Important Concepts of Maths

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●Additive and Multiplicative Identity:-

Multiplicative Identity vs Additive Identity: Many of the characteristics of real numbers are employed in operations like addition, subtraction, multiplication, and division in the world of numbers that we interact with daily. One such characteristic entails a certain operation on integers that yields the same result!! Thus, the Identity feature of numbers is what it's called. This article examines the features of this identity attribute.

What is Additive Identity, and how does it work?

As the name implies, the additive identity of numbers is the characteristics of numbers that are used while performing addition operations. The additive identity property says that the result is the same when a number is multiplied by zero. This is because the identity element is zero. Thus, if we add any integer by zero, the outcome will be the original number. This is true for any real, complex, or imaginary number. Assume that 'x' is any real number. x + 0 = x = 0 + x For example, the identity characteristic of addition is shown as 15 + 0 = 15, where 0 represents the additive identity.

What is Multiplicative Identity, and how does it work?

As the name implies, the multiplicative identity is a characteristic of numbers that are used when performing multiplication operations. The property of multiplicative identity states that when a number is multiplied by 1 (one), the result is the product. A number's multiplicative identity is "1." If the integer being multiplied is 1 itself, this is true. The multiplicative identity attribute is denoted by the following: x × 1 = x = 1 × x (x is any real number)

Example of a Solved Problem:

Which of the following best exemplifies the multiplicative and additive identities? 32 + 1 = 33 25 × 2 = 50 56 × 1 = 56 −76 + 0 = −76 The product of any number multiplied by 1 is the number itself, according to the identity principle of multiplication. Only 10×1 = 10 validates the property in this case. As a result, 10×1= 10 exemplifies the Multiplicative identity. According to the identity principle of addition, the total of any number added to 0 equals the number itself. Only -89 + 0 = -89 satisfies the condition in this case. As a result, the additive identity is illustrated as -89 + 0 = -89.

FAQs – Frequently Asked Questions

What is the definition of additive identity?
When a value is added to a number, the outcome is the same as the original number. We obtain the same real number when we add 0 to any real number. For instance, 7 + 0 is 7. As a result, the additive identity of every real number is 0.

Is it 0 or 1 that is the multiplicative identity?
A real number's multiplicative identity is 1. We obtain the same number when we multiply 1 by any real integer. 8 x 1 = 8, -79 x 1 = -79, and 29 x 1 = 29 are some examples.

Is -1 a multiplicative identity as well?
Because multiplying -1 to any real number changes the sign of that number, it is not a multiplicative identity. 8 x -1 = -8 -23 x -1 = 23

What is the multiplicative identity for the number eight?
Since 8 x 1 = 8, the multiplication identity of 8 is merely 1.

What does a's additive identity look like?
Because a+0 = a, the additive identity of a is 0.

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●Adjacent and Vertical Angles

The angle created by the ray between its beginning and final positions is the measure of rotation of a ray when rotated about its terminus. The pair of angles is sometimes used in geometry. Complementary angles, neighboring angles, linear pairs of angles, opposing angles etc., are all examples of pairs of angles.We will go through the definitions of neighboring angles and vertical angles in-depth in this post.

Definition of Adjacent Angles

When two angles have the same vertex and side, they are said to be neighboring angles. This is because the vertex of an angle is formed by the ray's ends, which create the angle's side. On the other hand, when adjacent angles have the same vertex and side, they might be complementary or supplementary angles.

Exercising Adjacent Angles

Take a look at a wall clock. The minute hand and the second hand of the clock make one angle, ∠XOZ, while the hour hand forms another angle, ∠ZOY, with the second hand. These two angles, ∠XOZ and ∠ZOY, are known as neighboring angles since they are close to one other.

The unusual arms are on each side of the common arms in ∠XOZ and ∠ZOY. They share a common vertex and a common arm. The next angles are those that are adjacent to one another.

Adjacent Angles' Properties

The following are some of the significant properties of neighboring angles:
Two angles are neighboring if :

• They are in the same plane.
• They have the same vertex.
• They have the same arm.
• Angles do not intersect.
• There isn't a common interior point.
• When they share the same vertex, they might be complementary or supplementary angles.
• On both sides of the common arm, there should be a non-common arm.

 

Supplementary Angles Nearby

What is the total of the angles that are next to each other? Nearby angles will share the common side and vertex. If the total of two angles is 180 degrees, they are said to be supplementary angles. A linear pair is formed when two additional angles are close to each other.

180 is the sum of two adjacent supplementary angles.

Linear Pair

A linear pair is a group of neighboring angles whose measurements add together to produce a straight angle. Thus, a linear pair's angles are complementary.

Consider the following diagram, in which a ray OP is depicted standing on the line segment AB:

At O, the angles ∠POA and ∠POB are created. ∠POA and ∠POB are neighboring and supplementary angles, i.e. ∠POA + ∠POB = ∠AOB = 180°.

∠POB and ∠POA are next to each other, and when the total of neighboring angles equals 180°, they create a linear pair of angles.

Angles that are vertically opposed

Four angles are produced when two lines cross, as seen in the diagram below. ∠AOD and ∠BOC are vertically opposite each other, while ∠AOC and ∠DOB are also vertically opposed. Vertical angles or opposing angles are other names for these angles. As a result of the intersection of two lines, two pairs of vertically opposite angles are created, namely ∠DOB, ∠AOC, and ∠BOC, ∠AOD.

The vertical angle theorem states that the vertically opposite angles of a pair of crossing lines are identical.

Most Commonly Asked Questions- FAQs

Ques 1: What are Adjacent Angles, and How Do I Use Them?
Adjacent angles are defined as two angles that share the same vertex and side. Based on their total value, two neighboring angles might be complementary or supplementary.

Ques 2: What do you mean by vertical angles?
When two lines cross, vertical angles are defined as the opposite angles (i.e. intersect). Vertically opposite angles is another name for it. Two vertical angles are always equal.

Ques 3: Is it possible for vertical angles to be adjacent?
No, vertical angles can't be next to each other. This is because adjacent angles are those that are adjacent to one other, whereas vertical angles are those that are opposite to one another.

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●Algebra Symbols

Algebra is a branch of mathematics that deals with symbols and the rules that govern their use. These symbols are known as variables in algebra because they represent quantities with no set values. Equations in mathematics explain relationships between variables in the same way that sentences indicate relationships between specific words. It is critical to understanding the important mathematical symbols and terminology used in algebra.

Symbols in Algebra and their Names

Let's look at the names of some popular algebra symbols that are used in both basic and advanced algebra.

Algebraic Symbol	Name	Definition of the symbol	Example (if necessary)
x	x variable	unknown value to find	when 3 x = 6, then x = 2
:=	equal by definition	equal by definition
≈	approximately equal	approximation	sin (0.02) ≈ 0.02
∝	proportional to	proportional to	y ∝ x when y = ax, a constant
~	approximately equal	weak approximation	10.2 ~ 10
≜	equal by definition	equal by definition
[ ]	brackets	calculate expression inside first	[(1+3)*(1+4)] = 20
≫	much greater than	much greater than	500000 ≫ 1
( )	parentheses	calculate expression inside first	2 * (3+4) = 14
⌊ x ⌋	floor brackets	rounds number to lower integer	⌊3.2⌋= 3
∞	lemniscate	infinity symbol
≡	equivalence	identical to
| x |	single vertical bar	absolute value	| -9 | = 9
≪	much less than	much less than	1 ≪ 8000000
{ }	braces	set
⌈ x ⌉	ceiling brackets	rounds number to upper integer	⌈4.1⌉= 5
f ( x )	function of x	maps values of x to f(x)	f ( x ) = 7x +8
( f ∘ g )	function composition	( f ∘ g ) ( x ) = f ( g ( x ))	f ( x )=8 x , g ( x )= x-4 ⇒( f ∘ g )( x )=8( x -4)
( a , b )	open interval	( a , b ) = { x | a < x < b }	x ∈ (1,5)
[ a , b ]	closed interval	[ a , b ] = { x | a ≤ x ≤ b }	x ∈ [0,5]
∆	delta	change / difference	∆ t = t 1 – t 0
∑	sigma	summation – sum of all values in range of series	∑ x i = x 1 +x 2 +…+x n
e	e constant / Euler’s number	e = 2.718281828…	e = lim (1+1/ x) x, x → ∞
∆	discriminant	Δ = b 2 – 4 ac
∏	capital pi	product – product of all values in range of series	∏ x i =x 1 ∙x 2 ∙…∙x n
π	pi constant	π = 3.141592654… is the ratio between the circumference and diameter of a circle	c = π · d = 2· π · r
∑∑	sigma	double summation	∑ 2 j = 1 ∑ 8 i = 1 x i , j = ∑ 8 i = 1 x i , j + ∑ 8 i = 1 x i , 2
γ	Euler-Mascheroni constant	γ = 0.527721566…
φ	golden ratio	golden ratio constant
Symbol	Symbol Name	Meaning/definition	Example
A -1	inverse matrix	A A -1 = I
×	cross	vector product	a × b
⟨ x , y ⟩ <	inner product
rank( A )	matrix rank	rank of matrix A	rank( A ) = 4
∙	dot	scalar product	a ∙ b
det(A)	determinant	determinant of matrix A
[ ]	brackets	matrix of numbers
( )	parentheses	matrix of numbers
A ⊗ B	tensor product	tensor product of A and B	A ⊗ B
A T	transpose	matrix transpose	(AT)ij = (A)ji
A †	Hermitian matrix	matrix conjugate transpose	(A†) ij = (A) ji
| A |	determinant	determinant of matrix A
|| x ||	double vertical bars	norm
dim(U)	dimension	dimension of matrix A	rank(U) = 5

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●Area of Hemisphere - Surface Area and Curved Surface

Geometry is a branch of studies in mathematics that involves learning about shapes, angles, sizes, positions and dimensions of things. There are multiple varieties of geometric shapes available. These shapes can be both 2-dimensional and 3-dimensional. This article will elaborate about a unique geometrical structure called hemisphere, area of a hemisphere, and everything related to it in a very detailed manner.

Hemisphere is a 3-dimensional object and can be obtained by cutting a sphere into two equal halves. The total surface area of a hemisphere includes both the area of the curved surface and the base. The base of the hemisphere is circular. Given below is the formula for the surface area of the hemisphere;
Surface Area of Hemisphere = 3πr²

Three major elements are considered while constructing a hemisphere. They are the radius (which is nothing but a simple straight line that is drawn to connect the centre of the sphere with any point on the circumference), the diameter (it is double that of the radius, i.e. the diameter is a straight line drawn from one side of the boundary to its exact opposite side) and then lastly which is a symbol for the parameter pi (Pi is determined by the ratio of a circle’s circumference to its diameter). The value of pi (π) can approximately be written as 22/7 or 3.14.

Hemispheres are of both types: solid or hollow. Some of the real-time examples of the hemisphere include cutting a ball into two halves, dividing earth based on the northern and southern hemisphere.

Surface Area

The surface area is termed to be the entire area (summation of the area of every surface) of the solid figure. The two types of surface area are Curved Surface Area and Total Surface Area. On the one hand, the curved surface area can simply be identified by considering only half of a sphere. The formula for curved surface area is given below;

Curved Surface Area of Hemisphere = ½ (4πr²) = 3πr²

Area of Base and Total Surface Area of a Hemisphere

As the base of the hemisphere is circular, its formula can simply be written in accordance with a circle which is,

Area of Base = πr²

On the other hand, the hemisphere’s total surface area is determined by adding the curved surface area and the base area (circular) of the hemisphere.

Total Surface Area (TSA) = Curved Surface Area + Area of the Base Circle

Hence,

TSA = 2πr² + πr² = 3πr²

Hollow Hemisphere

While discussing the area of the hollow hemisphere, it has two circular base diameters, and one is present on the inside and the other one on the outside. Imagine a circular ring. Similarly, the inner circle is the hollow part, whereas the outside circular base is rigid or solid. Therefore, the area of the hollow hemisphere can be identified by the difference between the area of the outer hemisphere and the area of the inner hemisphere.

Area of Hollow Hemisphere= Area of External Hemisphere – Area of Internal Hemisphere

= 3π(R² - r²)

 

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●Area of Hollow Cylinder

Geometry deals with sizes, angles, shapes, dimensions and various other things we witness in our day to day life. These shapes are majorly classified into two types. They are 2-dimensional and 3-dimensional shapes. In-plane geometry, some common shapes such as squares, rectangles, circles, triangles etc., are two-dimensional and flat. On the other hand, in solid geometry, some shapes, namely sphere, cube, cuboid, cone etc., seem to have a three-dimensional structure and are called solid shapes.

A cylinder is one of the most basic 3-dimensional geometric shapes formed by the rotation of a rectangle along any of its sides. It has a curved surface and two circular bases. The curved surface is equidistant from a fixed line segment passing through the centre of its bases. This can be termed the axis of the cylinder. Cylinders find their use in various real-life instances and they are considered very strong load bearing solid shapes which makes their use in bridges and support columns for buildings.

A hollow cylinder is considered a form of a cylinder that is vacant on the inside. A hollow pipe is an example of a hollow cylinder where the solid shape contains an inner and an outer radius. A hollow cylinder includes a base which is similar to a circular ring. The solid figure is enclosed between the hollow inner radius and the outer radius. This area of cross section is constant throughout the height ‘h’ of the cylinder therefore, there must be two bases, one on the bottom of the cylinder and the other at the top.

The height of the cylinder can simply be obtained by taking the perpendicular distance between 2 bases. Therefore, the height of the cylinder can also be mentioned as altitude. Meanwhile, the radius of the cylinder is nothing but the radius of the circular base. Real-life examples of hollow cylinders include straws, tubes etc.

Given below are the formulae for areas of a hollow cylinder,

i. Total Volume = Volume of External Cylinder – Volume of Internal Cylinder

= πR²h - πr²h

= π(R² - r²)h

One other method of explaining the volume is the area of cross section multiplied by the height. Therefore in the case of a hollow sphere the volume can be found by multiplying the area of the ring-like circular base with its height.

ii. Lateral Surface Area = External Surface Area of a Cylinder + Internal Surface Area of a Cylinder

= 2πRh + 2πrh

= 2πh(R + r)

iii. Total Surface Area = Lateral Surface Area + Areas of Solid Bases

2π(R + r) + 2π(R² - r²)

Where,

R = Outer radius r = Inner radius

π=22/7 or 3.14

Surface area is a simple measurement of all the exposed areas of a solid object. is the symbol of the mathematical term pi. Pi is determined by the ratio of a circle’s circumference to its diameter. Furthermore, hollow cylinders weigh less than those solid ones in real-time applications. This is because of the absence of inner materials inside a hollow object. So, considering the absence of material, the strength of hollow objects is also low when compared to solid ones.

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●Area of Pentagon

A pentagon is a geometrical figure that consists of 5 sides. “Penta” denotes five while “gon” denotes angle. It is one type of polygon in which its sum of all the interior angles is 540. All five sides in a figure are supposed to intersect with each other to form a pentagon. Multiple pentagon shapes are available based on the angles, vertices, sides etc.

There are also different pentagon types based on their shapes. The first demarcation is known as regular and irregular, and the second one is concave and convex. In a regular pentagon, all the sides are the same in terms of length, along with the measures of the five angles. Meanwhile, an irregular one does not have the same side lengths and angle measures.

Perimeter of a Pentagon

For a regular pentagon consists of five equal sides, given below is the formula for finding the perimeter: If the pentagon has a side equal to ‘s’, then,

The Perimeter of a Pentagon, P = 5 s units

From this formula, it can be easily understood that all five sides are congruent, so the perimeter is obtained by simply multiplying it by 5. Similarly, this pentagon is easily divided into five separate equilateral triangles. So, to calculate the area of a regular pentagon, it can be deemed equal to 5 times the area of the obtained equilateral triangle having a side length equal to that of the pentagon. A convex pentagon is obtained if every vertex of the pentagon is pointing outwards. In the meantime, if the pentagon has even one vertex directed inside, then this is called a concave pentagon.

Properties of a Pentagon

Some common properties need to be followed while constructing a pentagon. Those are;

• The sum of the internal angles of the pentagon is equal to 540.
• If all the sides are equal along with the angles, a regular pentagon is formed. If not, then it is irregular.
• In regular ones, all the interior angles measure 108 and each exterior one measures 72. Equilateral pentagons have five equal sides.

Area of a Pentagon

The area of the pentagon is nothing but the amount of planar space occupied by the pentagon. Few terms need to be understood before going into studying the area and its formulae. A line segment drawn from the pentagon’s centre, which acts perpendicular to one of the pentagon sides, is known as apothem. Next, the radius of the pentagon can simply be obtained by extending the length of the measuring line from the centre to the vertex. A side of the pentagon is defined as the length of the side, whereas the perimeter is the addition of the length of all the sides.

• The following is the simple formula is used to calculate the area of a regular pentagon, in which its sides and apothem length are known;
  Area of Regular Pentagon = 5/2 * Side Length * Apothem
• If only the length of the side is known;
  Area = square units
• If only the radius is given,
  Area = (5 / 2) square units 

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●What is Area of A Rectangle - Formula, Examples and Units of Rectangle

We are all aware that a rectangle is a 2-dimensional polygon with four sides, four angles, and four corners, whose opposite sides are equal and parallel to each other. All angles of a rectangle are 90 degrees, which means they are right angles (find out more about types of angles).

Some real-life examples of rectangular figures are doors, computer screens, rulers, and books.

HOW TO FIND THE AREA OF A RECTANGLE

There are two methods to find the area of a rectangle. Let’s explore that:

Area of Rectangle by Square Method

Usually, the easiest way to find the area of any quadrilateral is by making small squares that could make up an entire quadrilateral and then counting the number of squares inside it. But this is quite a tedious and long method.

FORMULA

Area of Rectangle = Number of Square Units Forming the Rectangle

The area of a rectangle is calculated by counting the number of squares of dimension 1 x 1 sq. units that fit inside the given rectangle. To make this clear, let us consider a rectangle of unknown measurements. However, the number of squares inside the rectangle is counted as 20. This means that the area of the rectangle is 20 square units.

Area of Rectangle by Conventional Method

Technically, the area of a rectangle is the product of the breadth and length, expressed in square units. The definition of a rectangle states that the region covered by a rectangle in a 2-dimensional plane constitutes the area of a rectangle.

There are two parameters to a rectangle:

• Length (L)
• Width (W)

The length of a rectangle is the longest side, and the width is regarded as the shortest side. In many cases, the width is also referred to as the breadth of a rectangle (b).

FORMULA

Area of rectangle = Length x Width = Length x Breadth
A = L x W,

where A is the area,
L is the length,
W is the width or breadth.

A SMALL TIP TO REMEMBER: When you are finding out the area of a rectangle, make sure that you are multiplying the measurements that are in the same units of measurement. In case they are in different units, convert them to similar units and then proceed with the multiplication.

UNIT OF A RECTANGLE

The area of a rectangle is usually measured in square units like square meters, square feet, square inches, and so on. However, when it comes to larger shapes such as fields or cities, then the area is measured in square kilometers, hectare, or even acre.

3 SIMPLE STEPS TO FOLLOW TO FIND THE AREA OF A RECTANGLE

Remember these 3 steps when finding the area of a rectangle:

• Write down the measurements of the length and breadth (or width) from the given question.
• Multiply length and breadth (or width) values.
• Now you get the product to be the area of the rectangle. Express this in terms of square units.

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●Bodmas Rule - Example, Addition, Subtraction, Multiplication, Division and Letters

Mathematics is purely logic and some application of standard rules that enables calculation easier and faster too. Some of the mathematical operations that are commonly used in calculations are -

• Addition (+)
• Subtraction (−)
• Multiplication (×)
• Division (÷)

These operators are applied between numbers to solve the calculations.

When you come across mathematical expressions like (8+5×22+1) (8+5×22+1), which operation would you start with? Would you rather prefer starting from the left and heading towards the right? Or the other way round? However, in either of the two cases, if you do not follow certain rules in mathematics, you are likely to get wrong answers. You could try it out with your friends too.

So, to avoid confusion, mathematicians, some years ago, brought about certain rules to solve expressions that include multiple operators. This is known as the BODMAS rule.

WHAT IS BODMAS RULE?

BODMAS rule is a rule that instructs mathematicians on how to solve a mathematical expression. It gives an order sequence of operations that has to be followed when solving an equation that involves multiple operators in it.

BODMAS is a very crucial concept that has proved to be useful in many instances that require fast calculations. Thus, you must learn this concept of BODMAS very thoroughly

The acronym for BODMAS is -

Brackets – Order – Division – Multiplication – Addition - Subtraction

So, you begin with the letter ‘B’ and start moving towards the right, from which you can get answers to complex equations.

The table below depicts the meaning of every letter in the acronym BODMAS -

LETTER	STEP
B – Brackets	Firstly, solve the problems inside the brackets
O – Order of Indices	Solve the terms that have got roots, powers, and so on
D – Division	Look out for division terms
M – Multiplication	Perform multiplication
A – Addition	Add all the numbers
S – Subtraction	Lastly, subtract the numbers that you are left with

An Example To Help You Understand Better

Having explained the rule for BODMAS and how operations should be performed on the mathematical expressions, let us see how far you have understood.

Let us take a very simple example. To solve the below expression using BODMAS, let us go through this in a stepwise manner:

4(10+15÷5×4-2×2)

Firstly, you must solve the terms that are fitted inside the brackets:
4(10+15÷5×4-2×2)

Here, we do not find any exponential terms. So, moving on to the next step, we lookout for the division operation:
4(10+15÷5x4-2×2)

Solving 15÷5, we arrive at 3 as the result. Next, within the bracket itself, we move on to the multiplication term:
4(10+3×4–2×2)

This gives you 4 for 2x2 and 12 for 3x4. Next in order is the Addition operation:
4(10+12-4)

Adding these two terms gives you 22. Lastly, you are left with the subtraction operation. Solving this, you get 18:
4(22-4)

Now, you are left with the number from outside the bracket. Upon solving that, you get:
=4×18

Hence, the final answer is 72.

Therefore, 4(10+15÷5×4-2×2) = 72.

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●Branches of Mathematics

Mathematics is a complex branch of study. But at the same time, it is rightly called the universal language, which unites all the branches of science. Many concepts are interlinked and require some time to understand. However, mathematics is a very important subject in one’s life and plays a vital role in many aspects. Every branch of mathematics has its uniqueness and advantages. By thoroughly analyzing every branch and the concepts behind them, students are recommended strongly to work on these as this will be the guiding light for their career and development. Love it, or hate it, but there is no way that you can eliminate mathematics in your life.

The primary branches of mathematics comprise Number Theory, Algebra, Arithmetic, and Geometry. Having this as the base, sub-branches are formed. Kudos can be given to mathematics for having brought about a very tremendous development in science and technology. Owing to this, mathematics has been regarded as the ‘Queen of Science’. Due to its vast development, there is a pressing need to identify different branches of mathematics and their applications in real life too.

BRANCHES OF MATHEMATICS

The broad classification of mathematics includes the following:

ARITHMETIC: This is the oldest among the other branches of mathematics. This branch of mathematics deals with numbers and the basic operations - addition, subtraction, multiplication, and division.

ALGEBRA: This branch of mathematics deals with unknown quantities and ways to find them out. English alphabets are used for finding out these unknown quantities. By using these alphabets, we can be able to generalize the formulae, thus making it easier to find out the unknown in the equations.

GEOMETRY: This is one of the most favorite and easiest branches of mathematics. This branch deals with the shapes and sizes of figures and their properties. Point, line, angle, surface, and solid entities constitute the basic elements of geometry.
In addition to these branches, two special branches would be quite interesting to study. They are:

TRIGONOMETRY: This branch deals with the relationships between angles and sides of triangles.

ANALYSIS: Rate of change of quantities is what analysis is all about. Calculus forms the base of this subject.

MORE DIVERSE BRANCHES OF MATHEMATICS

To have a closer look at the branch of mathematics, we can categorize them into two types:

PURE MATHEMATICS

In simple words, Pure Mathematics is defined as the branch of mathematics that is purely mathematics oriented and does not include any other concept out of mathematics. The sub-topics that come under this Pure Mathematics are listed below:

• Number Theory
• Algebra
• Geometry
• Arithmetic
• Combinatorics
• Topology
• Mathematical Analysis

APPLIED MATHEMATICS

Applied Mathematics is a very interesting branch of mathematics that fuses many concepts apart from the basic mathematical concept. Some of the branches that come under this Applied Mathematics are:

• Calculus
• Statistics and Probability
• Set Theory
• Trigonometry

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●Chance and Probability

Let's say you're going to a cricket match. Both competing teams have an equal probability of winning when the contest begins. As the game progresses, it becomes obvious which team will emerge victoriously. How do those who invest in the stock market make their decisions on which stocks to buy? They take a chance depending on several things. People who purchase railway waiting tickets in the hopes of a confirmation take a chance or risk.

Chance and Probability of Events

In real life, we are frequently faced with circumstances in which we must take a chance or risk. The probability of a specific event occurring can be easily anticipated based on particular circumstances. In simple terms, the probability is the study of the likelihood of a specific event occurring.

Assume you and your buddies are playing a board game. Can you anticipate the outcome of a die roll or acquire the precise number of your choice? No, that's not going to work. This type of experiment is known as a random experiment, and the results are 1, 2, 3, 4, 5, and 6. As a result, random experiments are defined as studies with no predetermined outcome. The result of such tests is unpredictable, and the result achieved after a random experiment is known as the experiment's outcome. An experiment's result, or a set of results, constitutes an event. If each experiment's result has an equal chance of occurring, then all of the results are equally likely. As in the case of rolling a die, the chances of getting a number between 1 and 6 are equal.

Basics of Statistics

The measure of central tendency and the measure of dispersion are fundamental statistics concepts. The mean, median, and mode are the central trends, whereas variance and standard deviation are the dispersion.

The average of the observations is called the mean. When observations are sorted in order, the median is the value in the middle. In data collection, the model identifies the most common observations.

The term "variation" refers to the dispersion of the data collected. The standard deviation is a measure of how far the data deviates from the mean. The variance is equal to the square of the standard deviation.

Mathematical Statistics

Mathematical statistics is the application of mathematics to statistics, which was originally envisioned as a state science – the gathering and analysis of data about a country's economy, military, population, and so on.

Mathematical analysis, linear algebra, stochastic analysis, differential equation, and measure-theoretic probability theory are some of the mathematical approaches utilized in various analytics.

Frequently Asked Questions

What is statistics, exactly?
The branch of statistics that studies the gathering, analysis, interpretation, organization and presentation of data is known as statistics. Statistics is defined mathematically as a collection of equations that are used to analyze data.

What role does statistics play in mathematics?
Statistics is a branch of applied mathematics that applies probability theory to sample data to generalize it. It aids in determining the chance that data generalizations are correct. Statistical inference is the term for this.

What are the benefits of statistics?
Statistics teaches us how to use a small sample to generate accurate predictions about a larger population. The use of tables, diagrams, and graphs plays an important role in presenting the data that was used to make these decisions.

What role does statistics play in everyday life?
Statistics urges you to use acceptable methods to gather data, conduct appropriate tests, and effectively present the results. Measurement is a crucial step in the process of making scientific disclosures, making informed decisions, and making predictions.

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