# Maths

Category: #### Pre-Algebra

All the positive whole numbers, their negative counterparts together with zero form the set of integers. Eg: -25, -9, 0, 6, 109 all are integers.
A variable denotes a number that varies. Variables are usually represented by letters in English alphabet. Eg: If ‘x’ apples and ‘y’ bananas are there, how many fruits are there in all? “x+y”. Here x+y is an expression. Algebraic Expressions combine numbers and variables with at least one arithmetic operation.
Not every number is a whole number like 15, 0, -8 etc. Decimals help us to represent fractional parts, or numbers in between whole numbers.
All the rational and irrational numbers together form the set of real numbers. These can be represented on the number line. If a number is not a real number, it is imaginary. Examples of real numbers are -27, 0, 6/15, 16.2779 etc. Square root of a negative number is an example for an imaginary number.
An exponent tells us how many times a particular number is multiplied by itself. It is a way of representing repeated multiplication. In other words, an exponent is a positive or negative number placed above and to the right of another number. It expresses the power to which the number is to be raised or lowered. In 4^3, 3 is the power and 4 is called the base.
Equation is an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. In this situation, variables are also known as unknowns and the values which satisfy the equality are known as solutions.
In an equation, each letter stands for a missing number. To solve an equation, you find the values of the missing numbers.
Multi-step equation is one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.
Math talks not only about equality! Inequality comes into picture when something is bigger or smaller. When solving for inequality involves negative numbers it may be a little bit tricky!
A fraction is a part of a whole. Normally we use fractions for measuring, rather than for counting. The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. Eg: In the fraction 3/4, the numerator, ‘3’ tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that ‘4’ equal parts make up a whole. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents.
Any number that can be represented as a ratio of two integers is called a rational number. The word rational has its relation with ‘ratio’. You get a rational number when you divide one integer with another [not being zero].
A “ratio” is just a comparison between two different numbers. Someone can look at a group of people, count noses, and refer to the “ratio of men to women” in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20. Expressing the ratio of men to women as “15 to 20” is expressing the ratio in words. There are two other notations for this “15 to 20” ratio; (1) Odds notation, as 15 : 20 (2) Fractional notation, as 15/20. Proportions are built from ratios. They have equivalent ratios. If I can drive 240 miles in 3 hours, I will be able to cover 320 miles in 4 hours! We say, 240 to 3 and 320 to 4 are in proportion.
Probability is the measure of the likeliness that an event will occur. It is a number between 0 and 1. The higher the probability of an event, the more certain we are that the event will occur. A simple example is the toss of a fair (unbiased) coin. Since the two outcomes are equally probable, the probability of “heads” equals the probability of “tails”, so the probability is 1/2 (or 50%) chance of either “heads” or “tails”.
Percent tells you how many parts per hundred. The word comes from the Latin phrase ‘per centum’, which means ‘per hundred’. In Math, we use the symbol % for percent. A Percent can also be expressed as a Decimal or a Fraction. Shops advertise discounts on products. These discounts are normally given in percentages. For example, “Up to 50% off marked prices”
A function is basically a relation between a set of inputs and outputs, satisfying certain conditions. When plotted on a graph, the linear functions will give us straight lines. We can also call it as a polynomial with degree 1. A linear function has an independent variable and a dependent variable, and will have a constant slope as well. A linear function has the following form y = f(x) = a + bx.
Here the independent variable is ‘x’ and the dependent variable is ‘y’. ’a‘ is the constant term or the ‘y’ intercept. It is the value of the dependent variable when x = 0. ’b‘ is the coefficient of the independent variable. It is also known as the slope and gives the rate of change of the dependent variable.
A non-linear function is a function which is not a linear function. Hence if you plot the graph of a non-linear function it will not be a straight line. The degree of the polynomial representing the function will be more than one, and its slope will be varying!
Measurement is nothing but finding a number that shows the size or quantity or amount of something. There are two main systems of measurement: Metric and US Standard. Measurements are most commonly made in the SI [International System of Units abbreviated for Systeme Internationale] system, which contains seven fundamental units: kilogram, metre, candela, second, ampere, kelvin, and mole. We use units of measure so frequently in daily life that they hardly think about what they are doing. A motorist goes to the gas station and fills
13 gallons (a measure of volume) into an automobile. To pay for the gas, the motorist uses dollars—another unit of measure, economic rather than scientific—in the form of paper money, a debit card, or a credit card.
Area tells us how much space on a two dimensional surface. In other words it is the amount of two-dimensional space taken up by an object. The concept has application in many fields like Science, Architecture, Engineering, Farming etc.
Volume is the amount of space a three-dimensional object occupies. It is nothing but the capacity. From measuring liquids to assessing drinking amounts, volume is necessary. Keep in mind that volume has nothing to do with weight.

If one of the angles of a triangle measures 90 degree, we call it a right angled triangle, or a right triangle. Sides of such a triangle satisfy the Pythagorean theorem: a^2+b^2=c^2, where the largest side is ‘c’ and is called the hypotenuse. ‘a’ and ‘b’ represent the lengths of the other two sides.

We can find many relationships between two or more angles, if certain conditions are met. Examples include complementary angles, supplementary angles, congruent angles etc.

Transformation refers to the movement of objects in the coordinate plane. In case there is change in the shape/ size, we refer to the original shape as the object and the transformed shape as the image. There are various ways to transform an object. Different types of transformation are translation, rotation, reflection, and dilation. You can relate to translation when you think of pushing a box from one side of the room to another. An example of a rotation in real life is Earth’s rotation around its axis. When you look into the mirror, you see your own reflection. When an eye doctor dilates your eye it enlarges the pupil, letting the eye doctor examine it closely. All these can be connected to the transformations in Math.

Polynomials are expressions that have constants, variables & exponents, which are combined using addition, subtraction, & multiplication. Each piece of the polynomial, or each part that is being added, is called a “term”.
Eg: (1) X^2 + 5X = 7

(2) 3X+1=0.
The exponents in a polynomial can only be positive, whole numbers. We cannot have fractional powers and variables as denominators in a polynomial. Hence ‘x’ to the power of -5, ‘x’ to the power of 1/3, and 7 over ‘x’ to the power of 2 are not polynomials.

#### Algebra-1

A variable is something that varies! It can have a variety of values. In Math it is denoted by a symbol that represents the unknown number. A pattern shows a particular trend! Through graphs we can represent functions, patterns etc., visually.
An equation is a mathematical way of looking at relationship between concepts or items. These concepts or items are the variables, and in an equation, each letter stands for a missing number. By solving an equation, we find the values of these missing numbers.

Solving inequalities is similar to solving equations. In both, we do most of the same things, but in the case of inequality we should pay special attention to the “direction” of inequality too.

A function defines certain relationships and tries to give it a mathematical form. The relationship is between terms or objects in one set with those in another. If for each term in the first set- the domain – there is one and only one corresponding term in the other set- range, we call that relation, a function. We can graph functions exactly like how we graph equations. It gives a pictorial representation of the function.

A system of linear equations comprises two or more linear equations. The solution of a linear system is the ordered pair that is a solution to all equations in the system.
One way of solving a linear system is by graphing. The solution to the system will then be the point at which the two equations intersect. We can solve linear equations by elimination method, cross multiplication method, and by substitution method too.

“Roots” (or “radicals”) are the “opposite” operation of applying exponents; you can “undo” a power with a radical. If you square 2, you get 4, and if you “take the square root of 4”, you get 2; if you square 3, you get 9, and if you “take the square root of 9”, you get 3. The radical expression has a radicand inside the root, and the power will be shown outside the radical sign.

A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. So, we can call it as “Polynomial Fraction”!

A system of equations is a collection of two or more equations with the same set of unknowns or variables. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. The equations in the system can be linear or non-linear.

A polynomial equation in a single variable, where the highest power of the variable is 2, is called a quadratic equation. There will be two “roots” or “solutions” for a quadratic equation. The solutions of the quadratic equation ax^2 + bx + c = 0 correspond to the roots of the function f(x) = ax^2 + bx + c, since they are the values of ‘x’ for which f(x) = 0. The name Quadratic comes from “quad” meaning square because the variable gets squared (like x2). It is also called an “Equation of Degree 2”. We can solve a quadratic equation using any of the three methods: (1) Factoring (2) Completing the square (3) Using the quadratic formula.

An exponent tells us how many times a particular number is multiplied by itself. We can consider function as a relation between two sets of numbers where, in the first set all the numbers are unique or different, and every number in the first set has at least one corresponding number in the second. There are functions where the base is a variable and exponent is a number. Eg: f(x) = x^2. However, exponential functions are functions where the base is a number and the power is an exponent. Eg: f(x) = 2^x

Matrix is a rectangular array of numbers. Matrices are widely used in solving problems in areas like Electronics, Robotics, Genetics, Linear Programming etc. The size of a matrix is defined by the number of rows and columns that it contains. A matrix with ‘m’ rows and ‘n’ columns are called an m × n matrix or m-by-n matrix, while ‘m’ and ‘n’ are called its dimensions.

Determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. The determinant tells us things about the matrix that are useful in systems of linear equations and helps us to find the inverse of a matrix, which is useful in calculus and more.

A set is a collection of objects which follow a common rule. The objects of the set are called elements. Eg: The set of even numbers, the set of movies I watched in the previous year, the set of girls in the class etc.

This is a technique widely used in “optimization”. It is the optimization of an outcome based on some constraints. It is trying to answer “What is the best?” and is very helpful in the decision making situations. The process could be maximizing or minimizing a function, based on a set of constraints, which could be equalities or inequalities. The general process for solving linear programming exercises is to graph the inequalities (called the “constraints”) to form a walled-off area on the x,y-plane (called the “feasibility region”).

#### Algebra-2

A quadratic function is a function in the form f(x) = ax^2 + bx + c, where a, b, and c are numbers with ‘a’ not equal to zero. Note that the highest power of the variable in a quadratic equation is always 2. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in “width” or “steepness”, but they all have the same basic “U” shape. If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.
A radical function is an expression that uses a “root” such as square root, cube root etc. The three terms that are closely associated are the radicand, the radical symbol and the degree of the radical. A radical function contains a radical expression with the independent variable (usually x) in the radicand. Radical equations where the radical is a square root are called square root functions.

A sequence, or a progression, is an ordered list of numbers. Each of the numbers is called a term or an element of the sequence. The sum of the terms of a sequence is called a series. While some sequences are simply random values, other sequences have a definite pattern that is used to arrive at the sequence’s terms. Two such sequences are the arithmetic and geometric sequences. If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. If the pattern is that of multiplying with a fixed number, it is a geometric sequence.

If ‘b’is any number such that b>0 and b≠1, and x>0, then y=logb(x) is called the logarithmic function. The function is read “log base b of x” [or, logarithm of x to base b]. In this definition logb(x) is called the logarithmic form and b^y = x is called the exponential form.

Periodic function is a function that repeats its values in regular intervals or periods. In other words, a function f(x) is said to be periodic with period p if f(x)=f(x+np)
for n=1, 2, 3,……It has a graph that repeats itself identically over and over as it is followed from left to right. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians.

Identity is an equation that is always true. Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle.

We get a conic section, when a plane intersects a cone. By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola. In the special case when the plain touches the vertex, it will be a point, line, or two intersecting lines.

What makes it complex is that it is the combination of a real and an imaginary number. A complex number is a number that can be expressed in the form a + bi, where ‘a’ and ‘b’ are real numbers and’ i’ is the imaginary unit, and it is the square root of  −1. In this expression, ‘a’ is the real part and ‘b’ is the imaginary part of the complex number. Remember, imaginary numbers when squared will give a negative result.

Statistics is concerned with the collection, analysis, interpretation, presentation, and organization of data. In other words, it is the science of analyzing data and of measuring, controlling and reporting uncertainties. The practice of statistics utilizes data from some population in order to describe it meaningfully, to draw conclusions from it, and make informed decisions. It finds a wide range of applications across different areas like Economics, Education, Engineering, Astronomy, Genetics, Biology, Psychology, Marketing, sports, and many others.

#### Geometry

If you want to paint the walls of your house, how will you determine the amount of paint required? Probably you will come across “surface area” here. The surface area of a solid object is a measure of the total area that the surface of the object occupies.
When you fill up your vehicle, how many gallons of gasoline are required? You look at the volume of the fuel tank of your vehicle! Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre.

A circle is the locus of all points equidistant from a central point. We can call it as a line, in the form of a closed loop where each point on it is at a fixed distance from a centre point.

Quadrilateral means “four sides” (quad means four, lateral means side). In Math, a quadrilateral is a polygon having four sides and four angles. Some of the common quadrilaterals are rectangle, square, parallelogram, rhombus etc.

Lines that are in the same plane, the same distance apart, and never touch are said to be parallel lines. No matter how far you extend them they never meet. If two lines in the same plane intersect each other at 90 degree, we say they are perpendicular to each other.

Two triangles are congruent or identical if they are of the same shape and size. In other words, if two triangles are congruent, their corresponding sides and corresponding angles will be equal.

Two triangles are similar if they have the same shape, but different size. If two triangles are similar, their corresponding angles will be equal and corresponding sides proportional.

The science of reasoning and proof is helpful around a wide range of situations in daily life. In the context of Math it deals with formal justification and reasoning for various concepts, theorems etc. A proof is nothing but a deductive argument.

Geometry deals with shapes and measurements. If plane geometry is for two dimensional shapes or objects, solid geometry is for the three dimensional. In coordinate geometry position of points on the plane is described using an ordered pair of numbers, which form the coordinates. There is wide application of this branch of Math in designs, architecture, space research, and many more areas.

#### Pre-Calculus

Trigonometry involves the study of angles and the angular relationships of both planar and three-dimensional figures. The trigonometric functions relate one non-right angle of a right triangle to the ratio of the lengths of any two sides, and vice versa. There are six trigonometric functions; three primary functions like the Sine, Cosine and Tangent, and three derived from the primary-Secant, Cosecant and Cotangent.
In trigonometry once we move from triangle ratios to functions and their graphs, we are entering the territory of analytic trigonometry. Here we examine trigonometric identities in terms of their positions on the x-y plane.

Vectors are used to represent quantities that have both a magnitude and a direction. Force and velocity are examples of vectors. In a graphical sense the vectors are represented by lines; the length of the line shows the magnitude, and its direction shows the direction of the vector. We can also say that a vector describes a movement from one point to another.

Parametric equations take an equation in two or more variables and define each variable in terms of one variable called the parameter. The parametric equations for a line are derived from its vector equations.

In Math we very often come across two important aspects -discovery and proof. Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. Generally the induction proof of a formula has three steps: (1) Shows that the formula is true for n+1 (2) Assumes that it is true for n (3) Proves that it is true for n+1

Permutation: Permutation means arrangement of things. The word arrangement is used, if the ‘order’ of things is considered. Permutation tells us the number of ways in which you can arrange items, when the order is important.
Combination: Combination refers to selection of things. The word selection is used, when the order of things has no importance. It tells you in how many ways you could choose a group from a larger group. Here the order does not matter.

Binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. It helps in expanding binomial expressions with large powers, easily.

Limit is the value that a function or sequence “approaches” as the input or index approaches some value. The limit of a sequence helps us to find out the value of the sequence at an infinite point. Similarly the limit of a function at a particular value tells us a lot of information regarding the function, around that value.

One of the concepts at the core of Calculus. The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. The geometrical approach views it as the slope of a curve at an exact point, while the physical approach says it is the rate of change.
The “simple” derivative of a function ‘f’ with respect to a variable ‘x’ is denoted either f^'(x) or (df)/(dx),

Integrals together with derivatives are the fundamental objects of calculus. An integral is a mathematical object that can be interpreted as an area or a generalization of area. There are definite and indefinite integrals.

#### Calculus

If there is a function f(x), then the limits analyze what value will the function approach when the variable is approaching a specific value.
f(x) → L as x → a means that f(x) approaches the number ‘L’ as ‘x’ approaches (but is not equal to) ’a‘ from both sides. A more precise way of describing it is that we can make f(x) be as close to ‘L’ as we like by making ‘x’ be sufficiently close to ‘a’. A function can be either continuous or discontinuous. With knowledge around limits and continuity, you are ready for Calculus.
The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. There are two ways to look at it: the geometrical and the physical. While Newton considered it as “rate of change” in Classical mechanics, the geometrical approach considered it from “slope of a curve” point of view.

The “simple” derivative of a function ’f’ with respect to a variable ‘x’ is denoted either f^'(x) or (df)/(dx),

Derivatives denote the rate of change. When there are multiple variables, but if the derivative is arrived at considering only one of the variables at a time, it is “partial”! Hence partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Note that we use slightly different notations for ordinary and partial derivatives.

Any calculus based optimization situation involving more than one variable uses partial derivatives. Partial differential equations are used to mathematically approximate many physical phenomena like fluid flows, nerve conduction, force in a spring etc. Economists use it in analyzing the economic behaviour, which might include more than one variable.

Integrals and derivatives are the basic objects in calculus. Widely used by physicists and engineers, integral refers to the area of certain types of regions. There are definite and indefinite integrals. Integration can be used to find areas, volumes, central points, and many more.

With line integral, instead of integrating over intervals, here it is done over a curve. Surface integral is used in general to refer to multiple integrals to integration over surfaces.

An integral is improper if either the interval of integration is not finite or the function to integrate is not finite. Improper integral is the limit of a definite integral as the endpoint of the interval(s) of integration approaches either a specified real number or, in some cases, as both endpoints approach limits. In other words, integrals that have discontinuous intervals or where the interval of integration is not finite are called improper integrals.

It is the third ‘limit process’ in calculus, after differentiation and integration. Infinite series is a “sum” of the terms in a series. An infinite series is an expression like this: S = 1 + 1/2 + 1/4 + 1/8 + … The dots mean that infinitely many terms follow.

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of trigonometric functions. Fourier series make use of the orthogonality relationships of the sine and cosine functions. It has got wide range of application in applied mathematics, and especially in physics and electronics. The concept was introduced first by the French mathematician, Joseph Fourier.

An important tool in the theory of partial differential equations, used in mathematical analysis.

Gamma function is a generalization of the factorial function. It is possible to show that the limit of the Gamma function at 0 from the right is infinity. Beta function is also called the Euler integral of the first kind.

Let ‘f’ be a function of a complex variable, defined over a domain. If z=x+iy , where ‘x’ and ‘y’ are real numbers, then ‘z’ is written in algebraic form. We can write f(z)in algebraic form as f(z)=u(x,y)+i v(x,y). Here f(z) is the function of the complex variable ‘z’.

Cartesian coordinates are not the only way to define a point in a two dimensional space. The polar coordinates of a point describe its position in terms of a distance from a fixed point (the origin) and an angle measured from a fixed direction which, interestingly, is not “north” (or up on a page) but “east” (to the right). That is in the direction ‘Ox’ on Cartesian axes.

So, we choose a fixed point ‘O’, known as “the pole”.