## Activity 15

Suggest appropriate units for each of the following:

(a) the age of the kitten when it is weaned;

(b) the distance between one train station and the

2.10 The complex exponential function

Consider the real exponential function *f* (*x*) = *e*^{x} (that is, *f* (*x*) = exp *x*). We now extend the definition of this function to define a function *f*(*z*) = *e*^{z} whose domain and codomain are Author(s):

After studying this course, you should be able to:

understand the arithmetical properties of the rational and real numbers

understand the definition of a

**complex number**perform arithmetical operations with complex numbers

explain the terms modular addition and modular multiplication

explain the meanings of a relation defined on a set, an equivalent relation and a partition of a set.

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

Course image: Stuart Rankin in Flickr made available under Creative Commons Attribution-NonCommercial 2.0 Licence.

All materials included in this course are

After studying this course, you should be able to:

create simple models, given a clear statement of the problem

identify the simplifying assumptions that underpin a model

identify the key variables and the parameters of a model

apply the input–output principle to obtain a mathematical model, where appropriate.

The sum of the angles of any triangle is 180°. This property can be demonstrated in several ways. One way is to draw a triangle on a piece of paper, mark each angle with a different symbol, and then cut out the angles and arrange them side by side, touching one another as illustrated.

## Question 1

Draw a line of symmetry on each of the shapes below.

There is another kind of symmetry which is often used in designs. It can be seen, for instance, in a car wheel trim.

Look at the trim on the left. It does not have line symmetry but

Two straight lines that do not intersect, no matter how far they are extended, are said to be **parallel**. Arrows are used to indicate parallel lines.

1.3.4 Vertically opposite angles

When two straight lines cross, they form four angles. In the diagram below, these angles are labelled *α*, *β*, *θ* and *φ* and referred to as alpha, beta, theta and phi. The angles opposite each other are equal. They are called **vertically opposite** angles. Here *α* and *β* are a pair of vertically opposite angles, as are *θ* and *φ*. Although such angles are called ‘vertically opposite’, they do not need to be vertically above and bel

## Question 1

A company carried out a survey, recording how staff in a particular office spent their working time. The table shows the average number of minutes spent in each hour on various activities.

You can use the fact that the sum of angles at a point is 360° to draw a pie chart.

## Example 4

Over a five-year period a mathematics tutor found that 16 of her students gained distinctions, 32 gained pass grades and 12 failed to complete the course. Draw a pie chart to re

## Activity 14

A piece of computer software is to be developed by a team of programmers. It is estimated that a team of four people would take a year. Which of the following times is the length of time taken by three program

Climate change: transitions to sustainability

Human societies have to take urgent action to end their dependences on fossil fuels. We have to alter the whole path of our development and decision making in order to make our societies both environmentally adaptable and sustainable. This free course, Climate change, takes on the task of trying to chart some of the ways in which it might be possible.Author(s):

The frozen planet

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Water for life

Atoms, elements and molecules are the building blocks of everything that makes up our world, including ourselves. In this free course, Water for life, you will learn the basic chemistry of how these components work together, starting with a chemical compound we are all very familiar with water.
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Water is arguably the most important physical resource as it is the one that is essential to human survival. Understanding the global water cycle and how we use water is essential to planning a sustainable source of water for the future. Globally, there are many areas that do not have enough water to support the current population adequately. Decisions will have to be made on the best way to use water in a world where there is climate change. This free course looks at Water in the UK where water

The science of nuclear energy

This free course, The science of nuclear energy, will delve into the science behind nuclear power and explain what happens inside a nuclear reactor and what it means for an element to be radioactive. It will explore some of the risks of producing nuclear power and examine the arguments for and against including it in future energy planning as well as looking at other potential future solutions.
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Environment: understanding atmospheric and ocean flows

What affects the atmospheric and ocean flows? This free course, Environment: understanding atmospheric and ocean flows, explores the mechanisms that are important; the most rapid carrier is the wind. The basic principle of global atmospheric circulation is simple: warm air rises and cold air sinks. How does this principle affect the atmosphere and flow of water in practical terms? Starting with an iconic environmental icon, the polar bear, you will learn how global flows of water, heat and pollu