵ (p,q) such that 

F(q)- F(p)/ q-p = F’(r)  or equivalently     F(q)-F(p) – F’(r)(q- p)

Which indicates

[int_{p}^{q}]  f(z)dz = f(r) (q-p)

This theorem is known as the First Mean Value Theorem for Integrals.The point f (r) is determined as  the average value of f (θ) on [p, q].

Along with the “First Mean Value Theorem for integrals”, there is also a “Second Mean Value Theorem for Integrals”

Let us learn about the second mean value theorem for integrals.

The number f (r) is known as the g(p)-weighted average of f (θ) on the interval [p, q].

Let f (θ) and g(θ) be continuous on [p,q]. Assume that g(θ) is positive, i.e. g (θ) ≥ 0 for any θ Є [p, q]. Then there includes r Є(p, q) such that

[int_{p}^{q}] f (z)g(z)dz = f(r) [int_{p}^{q}] g(z)dz

The number f (r) is called the g(θ)-weighted average of f (θ) on the interval [p, q].

The application of the second mean value theorem may define the Center of Mass of one-dimensional non-homogeneous objects such as a metal rod. If the object is homogeneous and placed on the x-axis from x = p to x = q, then its center of mass will be the midpoint.

 p+q/2

If the object is not homogeneous with  λ (θ) being the density function, then the total mass M is represented as 

M = [int_{p}^{q}]  λ (θ) dθ

The density-weighted average θc is stated through 

[int_{p}^{q}] θ  λ (θ)dθ = θc  [int_{p}^{q}]  θ  λ (θ)dθ = θc M,

Or equivalently

θc = 1/M [int_{p}^{q}] θ  λ(θ) dθ  

The point θc is the center of mass of the object.

Cauchy Mean Value Theorem

To define cauchy mean value theorem , we will consider two functions f and k functions  represented on [p,q] such both are continuous in closed interval [p,q] and also both are differentiable on open interval (p,q) k'(x) ≠ 0 for any x ∈ (p,q) then there includes at least one point r ∈ (p,q) such that

f’(r)/k’(r) = f(q) -f(p)/ k(q) -k(p)

If we consider k(x) = x for every x ∈ {p,q} in Cauchy’s mean value theorem, we get

f(q) -f(p)/q-p = f'(r) which is considered as Langrange’s mean value theorem. This is also known as extended mean value theorem.

Mean Value Theorem for Integral Problems

Here, you can see a mean value theorem for integrals problems with solutions.

1. A rod of length Z is placed on the x-axis from x = 0 to x = Z. Suppose that the density (x) of the rod is proportional to the distance from the x = 0 endpoint of the rod. Let us find the total mass M and the center of mass xc of the rod. We have (x) = gx, for some constant g> 0. We have

M [int_{z}^{0}] λ(x)dx = [int_{z}^{0}] k x dx = k  Z²/2 =  KZ²/2  

XC 1/M [int_{z}^{0}] λ(x)dx 1/M [int_{z}^{0}]kx² dx = 1/M k Z³/3 = 2/3Z

If the rod was homogeneous, then the center of mass would be placed at the middle point of the rod. Now it is closer to the ending point x = Z. This is not unexpected as there is more mass at this end.

Solved Example

1. For the function f(x) = ex , p= 0 and q = 1, then find the value of r in the mean value theorem.

Solution: f(q) –f(p)/q-p = f’r

= (eq )- (ep) / q-p = f’r

= e-1/1-0 = er

= c = log(e-1)

2. Assume that f(x) be continuous and increasing on (p,q). Compare,

f(p)(q-p) and [int_{p}^{q}] f(x)dx

Solution-The First
Mean Value Theorem for Integrals implies the existence of r Є (p,q) such that 

f(r)(q-p) = [int_{p}^{q}] f (x) dx

As f(x) is increasing, then f(p) ≤ f(r).Hence f (p)(q – p) ≤ f(r)(q-p) which implies

f(p)(q-p) ≤ [int_{p}^{q}] f (x) dx

Quiz Time 

1. The value of c for which mean value theorem f(x) for x in the interval (-1,1) is

1. ½

2. -½

3. 1

4. Non-existent in the interval.

2. Geometrically the mean value theorem assures that there is at least one point on the curve f(x) whose abscissa lies in (a,b) at which tangent is

1. Parallel to the x axis

2. Parallel to the u axis

3. Parallel to the line y = x

4. Parallel to the line joining the endpoints of the curve.

3. For the function f(θ) = [θ] ; θ Є [5,9], the mean value theorem 

1. Is applicable

2. ‘Not applicable as the function is continuous but not differentiable.

3. Not applicable because the function is differentiable but not continuous

4. Not applicable because the function is neither continuous nor differentiable

Facts

• The Mean Value Theorem was initially defined by famous Indian Mathematician and Astronomer Vatasseri Parameshvara Nambudiri. Later the theorem was proved by Augustin Louis Cauchy.

• Augustin Louis Cauchy proved the mean value theorem. The restricted form of this theorem was proved by Augustin Louis Cauchy in 1691. In 1823, Augustin Louis Cauchy stated and proved the modern form of mean value theorem.

Given any two points A and B, the line midpoint is point M that is located at halfway between points A and B.

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Observe that point M is equidistant from points A and B.

A line midpoint can only be found in a line segment. A line or ray cannot have a midpoint as the line is indefinite and can be extended indefinitely in both directions whereas a ray has only one end.

Let us now learn what is the midpoint of a line segment?

What is a Line Segment?

A line segment is a portion of a line that joins two different points.

It is the shortest distance between two points with a definite length that can be measured.

A line segment with two ending points XY is written as [overline{XY}].

Define Midpoint of a Line Segment?

A midpoint of a line segment is the point on a segment that bisects the segment into two congruent segments.

The midpoint of a line segment is the point on a segment that is at the same distance or halfway between the two ending points.

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The Midpoint of a Line Segment Formula

Let (a₁, b₁) and (a₂, b₂) be the ending point of the line segment. The midpoint formula of a line segment joining these two points is given as:

Midpoint Formula

Example:

Suppose we have two points 9 and 5 on a number line, the midpoint of a line will be calculated as:

[frac{9 + 5}{2} = frac{14}{2} = 7]

Let us learn to find the midpoint of a line segment joined by the ending points (-3, 3) and (5, 3).

Let (-3, 3) be the first endpoint, so a₁ = -3 and b₁ = 3. Similarly, Let (5, 3) be the second endpoint, so a₂ = 5 and b₂ = 3. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.

Using the midpoint formula, we get:

[(frac{a_{1} + a_{2}}{2}, frac{b_{1} + b_{2}}{2}) = (frac{-3 + 5}{2}, frac{3 + 3}{2}) = (frac{2}{2}, frac{6}{2} = (1, 3)]

Midpoint Theorem

The statement of the midpoint theorem says that the line segment joining midpoints of the two sides of a triangle is parallel to the third side of a triangle and equal to the half of it. Consider the △ABC given below. Let points D and E be the midpoints of AB and AC. Suppose that you join the points D and E.

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The midpoint theorem says that the line DE will be parallel to the BC and equal to exactly half of BC.

How to Find the Midpoint of a Line Segment?

The midpoint of a line segment can be determined using these two different methods. These are:

1. Counting Method.

2. Using the midpoint of a line segment formula.

Counting Method

If the line segment is vertical or horizontal, you can find the midpoint of a line segment by dividing the length of a line segment by 2 and counting that value from either of the two ending points.

Midpoint Formula Method

The midpoint of a line segment that lies diagonally across the coordinate axis can be found using the midpoint formula.

The midpoint (x,y) of the line segments with ending point A (x₁, y₁) and B(x₂, y₂) can be found using the following midpoint formula.

[(x, y) = (a, b) = (frac{x_{1} + x_{2}}{2}, frac{y_{1} + y_{2}}{2})]

Example:

Find the midpoint of segment AB, where coordinates of point A and B are (-3, 3) (1, 4) respectively.

Solution:

Using the midpoint formula, we get

[(frac{-3 + 1}{2}, frac{-3 + 4}{2}) = (frac{-2}{2}, frac{1}{2}) = (-1, frac{1}{2})]

Hence, the midpoint of segment AB is (-1, ½).

The Midpoint of a Line Segment Example with Solutions

1. The diameter of a circle given below has two ending points (2, 3) and (-6, 5). Determine the coordinates of the centre of the circle given below.

Solution:

The centre of a circle divides the diameter into two equal parts. Hence, the coordinates of the centre are the midpoints of a circle.

Let (2, 3) be the first endpoint, so a₁ = 2 and  b₁ = 3. Similarly, Let (-6, 5) be the second endpoint, so a₂ = -6 and b₂ = 5. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.

Using the midpoint formula, we get:

[(frac{a_{1} + a_{2}}{2}, frac{b_{1} + b_{2}}{2}) = (frac{2 + (-6)}{2}, frac{-3 + 3}{2}) = (frac{-4}{2}, frac{2}{2}) = (-2, 1)]

Hence, the coordinates of the centre of a circle are (-2, 1).

2. If (3, -2) is the midpoint of the line joining the points (1, x) and (5, 7). Find the value of x.

Solution:

Let (1, h) be the first endpoint, so a₁ = 1 and  b₁ = h. Similarly, Let (5, 7) be the second endpoint, so a₂ = 5 and b₂ = 7. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.

Using the midpoint formula, we get:

[(frac{a_{1} + a_{2}}{2}, frac{b_{1} + b_{2}}{2}) = (3, -2)]

[(frac{1 + 5}{2}, frac{h + 7}{2}) = (3, -2)]

[frac{7 + h}{2} = -2 = 7 + h = -4]

[h= -11]

Hence, the value of h is -11.

What is a Multiple?

• A multiple can be defined as a number obtained by multiplying a number by an integer. 

• The multiples of a whole number are obtained by calculating the product of any of the counting numbers and that of the whole numbers.

•  For example, to find the multiples of the number 7, we will multiply 7 by 1, 7 by 2, 7 by 3, etc. The multiples are the product of this multiplication. Any number that can be represented as in the form of 7n, where n is considered as an integer and a multiple of  9.

•  Any number in Mathematics is a multiple of itself (Number x 1 = Number).

•  Any number in maths is a multiple of 1 (1 x Number = Number).

•  Zero is always multiple of every number (0 x number = 0).

What is a Common Multiple?

A common multiple is defined as a number that is a multiple of two or more numbers in a given set. Let us understand through an example.

Let us take two numbers 9 and 27

The common multiples of 9 are 9,18,27,36,45,54,63,81

The common multiples of 27 are 27,54,81

Here, we can see 9 and 27 have common multiples like 27, 54, and 81.

So to find the multiples of the number 7, simply multiply this number by a number of the set of natural numbers as many times as we want. See below how to do this for the number 7, let’s find the multiples of 7.

7 x 0 = 0 so we can say 0 is a multiple of 7.

7 x 1 = 7 so we can say that 7 is a multiple of 7.

7 x 2 = 14 so we can say that 14 is a multiple of 7.

7 x 3 = 21 so we can say that 21 is a multiple of 7.

7 x 4 = 28 so we can say that 28 is a multiple of 7.

How to Find Multiples Using Multiplication?

We can find multiples by multiplying the number with any integer.

How to Find Multiple Using Division?

As we know, both multiplication and division are inverse operations. It implies both are linked with each other. We can find out using division whether a given number is multiple of another number or not.

Some of the Examples are Mentioned below:

1. 21 ÷ 7 = 3, so 21 can be divided evenly by 7 and is also a multiple of 7. Since, 3 * 7 = 21.

2. 49 ÷ 7 = 7, so 49 can be divided by 7 and is also a multiple of 7. Since, 7 * 7 = 49.

Some of the Multiples of 7 are

All the numbers that can be easily divided by the number 7 or a product of 7 are considered as the multiple of 7. The multiple is also known as factors and it relies on their usage in the equation also.

What is a Multiple of 7?

Any number that can be represented as in the form of 7n, where n is considered as an integer and a multiple of 7. So, if two values x and y are there, we can say that y is a multiple of x if y = nx for some integer n.

For Example, 70, 84, 91, and 700 are All Multiples of 7.

These values are known as multiple as we received these values by adding and subtracting the original value several times.

List of the Multiples of 7

Solved Examples

Question 1. Write any 5 multiples of the number 9.

Ans. Here are the five multiples of 9 – 9,18,27,36,45

Question 2. Write any 6 multiples of each of the following:

1. 15

2. 12

Ans.

1. The six multiples of 15  are- 15,30,45,60,75,90

2. The six multiples of 12 are- 12,24,36,48,60,72

This is all about the multiples of 7 and how to find them using the given methods. Focus on the concept so that you can easily find out the multiples of other numbers without any hassle.