4.3 Least Upper Bound Property In the examples just given, it was straightforward to guess the values of sup E and inf E. Sometimes, however, this is not the case. For example, if
In such cases, it i
then it can be shown that E is bounded above by 3, but it is not so easy to guess the least upper bound of E.
1.1 Rational numbers The set of natural numbers is
the set of integers is
and the set of rational numbers is
Author(s):
6.2 Getting the feel of big and small numbers Very small and very large numbers can be difficult to comprehend. Nothing in our everyday experience helps us to get a good feel for them. For example numbers such as 1099 are so big that if Figure 1 was drawn to scale, you would be dealing with enormous distances. How big is big? First express 1 000 000 000 in scientific notation as 109. Next, to find out how many times bigger 1099 is, use your calculator to divide 1099 by 109
Introduction This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics.
In order to complete this unit you will need
Learning outcomes By the end of this unit you should be able to: Section 1: Sets use set notation; determine whether two given sets are equal and whether one given set is a subset of another; find the union, intersection and difference of two given sets. Section 2: Functions determine the image of a given function; determine whether a given function is one-one
Learning outcomes After studying this unit you should: be able to perform basic algebraic manipulation with complex numbers; understand the geometric interpretation of complex numbers; know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations.
2.4.1 Try some yourself 1 Write the following as a number to a single power: (a) 26 ÷ 22 (b) 1010 ÷ 107 (c) 78 ÷ 74 Introduction This unit is an adapted extract from the course Mathematical methods and models
(MST209) This unit lays the foundations of Newtonian mechanics and in particular the procedure for solving dynamics problems. The prerequisite skills needed for this unit are the ability to solve first- and second-order differential equations, a knowledge of vectors, and an understanding of the concept of a Introduction This unit lays the foundations of the subject of mechanics. Mechanics is concerned with how and why objects stay put, and how and why they move. In particular, this unit – Modelling static problems – considers why objects stay put.
Please note that this unit assumes you have a good working knowledge of vectors. This is an adapted extract from the Open University course Author(s): Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. All materials included in this unit are derived from content originated at the Open University. Introduction This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane. This is an adapted extract from the Open University course Mathematical methods and models (MST209) Introduction This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations. This unit is an adapted extract from the course Mathematical methods and models (MST209 1 First-order differential equations The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Click 'View document' to open the workbook (PDF, 1.6 MB). Acknowledgements All materials included in this unit are derived from content originated at the Open University. 1.5 Exercises A vector a has magnitude
|a|Â =Â 7 and direction
θ = −70°.
Calculate the component form of a, giving the components
correct to two decimal places. 1.4.2 Displacements and bearings The displacement from a point P to a point
Q is the change of position between the two points, as
described by the displacement vector
If P and Q represent places on the
ground, then it is natural to use a bearing to describe the
direct 1.4.1 Bearings In the following subsections, we apply the vector ideas
introduced so far to displacements and velocities. The
examples will feature directions referred to points of the compass,
known as bearings. The direction of Leeds relative to Bristol can be described
as ‘15° to the East of due North’,
or N 15° E. This is an instance of a
bearing. Directions on the ground are typically given like
this, in terms of the directions North (N),
South (S), East (E) 1.3: Summing vectors given in geometric form The following activity illustrates how the conversion
processes outlined in the preceding sections may come in useful. If
two vectors are given in geometric form, and their sum is sought in
the same form, one approach is to convert each of the vectors into
component form, add their corresponding components, and then convert
the sum back to geometric form. 1.2: Converting to geometric form You have seen how any vector given in geometric form, in
terms of magnitude and direction, can be written in component form.
You will now see how conversion in the opposite sense may be
achieved, starting from component form. In other words, given a
vector
a = a
1
i + a
2
j,
what are its magnitude |a| and direction θ? The first part of this question is dealt with using
Pythagoras’ Theorem: the magnitude of a v 1.1: Converting to component form In some applications of vectors there is a need to move
backwards and forwards between geometric form and component form; we
deal here with how to achieve this. To start with, we recall definitions of cosine and sine. If
P is a point on the unit circle, and the line segment
OP makes an angle θ measured anticlockwise
from the positive x-axis, then
cos θ is the x-coordinate of
P and sin θ is the
y-coordinate of P (
Exercise 1
