Category: Maths Notes PPT

The dimension of an object/shape is basically a topological measure of the length, width, or height of its covering properties. Dimension is a measure of size extended in a specific direction. In simple terms, it is the number of coordinates required to specify a point on the object. For instance, a rectangle is a 2-dimensional figure, while a cube is 3-dimensional. The letter D in 2D, 3D denotes dimensions. The dimension of an object is sometimes also called its “dimensionality”.

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Types of Dimensions

Most commonly, there are 4 types of dimensions as given below:

1. Zero Dimension: A point consists of no extent in any direction, so it has no dimension.

2. One Dimension: A line has how many dimensions? A line has only one dimension, which is its length.

3. Two-Dimension: A piece of paper consists of two dimensions i.e. length and width.

4. Three-Dimension: Dices consist of three dimensions i.e., length, width, and height. Dimension implies the measurement of the size or distance of an object or space in only one direction.

Formula of Dimension

How many dimensions does a point have? Dimension is a measure of a line or point stretched out in one direction and we are aware that every shape around us consists of some dimensions. The theory of dimension does not contain any particular dimension formula. The value and the number of dimensions differ depending upon their size and shape. The following table depicts the dimension formulas of circles and polygons.

Dimension Formulas

Dimensions can be used to identify the unknown values of shapes using the help of formulas. For instance, if the perimeter of a square is 12 inches, we can determine its length with the help of the dimension formula: Perimeter = 6 × Length. Substituting the value of perimeter in the formula, we obtain 12 = 6 × Length. Thus, Length = 2 inches.

Different units are used to measure different quantities like length, tempurature, area etc. The following table shows the dimensions that are usually used for units like length, area, temperature, volume and weight.

Dimensions of Units

How Many Dimensions Does a Plane Have

Planes are 2 dimensional. However, they can exist in 3-dimensional spaces as well. The plane contains two dimensions: length and width. But since the plane is limitlessly large, the length and width are unable to be measured. Just as a line is defined as a 1-dimensional figure or one-dimensional shape having two points, a plane is defined by three points.

Dimensional Analysis

Dimensional analysis means to determine the physical quantities and evaluate the association between them. It helps us to study the nature of different objects like straightness and flatness mathematically.

Dimensionless Quantities

The quantities that contain no dimension are known as dimensionless quantities. Such quantities consist of a certain value. For instance, the constant Pi – π is a dimensionless quantity since it has a fixed value. Following are the types of Dimensionless quantities:

1. Dimensionless Quantities With Units: These include angular displacement –Joule’s constant – joule/calorie, radian etc.

2. Dimensionless Quantities Without Units: These include pure numbers, π, sin θ, cos θ, tan θ, e, etc.

Dimensional Formula

The expression displaying the powers to which the fundamental units are to be raised in order to get one unit of a derived quantity is known as the dimensional formula of that quantity.

If Q is the unit of a derived quantity denoted by the letter Q = MaLbTc, then MaLbTc is known as the dimensional formula and the exponents a, b and, c, are what we call the dimensions.

Fun Facts

• It is believed that the Universe consists of 10 dimensions; however, fortunately, the human eye can only apprehend three.

• The meaning of dimension differs in mathematics and
  physics.

• Dimension is the point or value, measuring the object.

Distance refers to a mathematical quantity that shows how far the two points lie from each other. Indeed, distance is one of the essential mathematical quantities. It plays a significant role in advanced mathematics and physics. It helps to determine the velocity of a moving object, magnitude, and direction of gravitational and electrical forces, and it helps with signal processing too. In mathematics, the distance formula is used for finding the distance between two points in a coordinate plane. The distance between the two points formula can be evaluated when you know the coordinates of the two points in a plane. By inserting those points in the formula, you can quickly find the distance between two points. In this article, you can learn about the distance between two points in 3D, its formula, and examples.  

Distance Between Two Points Formula

Typically, in 2D space, each point in the space gets qualified by two parameters: x-coordinate and y-coordinate. You require a pair of the coordinate axis to locate the exact position of a point in a plan. The combination of x and y coordinates gets expressed in the form of an ordered pair such as, (x, y). So, the coordinates of a point, say M, can get expressed as, M (x, y). That ordered pair (x, y) gives you the coordinate of the point. 

Before you learn to find the distance between two points in 3D, you must know the basic distance formula, which is as below. 

Considering two points M [(x_{1}, y_{1})] and N [(x_{2}, y_{2} )] on the given coordinate axis, you can find the distance between them using the formula:

Steps to find the distance between two points:

• First, you need to take coordinates of two points like [(x_{1}, y_{1}) and N (x_{2}, y_{2} )].

• Then, you have to use the distance formula, which is [sqrt{(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}}].

• Now, you have to calculate the vertical and horizontal distance between the two points. The horizontal distance [(x_{2} – x_{1})] represents the points on the x-axis, and the vertical distance [(y_{2} – y_{1})] denotes the points on the y-axis.  

• Next, you have to square both the values obtained from [(x_{2} – x_{1})] and [(y_{2} – y_{1})]. 

• Now, all you need to do is add both the values, which look like, [(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}].

• Finally, you need to find the square root of the obtained value.

• The value you get in the end is the distance between two points in the coordinate plane.  

Coordinates of A Point

The distance between the two pints can be figured out if one has the coordinates of the two points in the given XY plane. In Euclidean geometry, we often come across points that are positioned in the plane. These points are known to be defined by the coordinates along the x and y-axis. These coordinates are a pair of values that precisely define the location of the point in the given coordinate plane. 

Distance Between the Two Points With the Help of the Pythagorean Theorem

Taking the following scenario; 

A girl is walking towards the north till 30 meters from where she takes a turn to the east and walks for an additional 40 meters. How can we calculate the shortest between the initial and the final place?

Let us say that the initial point is A while the final point is C. we know that the distance between AB is 30 meters and BC is 40 meters. 

Therefore, the shortest distance between the given points A and C is AC. to calculate this distance, we take the help of the Pythagorean theorem, as a result, we get the answer as 50m.

Thus, we have obtained the distance between the starting point to the ending point. In a similar fashion, the distance between two points in a given coordinate plane can also be derived with the help of the Pythagorean theorem or even the right angle theorem formula. 

Distance Between Two Points in 3D

The following study can get extended to find out the distance between two points in space. We can determine the distance between two points in 3D using a formula as derived below. 

For now, refer to fig. 1. Here, point [ P (x_{1}, y_{1}, z_{1})]  and [ Q (x_{2}, y_{2}, z_{2}) ] refer to a system of rectangular axes OX, OY, and OZ.

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From the points P and Q, you need to draw planes parallel to the coordinate plane. Then, you get a rectangular parallelepiped with PQ as the diagonal. 

As you can see in the figure, ∠PAQ is forming a right angle. It enables us to apply the Pythagoras theorem in triangle PAQ. 

So, now you get [PQ^{2} = PA^{2} + AQ^{2} …(1) ]

Also note that, in triangle ANQ, ∠ANQ is a right angle. Now, you need to apply the Pythagoras theorem to ΔANQ as well.  

Now, you obtain [AQ^{2} = AN^{2} + NQ^{2} …(2)]

From equation (1) and equation (2), you get [PQ^{2} = PA^{2} + NQ^{2} +AN^{2}].

As you know the coordinates of the points P and Q, PA = [(y_{2} – y_{1})], AN = [(x_{2} – x_{1})] and NQ = [(z_{2} – z_{1})].

Hence, [PQ^{2} = (x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2} + (z_{2} – z_{1})^{2}].

Finally, the formula to obtain the distance between two points in 3D is –

[PQ = sqrt{(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2} + (z_{2} – z_{1})^{2}}]

That formula can give you the distance between two points P (x1, y1, z1) and Q (x2, y2, z2) in 3D.

Also note that the distance of any point Q (x, y, z) in space from origin O (0, 0, 0), can get expressed as, [OQ = sqrt{x^{2} + y^{2} + z^{2}}].

Applications

The distance formula has numerous applications in our day-to-day life. For instance, the pilot in the plane can use the distance formula to calculate the distance between their plane and any other plane. The first step involved is the finding of the coordinate and then applying the distance formula to get the desired results. 

Note

It is to be kept in mind that the distance between two points can never have a negative value. This is because;

• Distance is a term that is used to represent how far two points are from one another. And as it is a physical quantity, it can never be negative. 

• The use of the distance formula conveys that it represents its outcome as the square root of the addition of the two given positive numbers. As we know that the addition of two positive numbers always gives a positive number.

• Even in a case where the distance between the two points equals zero, it is still a nonnegative integer. And thus, the distance between the two points cannot be negative. 

Solved Examples

1. Find the distance between two points given by A (6, 4, -3) and B (2, -8, 3).

Answer: Here, we need to use the distance formula to find the distance between points A and B. 

You have, [AB = sqrt{(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2} + (z_{2} – z_{1})^{2}}] 

[AB = sqrt{(6 – 2)^{2} + (4 – (-8)^{2} + (-3 -3)^{2}}]

[AB = sqrt{16 + 144 + 36}]

Finally, AB = 14; so the distance between points A and B is 14.

A dot plot, also known as a dot diagram, is a statistical chart consisting of data points on a relatively simple scale. Dot plots are considered as one of the easiest statistical plots, used for small data sets. Dot plots are used for highlighting clusters, gaps, skews in distribution, and outliers. When working with datasets larger than 20 or 30, it is better to use another statistical chart such as a histogram, as the dot plot will be less manageable.

Constructing a Dot Diagram

While constructing a dot diagram, data points are stacked in a column over a category. The height of the column denotes the frequency of observation in a given category. The categories of dot diagram given below are the numbers from 0 to 9, where 0, 1, and 5 occur three times, 2 occurs 7 times, 3 and 4 occurs six times, 6 occurs 4 times, 7 and 9 occurs 9 times, and 8 occurs two times.

The dot diagram given above represents the total number of data points and how the data points are distributed among certain categories.

Analyzing Dot Plot

A dot plot or dot diagram consists of a horizontal scale (a number line) on which dots are arranged to represent the numerical values of the data set. If the values in a data set repeat the dots are accumulated at that location. One dot is plotted for each repetition. We can say that dot diagrams represent the distribution of data.

Analyzing the dot plot of a given data set can be done easily in comparison to a long list of numbers. By looking at the dot plot diagram given below, you can see that dots are evenly spread with a peak at around 8 letters. The centre of the graph is between 6 and 7 letters. The dots are spread from 3 to 9 letters.

With a dot plot, you can easily count the number of times each category, or number is repeated. A dot plot is an optimum way of organizing numbers and values.

In a dot plot, you can easily count the number of times each category, or number is repeated. A dot plot is an optimum way of organizing numbers and values.

Dot Plot Interpretation

As you know, a dot plot is made up of dots plotted on a graph. Considering this, let us learn dot plot interpretation.

• Each dot plotted in a dot graph denotes a specific number of observations from a given set of data (unless specified, assume that each dot represents one observation. If a dot represents more than one observation, that should be written on the plot.

• The dots are arranged in a column over category, so that height of the column represents the absolute or relative frequency of observation in the category.

• The pattern of data formed in a dot plot can be clearly stated in terms of symmetry and skewness only if the categories are quantitative, but if the categories are qualitative, a dot plot is not suitable to describe those terms.

In comparison to other types of graphs, dot plots are often used to plot frequency counts within a small number of categories, generally a small set of data.

Types of Dot Plot

There are two types of dot plot namely:

1. Wilkinson Dot Plot

2. Cleveland Dot Plot

Wilkinson Dot Plot

The Wilkinson Dot Plot represents the distribution of continuous data points, like a histogram. For example, the tempurature in a room throughout the day.

A histogram is shown in Wilkinson Dot Plot but represents the individual data points like P, Q, R, S shown below rather than bins. These points can consider any values like Mathematics marks of four friends, or the number of minutes taken by four kids to finish their lunch.

Cleveland Dot Plot

Cleveland dot plot is a good alternative to a simple bar chart, specifically if you have more than a few items.

It does not take much for a graph to look cluttered. In the same amount of space, many more values can be plotted in a dot plot and it is easier to read as well.

Although this type of dot plot is similar to a bar chart but uses location rather than the duration of the data encoding values.

Just like the height of the bar chart represents the number of items, the position of dots in the graph represents the number of items for that category or bin.

For example, the Cleveland dot plot given below represents the trend in literacy rate in Post independent India.

For example,

Dot Plot Examples

1. The dot plot given below represents the number and types of pizzas ordered for the party.  By analyzing the dot plot, determine the number of pizzas ordered for the party, How many of these pizzas were pepperoni or cheese?

Solution: 

To determine the number of pizzas ordered for the party, add all of the data points together on a dot plot. In total, 22 pizzas were ordered for the party.

To find the numbers of pizzas ordered for a certain category, sum up all the points in those categories. There are 7 pepperoni pizzas and 5 cheese pizzas. In total 12 pizzas are of cheese or pepperoni.

2. The dot plot given below represents the height of each kid at Mr. George’s daycare. Each dot represents the height of different kids. Determine the height of the shortest kid.

Solution:

The height range of kids on the axis is from 83 – 86.

The 3 dots are plotted on bin – 83 which implies that there are 3 kids whose height is 83.

Therefore, the height of the shortest kid is 83.

Talking about a part, it is a fraction of the whole substance. It is a fragment or a portion of something which we can define as something less than the entire object. Similarly, a part has been defined by different terms, including a fraction, element, share, portion, piece, and many more. When it is about defining a part, there comes a new term called equal parts. The article below will understand what a part is, how parts make up a new substance, its basic definition, and some fun facts. 

What is the Definition of Part?

Let us define part with the help of a basic example of a circle. When we draw a line joining one point on the circle to the other point, it will be divided into parts. These may be equal or unequal. If these parts are unequal, then they will form a big and a small part. Look in the figure below:

In the diagram above, AB is the line drawn from one point to another point of the circle. With this line, the circle is divided into two uneven parts of maths. It clearly explains the part definition. 

I am considering the same circle. Now we will learn how it is divided into equal parts. Let the line pass through the centre of the circle that is a diameter. Look for the below diagram:

In the diagram above, a line is drawn passing through the centre of the circle. Thus it will divide the circle into two equal parts. 

It is even easy to divide a circle into different uniforms and even parts. The fact is, angle subtended by arcs must be equal. As if the arcs subtend 120° in the centre of the circle, then the circle is divided into three equal parts. 

Counting and Comparing Uniform Parts

Before we say parts are equal, we need to look into various facts, including looking and comparing each part with others. When you find parts in the same shape and size, they are equal and identical. 

One basic example that we generally study before diving into the concept of even parts is folding a sheet of paper. Let us take a sheet of paper and divide it from the centre by drawing a straight line. Make a fold along this line, the sheet will be divided, and it is to define parts in even shape and size. 

Now try the same activity and explain it to your friends by dividing this paper into non-uniform parts of maths. 

Now try to find what common problems your friend has faced. The fact is, students fail to recognise if the parts are equal or of identical size. It could happen if we place one shape in the inverted form to another. However, it is their mental ability to catch the right results. 

More Examples for Part Definition

1: Count and Tell if Parts are Uniform and Equal.

In the above picture, there are three squares. In the first example, a square is divided into 2 Halves, a second into 3 thirds, and a third into 4 fourths.

2: According to the part definition of uniform part, check if the below diagram gives you equal shapes or not.

In the above picture, there are three figures (A), (B), and (C). In figure (A), there are four equal squares. In figure (B), 6 equal triangles make one hexagon, and in figure (C), there are two unequal parts as the division is uneven. 

3: Count the equal shapes and find the numbers to find the total.

According to the above question, the first example has 1+2= 3 apples. 

Second example has 2 + 2 = 4 apples.

Third example has 3 + 1 = 4 apples. 

Fun Facts For Parts of Maths

• You divide if a shape has equal divisions by looking at its parts and comparing if these are equal.

• If a shape has three uniform parts, it is called thirds.

• To name the number of equal shapes, count them, and give the total count ending with “s” (valid till number four).  

• When we divide an object into two parts, there is a possibility of equal or unequal. 

• In the concept of a fraction as a whole, the numerator must be divisible by the denominator. 

If your mother gives two chapatis to your elder brother, and one to you, is it unfair? NO, right? Because he is older than you, he needs more food. After having food, both of your stomachs filled completely. This is what we call ‘Equivalent’. 

An equivalent is basically the numbers or quantities which are the same or logically the same. ‘Logically same’ means that numbers might be different but eventually the same part of a whole. To know more about the term ‘Equivalent’ in Mathematics, read the complete article.

Equivalent- Meaning in Mathematics

The term “equivalent” in math refers to two meanings, numbers, or quantities that are the same. The equivalence of two such quantities shall be denoted by a bar over an equivalent symbol or Equivalent Sign. It also means a logical equivalence between two values or a set of quantities. Equivalence is similar but more universal than equality. If two sets of equations have the same solutions, we might consider them similar, but they are not identical.

How to Define Equivalent in Mathematics?

In mathematics, Equivalent meanings are used in two different ways. First, within the framework of a particular mathematical theory (An Equivalent Example would be Euclidean geometry), a notion (e.g. ellipse or minimal surface) may have more than one meaning. In the context of a given mathematical form, these concepts are identical (Euclidean space, in this case). Second, there could be a mathematical framework. In the prior example, the equivalence of two definitions implies that a mathematical entity follows one definition if and only if it meets the other definition. In the above case, the sense of equivalence is more complex since the structure is more abstract than the entity. Several different artifacts can follow the same structure.

Representation of Equivalent Symbol

If A ⇒ B and B ⇒ A (i.e., A ⇒ B & B ⇒ A, where ⇒ denotes implies), then A and B are said to be equal, a relationship that is symbolically written as A = B in this job. 

• Equivalent Expressions: An algebraic expression is an expression composed of variables, coefficients, constants, and mathematical functions, such as addition, subtraction, multiplication, and division. Generally, if two objects are the same, they’re considered identical. Similarly, in mathematics, equivalent expressions are expressions that are the same, even though the expression looks different. Look at 3 × 3 + 1 and 5 × 2 expressions. They’re both equivalents to 10. That is, they are equivalent expressions.

How to find that two mathematical expressions are equivalent expressions or not? 

Ans. In general, to prove that two mathematical expressions are equivalent we keep them as equal. Later, we evaluate the left-hand side expression and right-hand side expression, and if both sides are equal then the given expressions are equivalent expressions. (To solve the basic mathematical functions we use the BODMAS–Bracket OF Division Multiple Addition Subtraction rule.)

For Example: Check whether 3 × 9 + 5 × 2 is equivalent to 7 × 3 + 4 × 4 or not.

Solution: To check the equivalency of both the equations, we put both of them equal to each other.

3 × 9 + 5 × 2 = 7 × 3 + 4 × 4

27 + 10 = 21 + 16

37 = 37

Therefore, L.H.S. = R.H.S.

So, the given expressions are equivalent to each other. 

Remember: We also need to give attention to the units obtained on both sides. 

• Equivalent Fractions:  Equivalent fractions say that when all results are the same fraction after simplification, two or more than two fractions are said to be equal. Let’s assume, after simplifying these fractions, a/b and c/d are two fractions, both result in identical fractions, say e/f, then they are equal to each other.

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Equivalent fractions are basically the same part of a whole. For example: [frac{1}{2}, frac{2}{4}, frac{3}{6}…,frac{36}{72}…,]are the example of equivalent fractions. If we solve and simplify each term, it results [frac{1}{2}] .

By multiplying the numerator and denominator with the same number, you can find equivalent fractions of any given fraction. And, by dividing the numerator and denominator with the same number, you can also find the equivalent fraction.

• Equal and Equivalent Sets: Equivalent sets should have the same cardinality, to be equivalent. This suggests that one-to-one correspondence can be made between the elements of both sets. Here, one-to-one correspondence implies that an element remains in set B with each element in set A before the sets are exhausted. What is an Equal set? In the case of equal sets only if each element of set A is also an element of set B will the two sets A and B be equal. They are often said to be equivalent if two sets are each other’s sub-sets. This shall be expressed by:

A = B 

A ⊂ B and B ⊂ A ⇔ A = B

(Images will be uploaded soon)

The first image represents an equal set and the second image represents an equivalent set.

In simple words, if A={2, 4, 6, 8, 10}, B={@, #, %, &, !}, C={1, 2, 3, 4, 5, 6}, D={2, 4, 6, 8, 10}, then set A, set B and set D are said to be equivalent sets, because all three sets have 5 elements. Therefore equivalent sets are those sets that have an equal number of elements, irrespective of what they are. 

Remember: All equal sets are equivalent sets. Also, all null sets (sets that have no elements) are equivalent sets.

• Equivalent Ratios: Two ratios that express the same relationship between numbers are identical ratios or identical ratios. We multiply or divide the terms of the given ratio by the same non-zero number to get a ratio equal to a specified ratio. Equivalent ratios have varying numbers, but the same relationship is reflected.

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How to find that two ratios are equivalent or not? 

Two or more than two ratios can be compared with each other. In order to find whether they are equivalent or not, first, we convert them into like fractions. After converting, denominators of al
l the fractions become equal and if the numerators of all the fractions also become equal, the fractions are said to be equivalent fractions. 

Also, if the two fractions a/b and c/d become equivalent, then the four quantities a, b, c, and d are said to be in proportion. It can be written as a:b::c:d.

Solved Examples

1. Are 4/5 and 16/20 equal ratios?

Solution: If we simplify 16/20 we get 4/5, as 16 and 20 both when divided by 4 gives 4 and 5 respectively.

Therefore the above ratios are equivalent.

2. Identify a fraction that is equivalent to 7/9.

Solution: In Equivalent Calculations Maths, any equivalent fraction can be obtained by multiplying and dividing the same number into the fraction. Therefore let us multiply 2 to both numerator and denominator.

On multiplying 2 to both numerator and denominator, we get,

(7 × 2)/(9 × 2) = 14/18.

3. Check whether 7 × 6 + 66 ÷ 11 – 5 × 2 is equivalent to 7 × 3 + 24 ÷ 2 + 9 × 3 or not.

Solution: To check equivalency of both the equations, we put both of them equal to each other.

 7 × 6 + 66 ÷ 11 – 5 × 2 = 7 × 3 + 24 ÷ 2 + 9 × 3

42 + 6 – 10  = 21 + 12 + 27

38 ≠ 60

Therefore, L.H.S. ≠ R.H.S.

So, the given expressions are not equivalent to each other.

Any number which is exactly divisible by 2 is called an even number. i.e. if a number when divided by 2 leaves no remainder, then the number is called an even number.

If the number when divided by 2 leaves a remainder then the number is an odd number.

Let’s see some examples and check if the number is even or not.

1. 2 – if we divide the given number 2 by 2 i.e 2 ÷ 2 then the remainder is 0. So the given number 2 is an even number.

2. 10 – again if we divide the given number 10 by 2 i.e. 10 ÷ 2 then the remainder is 0. So the given number 10 is an even number.

3. 5 – if we divide 5 by 2 i.e. 5 ÷ 2 then remainder comes out as 1, so the given number 5 is not an even number. It is an odd number.

4. 11 – check this by the same method. You will find 11 ÷ 2 gives remainder 1. So 11 is not an even number. It is an odd number.

So, examples of Even numbers are :–  2, 4, 6, 8, 10, 12, 14, 42, 50, 98, 100 etc. 

How to Check if a Given Number is Even or Not Without Dividing it By 2.

A number can be checked if it is even or not without dividing it by 2. This can be done by just checking the units place of a given number. If the units place of a given number contains digits 0, 2, 4, 6 and 8. Then the given number is an even number.

• Unit digit of Even number contains either of the following digits:  0, 2, 4, 6 or 8.

• Unit digit of Odd number contains either of the following digits: 1, 3, 5, 7 or 9.

By this method, you can check if the given large number is Even or Odd. Let’s consider two numbers- (1) 72580 and (2) 2311. You can now clearly check that in first number 72580 the unit digit is 0. So, 72580 is even. And for the second number 2311, the unit digit is 1. So, 2311 is odd.

List of Even numbers up to 100. 

There are 50 even numbers between 1 to 100. They are given in the below table.

Properties of Even numbers

1. Property of Addition of Even Numbers

1. Even + Even = Even

Eg.  12 + 14 = 26. 

       10 + 8 = 18.

2. Even + odd = odd

Eg.  12 + 11 = 23. 

       10 + 7 = 17.

2. Property of Subtraction of Even Numbers

a ) Even – Even = Even

Eg.  18 – 14 = 4. 

   10 – 4 = 6.

b ) Even – odd = odd

Eg.  12 – 11 = 1. 

22. – 7 = 15.

3. Property of Multiplication of Even Numbers

Even x Even = Even

Eg.  8 x 2 = 16. 

12 x 4 = 48.

Even x odd = Even

 Eg.  2 x 3 = 6. 

 8 x 7 = 56.

Solved Examples

Q1.  Choose even numbers from the given set of numbers. 11, 8, 91, 07, 31, 143, 500, 252, 581, 1794.

Ans. 

Even numbers are 8, 500, 252 and 1794.

Q2. Write first 10 even numbers.

Ans. 

2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Q3. What is the smallest even number?

Ans. 

2 is the smallest even number.

Q4. How can you check a given is Even or Odd?

Ans.

Unit digit of Even number is either 0, 2, 4, 6 or 8. While, Unit digit of Odd number is either 1, 3, 5, 7 or 9.   

Q5. What will be the resulting number (Even or Odd) for the following operations

1. 67347 + 9846

2. 58731 x 90754

3. 11235 – 10110

4. 23421 x 12895

5. 99981 + 85789    

Ans.

A) odd B) Even C) odd D) odd E) Even