# Glossary of Common Maths Terms

Here is a list of common Maths terms that are often used in O Level and A Level Maths:

## Sequences and Series

A **sequence** is an ordered list of numbers or other objects. A sequence can be finite, meaning it has a fixed number of elements, or infinite, meaning it goes on indefinitely. For example, the sequence 1, 2, 3, 4, 5 is a finite sequence of five elements, while the sequence 1, 2, 3, 4, 5, … is an infinite sequence. Sequences are often denoted using a special notation, such as a(n) or an, where n is the position of the element in the sequence. For example, in the sequence 1, 2, 3, 4, 5, the element at position 1 is 1, the element at position 2 is 2, and so on. There are different types of sequences, including arithmetic sequences, which have a common difference between consecutive elements, and geometric sequences, which have a common ratio between consecutive elements. Sequences can be used to model real-world phenomena, such as population growth or the decay of radioactive elements. They are also used in various fields, including mathematics, engineering, and computer science.

## Vectors

A **vector** is an object that has both magnitude and direction. Vectors are often used to represent quantities that have both magnitude and direction, such as velocity, force, and acceleration. Vectors can be represented by directed line segments, with the length of the line segment representing the magnitude of the vector and the direction of the line segment representing the direction of the vector. Vectors can also be represented by ordered pairs of numbers or by column matrices. Vectors can be added and subtracted using vector arithmetic. When two vectors are added, the result is a vector that has the same direction as one of the original vectors and a magnitude equal to the sum of the magnitudes of the original vectors. When one vector is subtracted from another, the result is a vector that has the same direction as the first vector but a magnitude equal to the difference of the magnitudes of the original vectors. Vectors can also be multiplied by scalars, which are non-negative numbers. When a vector is multiplied by a scalar, the result is a vector with the same direction as the original vector but a magnitude that is scaled by the scalar. Vectors are used in a variety of fields, including physics, engineering, and computer science. The **dot product** (also known as the scalar product or the inner product) is a binary operation that takes two vectors and returns a scalar. The dot product is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. The dot product can be written in the form: a β’ b = |a| |b| cos(ΞΈ) where a and b are the two vectors, |a| and |b| are the magnitudes of the vectors, and ΞΈ is the angle between the vectors. The dot product has several properties, including: Commutative: The dot product is commutative, meaning that the order of the vectors does not matter. That is, a β’ b = b β’ a. Distributive: The dot product is distributive over vector addition. That is, a β’ (b + c) = a β’ b + a β’ c. Associative: The dot product is associative, meaning that the grouping of the vectors does not matter. That is, (a β’ b) β’ c = a β’ (b β’ c). The dot product is used in a variety of fields, including physics, engineering, and computer science. It is used to compute the projection of one vector onto another, to test whether two vectors are orthogonal (perpendicular) or parallel, and to compute the angle between two vectors. The **vector product** (also known as the cross product or the outer product) is a binary operation that takes two vectors and returns another vector. The vector product is defined as a vector that is perpendicular to both of the original vectors and has a magnitude equal to the product of the magnitudes of the original vectors and the sine of the angle between them. The vector product can be written in the form: a Γ b = |a| |b| sin(ΞΈ) n where a and b are the two vectors, |a| and |b| are the magnitudes of the vectors, ΞΈ is the angle between the vectors, and n is a unit vector that is perpendicular to both a and b and points in the direction determined by the “right-hand rule.” The vector product has several properties, including: Anticommutative: The vector product is anticommutative, meaning that the order of the vectors matters. That is, a Γ b = -(b Γ a). Distributive: The vector product is distributive over vector addition. That is, a Γ (b + c) = a Γ b + a Γ c. Non-associative: The vector product is non-associative, meaning that the grouping of the vectors matters. That is, (a Γ b) Γ c β a Γ (b Γ c). The vector product is used in a variety of fields, including physics, engineering, and computer science. It is used to compute the area of a parallelogram, to determine whether three points are coplanar, and to compute the distance between a point and a plane.

## Introduction to Complex Numbers

A **complex number** is a number of the form a + bi, where a and b are real numbers and i is the square root of -1. The number a is called the real part of the complex number, and the number b is called the imaginary part. For example, the complex number 3 + 2i is composed of the real part 3 and the imaginary part 2. The real part of a complex number represents the position of the number on the real number line, and the imaginary part represents the position of the number on the imaginary number line, which is perpendicular to the real number line. Complex numbers can be added, subtracted, multiplied, and divided using the same rules as for real numbers, with the exception that i2 = -1. For example, the sum of the complex numbers 3 + 2i and 1 + 4i is 4 + 6i, and the product of the complex numbers 3 + 2i and 1 + 4i is -5 + 10i. Complex numbers are used in a variety of fields, including mathematics, engineering, and physics. They are used to represent quantities that have two components, such as position and velocity, and they are used to solve equations that have no real solutions. The **Cartesian form** of a complex number is a representation of the complex number in the form a + bi, where a and b are real numbers and i is the square root of -1. This form is named after the mathematician RenΓ© Descartes, who is credited with the development of analytic geometry, which combines algebra and geometry using the Cartesian coordinate system. For example, the complex number 3 + 2i is written in Cartesian form, where 3 is the real part and 2 is the imaginary part. The complex number -4 – 5i is also written in Cartesian form, where -4 is the real part and -5 is the imaginary part. The Cartesian form is the most common way of representing complex numbers, and it is used in a variety of fields, including mathematics, engineering, and physics. It allows complex numbers to be represented graphically using the complex plane, which is a plane that consists of the real number line and the imaginary number line. The **polar form** of a complex number is a representation of the complex number in the form r cis(ΞΈ), where r is the magnitude (or absolute value) of the complex number, cis(ΞΈ) is a notation for the complex number with magnitude 1 and argument ΞΈ, and ΞΈ is the argument (or angle) of the complex number. The polar form of a complex number is often used to represent the complex number in terms of its magnitude and argument, rather than its real and imaginary parts. The magnitude of a complex number represents the distance of the complex number from the origin of the complex plane, and the argument represents the angle between the positive real axis and the line connecting the origin to the complex number. For example, the complex number 3 + 2i can be written in polar form as sqrt(13) cis(53.13010235415598). The magnitude of this complex number is sqrt(13), and the argument is 53.13010235415598 degrees. The polar form is useful for performing operations on complex numbers, such as multiplication and division, and it is also useful for representing complex numbers in a graphical form using the complex plane.

## Calculus

**Differentiation** is a mathematical process that involves finding the rate of change of a function at a given point. It is a fundamental concept in calculus, and it is used to model how a quantity changes over time or in response to changing conditions. To differentiate a function, you need to take the derivative of the function, which is a measure of the slope of the function at a given point. The derivative of a function can be found using the derivative rules, which involve taking the limit of the difference quotient as the change in the independent variable approaches zero. There are different types of derivatives, including: Ordinary derivatives – These are derivatives of functions with respect to a single variable. Partial derivatives – These are derivatives of functions with respect to multiple variables. Implicit derivatives – These are derivatives of functions that are defined implicitly, rather than explicitly. Total derivatives – These are derivatives of functions that depend on both a variable and one or more parameters. Differentiation has many applications in various fields, including physics, engineering, economics, and biology. It is used to model physical phenomena, such as the motion of objects and the flow of fluids, and it is also used to optimize systems, such as finding the maximum or minimum value of a function. **Maclaurin’s series** is an expansion of a function in a power series about the point x = 0. It is named after the mathematician Colin Maclaurin, who developed the technique of Taylor series expansion. To find the Maclaurin series expansion of a function f(x), you first need to find the derivatives of the function at x = 0. The Maclaurin series expansion of the function is then given by: f(x) = f(0) + f'(0)x + f”(0)x^2/2! + f”'(0)x^3/3! + … where f'(0), f”(0), etc. are the first, second, etc. derivatives of the function at x = 0, and x^n/n! is the nth term in the expansion. The Maclaurin series expansion of a function can be used to approximate the function for small values of x. It can also be used to find the Taylor series expansion of the function about any other point by shifting the expansion to that point. Maclaurin’s series is used in a variety of fields, including mathematics, physics, and engineering, to approximate functions and to analyze the behavior of functions near a point. **Integration** is a mathematical process that involves finding the area under a curve or the volume inside a three-dimensional object. It is the inverse of differentiation and is a fundamental concept in calculus. There are different types of integration, including: Indefinite integration – This is the process of finding a function whose derivative is a given function. Definite integration – This is the process of finding the area under a curve or the volume inside a three-dimensional object. Multiple integration – This is the process of finding the volume inside a three-dimensional object that is bounded by multiple curves. To perform integration, you need to use integration rules and techniques, such as the fundamental theorem of calculus, substitution, and integration by parts. Integration has many applications in various fields, including physics, engineering, economics, and biology. It is used to model physical phenomena, such as the flow of fluids and the movement of particles, and it is also used to solve optimization problems, such as finding the maximum or minimum value of a function. In calculus, a **definite integral** is a type of integral that involves finding the area under a curve or the volume inside a three-dimensional object over a specified range. Definite integrals are used to compute the exact size or measure of a continuous quantity, such as distance, volume, or mass. To compute a definite integral, you need to specify the limits of integration, which are the lower and upper bounds of the integration range. The definite integral is then defined as: β«a^bf(x)dx = F(b) – F(a) where f(x) is the function being integrated, a and b are the limits of integration, and F(x) is the indefinite integral of f(x). Definite integrals can be computed using integration techniques and rules, such as the fundamental theorem of calculus and integration by substitution. They can also be approximated using numerical methods, such as the trapezoidal rule and Simpson’s rule. Definite integrals are used in a variety of fields, including physics, engineering, economics, and biology. They are used to model physical phenomena, such as the flow of fluids and the movement of particles, and they are also used to solve optimization problems, such as finding the maximum or minimum value of a function. A **differential equation** is an equation that involves an unknown function and its derivatives. It is used to model the behavior of a system that changes over time or in response to changing conditions. Differential equations can be ordinary differential equations, which involve a single independent variable, or partial differential equations, which involve multiple independent variables. They can also be linear or nonlinear, depending on whether the highest derivative in the equation is linear or nonlinear. To solve a differential equation, you need to find a function that satisfies the equation and meets certain boundary conditions. There are different methods for solving differential equations, including analytical methods, such as separation of variables and integration factor, and numerical methods, such as Euler’s method and Runge-Kutta methods. Differential equations are used in a variety of fields, including physics, engineering, economics, and biology. They are used to model physical phenomena, such as the motion of objects and the flow of fluids, and they are also used to describe the behavior of systems, such as electrical circuits and population dynamics.

## Probability and Statistics

**Probability** is the branch of mathematics that deals with the study of random events and the likelihood of their occurrence. It is used to model and analyze uncertain situations, such as the outcome of a coin flip or the results of a survey. The probability of an event is a number between 0 and 1 that represents the likelihood of the event occurring. An event with a probability of 0 cannot occur, while an event with a probability of 1 is certain to occur. The probability of an event occurring is calculated by dividing the number of ways the event can occur by the total number of possible outcomes. There are different types of probability, including: Classical probability – This is the probability of an event occurring based on the symmetry of the situation. Empirical probability – This is the probability of an event occurring based on past observations or experiments. Subjective probability – This is the probability of an event occurring based on an individual’s personal judgment or belief. Probability theory has many applications in various fields, including economics, finance, and insurance. It is used to model and analyze uncertain situations, such as the performance of investments and the risk of natural disasters, and it is also used to design experiments and make predictions. A **discrete random variable** is a random variable that can take on a countable number of distinct values. The values that a discrete random variable can take on are called its outcomes, and the probability of each outcome occurring is known as the probability mass function (PMF). The probability mass function of a discrete random variable is a function that assigns a probability to each of the possible outcomes of the random variable. The probability mass function satisfies the following conditions: The probability of each outcome is between 0 and 1. The sum of the probabilities of all possible outcomes is 1. Examples of discrete random variables include the number of heads in a series of coin flips, the number of defective items in a batch of products, and the number of customers that visit a store in a given day. Discrete random variables are used in a variety of fields, including economics, finance, and computer science, to model and analyze uncertain situations. They are often used to represent the outcomes of experiments and to make predictions about future events. In probability and statistics, the **normal distribution** is a continuous probability distribution that is defined by a symmetric bell-shaped curve. It is often referred to as the bell curve because of its shape. The normal distribution is characterized by its mean, which is the peak of the curve, and its standard deviation, which determines the width of the curve. The normal distribution is often used to model and analyze data that are continuous and follow a symmetric distribution. It is widely used in various fields, including biology, economics, and engineering, to represent the distribution of a wide variety of variables, such as height, weight, and intelligence. The normal distribution has several useful properties, including: It is symmetric around its mean. It is completely determined by its mean and standard deviation. The area under the curve is 1, which means that the probability of a value occurring is always between 0 and 1. Approximately 6
The normal distribution is often used in statistical analysis to test hypotheses and make predictions about future events. It is also used in many fields to model the distribution of variables and to evaluate the performance of statistical models. **Sampling** is the process of selecting a subset of a population for the purpose of studying and analyzing the characteristics of the population. It is an important tool in research and statistical analysis, as it allows researchers to study a smaller and more manageable group of individuals rather than the entire population. There are different types of sampling methods, including: Simple random sampling – This is a sampling method in which each member of the population has an equal chance of being selected. Stratified sampling – This is a sampling method in which the population is divided into groups (or strata) and a simple random sample is taken from each group. Cluster sampling – This is a sampling method in which clusters of individuals are selected and all members of the selected clusters are included in the sample. Systematic sampling – This is a sampling method in which members of the population are selected at fixed intervals. The accuracy of the results obtained from a sample depends on the sampling method used and the size of the sample. In general, a larger sample size is more representative of the population and is more likely to produce accurate results. Sampling is used in a variety of fields, including market research, public opinion polling, and scientific research, to study and analyze populations and to make inferences about the characteristics of the population. **Hypothesis testing** is a method for evaluating the validity of a hypothesis about a population parameter. It involves making a statistical inference about the population based on a sample from the population and testing the hypothesis using statistical techniques. To perform hypothesis testing, you need to specify: The null hypothesis, which is the hypothesis that you are trying to refute or disprove. The null hypothesis is usually a statement of no difference or no effect. The alternative hypothesis, which is the hypothesis that you are trying to confirm or support. The alternative hypothesis is usually a statement of difference or effect. The level of significance, which is the probability of rejecting the null hypothesis when it is true. This is usually set at a level of 0.05, which means that there is a
To test the hypothesis, you need to collect a sample from the population and compute a test statistic, which is a numerical measure of the deviation of the sample from the null hypothesis. You then need to compare the test statistic to a critical value, which is a value that is determined by the level of significance and the characteristics of the sample. If the test statistic is greater than the critical value, you can reject the null hypothesis and conclude that the alternative hypothesis is true. Hypothesis testing is used in many fields to make inferences about populations and to evaluate the validity of claims and hypotheses. It is an important tool for making decisions and for conducting research. **Correlation** is a measure of the strength and direction of a linear relationship between two variables. It is used to describe the degree to which two variables are related and to predict the value of one variable based on the value of the other variable. There are different types of correlations, including: Positive correlation – This is a correlation in which both variables increase or decrease together. Negative correlation – This is a correlation in which one variable increases as the other variable decreases. No correlation – This is a correlation in which there is no relationship between the variables. The strength of a correlation is measured using the correlation coefficient, which is a numerical measure of the degree of correlation between two variables. The correlation coefficient can range from -1 to 1, with -1 indicating a strong negative correlation, 0 indicating no correlation, and 1 indicating a strong positive correlation. Correlation is used in many fields to analyze the relationship between variables and to make predictions about the behavior of one variable based on the value of the other variable. It is important to note that correlation does not imply causation, meaning that the presence of a correlation between two variables does not necessarily mean that one variable causes the other. **Regression** is a statistical method for estimating the relationship between a dependent variable and one or more independent variables. It is used to predict the value of the dependent variable based on the values of the independent variables. There are different types of regression, including: Simple linear regression – This is a regression method that involves one independent variable and one dependent variable. Multiple linear regression – This is a regression method that involves multiple independent variables and one dependent variable. Polynomial regression – This is a regression method that involves fitting a polynomial function to the data. Nonlinear regression – This is a regression method that involves fitting a nonlinear function to the data. To perform regression, you need to estimate the regression equation, which is a mathematical formula that describes the relationship between the dependent variable and the independent variables. You then need to use the regression equation to make predictions about the value of the dependent variable based on the values of the independent variables. Regression is used in many fields to analyze the relationship between variables and to make predictions about the behavior of one variable based on the values of other variables. It is an important tool for making decisions and for conducting research.