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Pure Maths

**Pure maths **is the branch of mathematics most concerned with mathematical concepts. It is the most abstract area of mathematics, but it still has many areas which are applicable in everyday life. It's also a broad topic, so it's important to spend a lot of time on it! In this course, you will learn a variety of skills such as calculus, geometry and algebra. The concepts can then be applied to different fields such as social sciences, logic, engineering, biology, chemistry or physics.

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- Applied Mathematics
- Calculus
- Decision Maths
- Discrete Mathematics
- Geometry
- Logic and Functions
- Mechanics Maths
- Probability and Statistics
- Pure Maths
- ASA Theorem
- Absolute Convergence
- Absolute Value Equations and Inequalities
- Abstract algebra
- Addition and Multiplication of series
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebra of limits
- Algebra over a field
- Algebraic Fractions
- Algebraic K-theory
- Algebraic Notation
- Algebraic Representation
- Algebraic curves
- Algebraic geometry
- Algebraic number theory
- Algebraic topology
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Associative algebra
- Average Rate of Change
- Banach algebras
- Basis
- Bijective Functions
- Bilinear forms
- Binomial Expansion
- Binomial Theorem
- Bounded Sequence
- C*-algebras
- Category theory
- Cauchy Sequence
- Cayley Hamilton Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Clifford algebras
- Cohomology theory
- Combinatorics
- Common Factors
- Common Multiples
- Commutative algebra
- Compact Set
- Completing the Square
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Congruence Equations
- Conic Sections
- Connected Set
- Construction and Loci
- Continuity and Uniform convergence
- Continuity of derivative
- Continuity of real valued functions
- Continuous Function
- Convergent Sequence
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Coupled First-order Differential Equations
- Cubic Function Graph
- Data Transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Derivative of a real function
- Deriving Equations
- Determinant Of Inverse Matrix
- Determinant of Matrix
- Determinants
- Diagonalising Matrix
- Differentiability of real valued functions
- Differential Equations
- Differential algebra
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Dimension
- Direct and Inverse proportions
- Discontinuity
- Disjoint and Overlapping Events
- Disproof By Counterexample
- Distance from a Point to a Line
- Divergent Sequence
- Divisibility Tests
- Division algebras
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Eigenvalues and Eigenvectors
- Ellipse
- Elliptic curves
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Equicontinuous families of functions
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Fermat's Little Theorem
- Field theory
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding The Area
- First Fundamental Theorem
- First-order Differential Equations
- Forms of Quadratic Functions
- Fourier analysis
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Gram-Schmidt Process
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs And Differentiation
- Graphs Of Exponents And Logarithms
- Graphs of Common Functions
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Grothendieck topologies
- Group Mathematics
- Group representations
- Growth and Decay
- Growth of Functions
- Gröbner bases
- Harmonic Motion
- Hermitian algebra
- Higher Derivatives
- Highest Common Factor
- Homogeneous System of Equations
- Homological algebra
- Homotopy theory
- Hopf algebras
- Hyperbolas
- Ideal theory
- Imaginary Unit And Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Injective linear transformation
- Instantaneous Rate of Change
- Integers
- Integrating Ex And 1x
- Integrating Polynomials
- Integrating Trigonometric Functions
- Integration
- Integration By Parts
- Integration By Substitution
- Integration Using Partial Fractions
- Integration of Hyperbolic Functions
- Interest
- Invariant Points
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Inverse of a Matrix and System of Linear equation
- Invertible linear transformation
- Iterative Methods
- Jordan algebras
- Knot theory
- L'hopitals Rule
- Lattice theory
- Law Of Cosines In Algebra
- Law Of Sines In Algebra
- Laws of Logs
- Leibnitz's Theorem
- Lie algebras
- Lie groups
- Limits of Accuracy
- Linear Algebra
- Linear Combination
- Linear Expressions
- Linear Independence
- Linear Systems
- Linear Transformation
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition And Subtraction
- Matrix Calculations
- Matrix Determinant
- Matrix Multiplication
- Matrix operations
- Mean value theorem
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modelling with First-order Differential Equations
- Modular Arithmetic
- Module theory
- Modulus Functions
- Modulus and Phase
- Monoidal categories
- Monotonic Function
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplicative ideal theory
- Multiplying And Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Non-associative algebra
- Normed spaces
- Notation
- Number
- Number Line
- Number Systems
- Number Theory
- Number e
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations With Matrices
- Operations with Decimals
- Operations with Polynomials
- Operator algebras
- Order of Operations
- Orthogonal groups
- Orthogonality
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Hyperbolas
- Parametric Integration
- Parametric Parabolas
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Pointwise convergence
- Poisson algebras
- Polynomial Graphs
- Polynomial rings
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Determinants
- Properties of Exponents
- Properties of Riemann Integral
- Properties of dimension
- Properties of eigenvalues and eigenvectors
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic forms
- Quadratic functions
- Quadrilaterals
- Quantum groups
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Ratio and Root test
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Rearrangement
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Reduced Row Echelon Form
- Reducible Differential Equations
- Remainder and Factor Theorems
- Representation Of Complex Numbers
- Representation theory
- Rewriting Formulas and Equations
- Riemann integral for step function
- Riemann surfaces
- Riemannian geometry
- Ring theory
- Roots Of Unity
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Products
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Fundamental Theorem
- Second Order Recurrence Relation
- Second-order Differential Equations
- Sector of a Circle
- Segment of a Circle
- Sequence and series of real valued functions
- Sequence of Real Numbers
- Sequences
- Sequences and Series
- Series Maths
- Series of non negative terms
- Series of real numbers
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Similarity and diagonalisation
- Simple Interest
- Simple algebras
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Spanning Set
- Special Products
- Special Sequences
- Standard Form
- Standard Integrals
- Standard Unit
- Stone Weierstrass theorem
- Straight Line Graphs
- Subgroup
- Subsequence
- Subspace
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Summation by Parts
- Supremum and Infimum
- Surds
- Surjective functions
- Surjective linear transformation
- System of Linear Equations
- Tables and Graphs
- Tangent of a Circle
- Taylor theorem
- The Quadratic Formula and the Discriminant
- Topological groups
- Torsion theories
- Transformations
- Transformations of Graphs
- Transformations of Roots
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Uniform convergence
- Unit Circle
- Units
- Universal algebra
- Upper and Lower Bounds
- Valuation theory
- Variables in Algebra
- Vector Notation
- Vector Space
- Vector spaces
- Vectors
- Verifying Trigonometric Identities
- Volumes of Revolution
- Von Neumann algebras
- Writing Equations
- Writing Linear Equations
- Zariski topology
- Statistics
- Theoretical and Mathematical Physics

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Jetzt kostenlos anmelden**Pure maths **is the branch of mathematics most concerned with mathematical concepts. It is the most abstract area of mathematics, but it still has many areas which are applicable in everyday life. It's also a broad topic, so it's important to spend a lot of time on it! In this course, you will learn a variety of skills such as calculus, geometry and algebra. The concepts can then be applied to different fields such as social sciences, logic, engineering, biology, chemistry or physics.

Let's take a look at some important themes that are used throughout pure mathematics.

You can find the following topics on StudySmarter:

Proof is a logical argument that shows whether or not a mathematical statement is true. Mathematical proofs are **robust** - that is, they should consider every possibility, and there should be no holes in the logic. Proofs are built on **axioms **and are incredibly important - without them, we would not know that mathematical results are accurate.

You can use many different techniques to prove statements, and StudySmarter articles cover the key ones you need to know.

Algebra is an abstract representation of numbers that allows you to make general statements consisting of mathematical operations. For example, suppose you wanted to write a mathematical statement involving an unknown quantity. In that case, you could refer to it as *x* and see how this variable would change as the statement changes.

There are a few direct subtopics that we explore on StudySmarter, but you can use algebraic concepts across all areas of mathematics - algebra is an extremely powerful tool, and a solid understanding of it is incredibly valuable. Even if you find it confusing, our guides will walk you through the fundamentals and allow you to apply them to questions similar to those you may see in your exams.

Functions are used to apply a particular operation or set of operations to an input value, giving an output value. They are often expressed as *f(x)*, meaning that the function has a variable as a value, usually “*x*” for simpler functions. A simple example is \(f(x) = x+2\), this function just adds two to an input represented by *x*.

Functions are closely linked with graphs, and being able to plot them is an essential skill for your exams.

Coordinate geometry is the study of geometry that uses a coordinate system, usually in two dimensions with *x *and *y * axes, as pictured below. Coordinate geometry can also include systems in three dimensions. Functions can be represented as graphs on the coordinate systems. Again, graphs are a big part of this topic!

Example of a 2D coordinate system

Sequences are a list of numbers that follow a pattern or rule, usually linked by some common function. Binomial expansion is also a form of sequence closely linked to **factorials**. You will need to know how to generate terms of a sequence, identify the common function and find their sums (the sum of a sequence is known as the **series**).

Trigonometry is the area of mathematics involving angles and the geometric lines of shapes, most commonly triangles. Its applications are wider than you might expect – they aren't limited to triangles – and it is one of the most important areas of mathematics you need to understand for your exams. It includes trigonometric functions like sine, cosine and tangent and their reciprocals, radians (an alternative form of measuring angles to degrees), and other important rules.

Exponentials are functions of the form *N*^{x}, which increase or decrease rapidly as x increases, as number N is raised to a power of x. An example of an exponential function is \(f(x) = 2^x\) and there is a special function e^{x}.

Logarithms are the inverse function of an exponential. They can be used to find the power to which a number was raised to get another number. For any exponential, we have a logarithm in the form \(\log_a(b)\). For example, \(\log_2(8) = 3\) because \(2^3 = 8\). Again, there is a special function called the natural logarithm, which is the inverse of *e* - expressed as "\(\ln(x)\)".

Differentiation is a method of finding rates of change, ie gradients of functions. We can find this by drawing gradient lines of the graphs. This isn't always easy or precise, so we can also do it analytically. The result of a differentiation calculation is called the derivative of a function. The process of differentiation is represented by \(\frac{dy}{dx}\). This is equivalent to “change in *y* divided by change in *x*”, and *x* and *y* can be substituted for any variable.

There are a few different rules you will need to remember to help solve more complex problems, some of which are below:

Product rule, used when two functions are multiplied by each other, e.g. \(f(x) g(x)\).

Quotient rule, used when two functions are divided by each other, e.g. \(\frac{f(x)}{g(x)}\).

The chain rule, used for composite functions, e.g. \(f(g(x))\).

You will also need to know how to derive trigonometric functions. For example, the derivative of \(\sin(x)\) is \(\cos(x)\).

Integration is a method for finding the area under a graph and is the inverse operation of derivation. An integral is represented by the \(\int\) symbol. This kind of integral is called an indefinite integral - a definite integral refers to the area of a given range and is represented in the format \(\int^a_b\), where a and b signify the desired range of values.

Once again, there are key methods that you can use to solve more complex problems, such as integration by parts, and you can memorize some standard results.

Numerical methods are ways to approximate mathematical solutions that cannot be found easily. Some examples where we can use these methods are when finding the roots of equations and integration.

One example is the Newton-Raphson method, an algorithm that repeatedly attempts to improve its accuracy with each iteration.

Numerical methods have extensive applications and are very important not only in mathematics but also in engineering. Some real-world examples are listed below:

Solving problems in naval engineering, aerospace and structural mechanics.

Machine learning algorithms.

Weather prediction.

Price estimations such as the ones done by flight companies.

As a mathematics student, knowledge of numerical methods will help you solve problems and gain a good foundation of tools used in the public and private sectors.

Vectors are quantities that have both magnitude and direction, and you can use them to show the position of a point in relation to another point. Coordinates are useful to represent vectors, as shown in the diagram below!

Example of Coordinates as Vector Representation

Vectors can be expressed using unit vectors i and j (representing the x and y directions, respectively). One example is given below for the vector \(v = \left[ \begin{array}{c} 1\\2 \end{array} \right]\).

\(v = \left[ \begin{array}{c} 1\\2 \end{array} \right] = 1i + 2j\)

Systems of vectors can also be expressed as column vectors when we have two or more vectors, and you can see an example below.

\(u = 3i, \space v = 4i, \space w = 7i\)

We can express these vectors as a column.

\(\left[ \begin{array} {c} u\\ v \\ w \end{array} \right] = \left[ \begin{array} {c} 3 \\ 4 \\ 7 \end{array} \right] i\)

\(\left[ \begin{array} {c} 3i \\ 4i \\ 7i \end{array} \right]\)You will also need to know the basics of 3D vectors at A level, where the z-direction (represented by k) is introduced.

Example of Coordinate System in 2D and 3D

Pure mathematics is an important area that is applied to other parts of mathematics and is most concerned with mathematical concepts.

Proof is a logical argument that shows whether or not a mathematical statement is true.

Algebra is an abstract representation of numbers that allows you to make general statements consisting of mathematical operations.

Functions are used to apply a particular operation or set of operations to an input value, giving an output value.

Coordinate geometry is the study of geometry that uses a coordinate system.

Sequences are a list of numbers that follow a pattern or rule, usually linked by some common function. Series are the sum of sequences.

Trigonometry is the area of mathematics involving angles and the geometric lines of shapes, most commonly of triangles.

Exponentials are functions of the form

*N*, which increase or decrease rapidly as^{x}*x*increases, as number*N*is raised to a power of*x*, whilst logarithms are the inverse function of an exponential.Differentiation is an analytical method for finding rates of change, ie gradients of functions. And integration is the inverse operation of derivation for finding the area under a graph.

Numerical methods are ways to approximate mathematical solutions that cannot be found easily.

Vectors are quantities that have both magnitude and direction, and you can use them to show the position of a point in relation to another point.

An example of pure maths is trigonometry.

What is algebra?

Algebra is a branch of mathematics that represents problems as mathematical expressions, using letters or variables (ie x, y or z) to represent unknown values that can change. The purpose of algebra is to find out what the unknown values are, by using predefined rules to manipulate each mathematical expression.

What is the distributive property of algebra?

a × (b + c) = a × b + a × c

What is the commutative property of multiplication of algebra?

a × b = b × a

What is the associative property of addition of algebra?

a + (b + c) = (a + b) + c

What is the additive inverse property of algebra?

a + (-a) = 0

What are the steps to solve linear algebraic equations?

Step 1: each side of the equation must be simplified by removing parentheses and combining terms

Step 2: add or subtract to isolate the variable on one side of the equation

Step 3: multiply or divide to obtain the value of the unknown variable

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