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# Pure Maths

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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.

## What are the topics in pure mathematics?

You can find the following topics on StudySmarter:

### Proof

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

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 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

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 , meaning that the function has a variable as a value, usually “” for simpler functions. A simple example is , this function just adds two to an input represented by

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

### Coordinate geometry

Coordinate geometry is the study of geometry that uses a coordinate system, usually in two dimensions with and 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 and series

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

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

Exponentials are functions of the form, 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 and there is a special function

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 . For example, because . Again, there is a special function called the natural logarithm, which is the inverse of - expressed as

### Differentiation

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 . This is equivalent to “change in divided by change in ”, and and 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. .

• Quotient rule, used when two functions are divided by each other, e.g. .

You will also need to know how to derive trigonometric functions. For example, the derivative of is

### Integration

Integration is a method for finding the area under a graph and is the inverse operation of derivation. An integral is represented by the 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 , 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

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 estimation 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

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

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

We can express these vectors as a column.

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 Maths - Key takeaways

• 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 , which increase or decrease rapidly as increases, as number is raised to a power of , 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.

Within pure maths, you study the following topics: Proof, Algebra, Functions, Coordinate Geometry, Sequences and Series, Trigonometry, Exponentials and Logarithms, Differentiation, Integration, Numerical Methods and Vectors.

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

An example of pure maths is trigonometry.

## Final Pure Maths Quiz

Question

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.

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Question

Who invented algebra?

Algebra was invented by Abu Ja'far Muhammad ibn Musa al-Khwarizmi

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Question

What is the distributive property of algebra?

a × (b + c) = a × b + a × c

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What is the commutative property of multiplication of algebra?

a × b = b × a

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What is the associative property of addition of algebra?

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

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What is the additive inverse property of algebra?

a + (-a) = 0

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Question

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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Question

What are the steps to solve word problems in algebra?

Step 1: Assign variables to unknown values

Step 2: Construct the equations

Step 3: Solve the equations

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Question

Solve the equation: x + 2 = 5

x = 3

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Solve the equation: 2x - 5 = 1

x = 3

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Solve the equation: x/3 = 9

x = 27

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Solve the equation: 2(x + 6) = 4(x + 1)

x = 4

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Solve the equation: 5x - 2 = 3x + 8

x = 5

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Question

Mary is 15. If we know that Ben's age is three times Mary's age minus 4, how old is Ben?

Ben is 41 years old

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What is a sequence?

A sequence is a set of numbers that follow a specific rule

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What is a series?

A series is the sum of a sequence

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What are the two types of sequences?

Arithmetic and geometric

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Question

When do you use the sigma notation?

This notation is used to identify the sum of the series.

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What are the three trigonometric functions?

Sine, cosine and tangent

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What are the names of the three sides of a right-angled triangle?

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How do you find the length of a side in a non-right-angled triangle?

You can use the sine or cosine rule.

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When would you use the sine rule?

You would use the sine rule when finding the missing side length or angle when you have two opposite sides and angles.

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What is the difference between a vector and scalar?

A vector has direction and magnitude, whereas a scalar only has magnitude.

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Question

Find the missing vertex of the parallelogram given by (7, 1), (3, 1), (2, 0), (123)

(6, 0)

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Question

What are powers?

Power is the exponent that a variable or number is being raised to, which in practice means that the number or variable is multiplied by itself as many times as the value of the power or exponent.

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What are roots?

Roots are the opposite of powers, they find what number multiplied n times equals the number inside the root, where n is the index of the root.

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How many solutions do odd roots have?

One solution

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How many solutions do even roots have?

Two solutions

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Question

Calculate $$2^5$$.

$$32$$.

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Evaluate $$\sqrt{243}$$.

$$9\sqrt{3}$$.

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Question

Write $$x^{\frac{5}{4}}$$ as a root.

$$\sqrt[4]{x^5}$$.

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Write $$\sqrt[3]{x^8}$$ as a power.

$$x^{\frac{8}{3}}$$.

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Evaluate or simplify $$(3x^3y)^2$$.

$$9x^6y^2$$.

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Evaluate or simplify $\frac{x^{2y+1}}{x^{2y-1}}.$

$$x^2$$.

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Question

What are surds?

Surds are expressions that contain a square root, cube root or other roots, which produce an irrational number as a result, with infinite decimals. They are left in their root form to represent them exactly.

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How do you multiply brackets containing surds?

To multiply brackets containing surds, each term in the first bracket must be multiplied by each term in the second bracket.

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How do you simplify surds?

The steps to simplify surds are:

• Write the number inside the root as the multiplication of its factors. One of the factors should be a square number
• Split the factors into separate roots
• Simplify the terms
• Take out the multiplication symbol

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Question

How do you rationalise the denominator of surds?

• If the denominator is a surd, then multiply the numerator and denominator by that surd.
• If the denominator has two terms, one rational and a surd, then multiply the numerator and denominator by the expression conjugate of the denominator.

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Question

What is the correct simplification of $$\sqrt{20}$$?

$$2\sqrt{5}$$.

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Simplify $$\sqrt{300}$$.

$$10\sqrt{3}$$.

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Simplify $$\sqrt{243}$$.

$$9\sqrt{3}$$.

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Question

Which of these is equivalent to $$7\sqrt{2}$$?

$$sqrt{98}$$.

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Question

Simplify $$\sqrt{32}$$.

$$4\sqrt{2}$$.

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Question

How would you rationalize the fraction $\frac{4}{\sqrt{2}}?$

By multiplying the fraction by $$\frac{\sqrt{2}}{\sqrt{2}}$$.

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Question

How can you simplify $\frac{ \sqrt{5} + 3}{\sqrt{5} - 2}?$

By multiplying both the numerator and denominator by $$\sqrt{5} + 2$$.

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Question

Can you add $$\sqrt{2}$$ and $$\sqrt{3}$$?

No, the numbers inside the square roots are not the same.

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Question

Can you add $$\sqrt{2}$$ and $$\sqrt{50}$$?

Yes, if you simplify the $$\sqrt{50}$$ first.

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Question

What is a function?

A function involves an input and an output.

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Question

What are the types of functions?

Composite and inverse functions.

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Question

Which function do you solve first in this composite function fg(x)?

g(x)

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