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8 Dealing with remainders

How do you deal with divisions where there is a remainder and there are no more digits you can carry to? In most cases you will need to express your answer as a decimal number rather than as a whole number plus a remainder.

Take the example of 518 divided by 8.

Figure 15
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3.5 Maths in archaeology

In several different parts of the world, footprints from prehistoric human civilisations have been found preserved in either sand or volcanic ash. From these tracks it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running by using the following three formulas:

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4.6 Hyperbola (e > 1)

A hyperbola is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d, where e > 1. We obtain a hyperbola in standard form if

  1. the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0;

  2. the directrix d is the line with equation x = a/e.


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4.2 Circles

Recall that a circle in 2 is the set of points (x, y) that lie at a fixed distance, called the radius, from a fixed point, called the centre of the circle. We can use the techniques of coordinate geometry to find the equation of a circle with a given centre and radius.
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3.1 Definition, properties and some applications

In the previous section we saw how to add two vectors and how to multiply a vector by a scalar, but we did not consider how to multiply two vectors. There are two different ways in which we can multiply two vectors, known as the dot product (or scalar product) and the vector product. They are given these names because the result of the first is a scalar and the result of the second is a vector. (We shall not consider vector products in this course.)

In the audio sec
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2.4 Components and the arithmetic of vectors

We introduce now a different method of representing vectors, which will make the manipulation of vectors much easier. Thus we shall avoid having to solve problems involving vectors by drawing the vectors and making measurements, which is very time-consuming and never very accurate.

We can think of a vector as a translation, that is, as representing a movement by a certain amount in a given direction. Then we can use the Cartesian axes in the plane or in Author(s): The Open University

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2.2 Multiplication by a scalar

In the collection of vectors sketched in Section 2.1, although v is not equal to c, the vectors v and c are closely related: c is a vector in the same direction as v, but it is twice as long as v. Thus it is natural to write 2v for c, since we can think of a journey repre
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1.8 Intersection of two planes

We saw earlier that two arbitrary lines in 2 may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in
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1.3 Parallel and perpendicular lines

We often wish to know whether two lines are parallel (that is, they never meet) or perpendicular (that is, they meet at right angles).

Two distinct lines, y = m1x + c1 and y = m2x + c2, are parallel if and only if they have the same gradient; that is, if and only if m1 = m2. For example, the lines y = −2x + 7 and
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Learning outcomes

By the end of this unit you should be able to:

  • Section 1

  • recognise the equation of a line in the plane;

  • determine the point of intersection of two lines in the plane, if it exists;

  • recognise the one-one correspondence between the set of points in three-dimensional space and the set of ordered triples of real numbers;

  • recognise the equation of a plane in three dimensions.


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1 The importance of evidence

The gathering, presentation and assessment of evidence are crucial and indeed inescapable parts of the practice of social science, hence the crucial role of evidence in the circuit of knowledge (see Figure 1).

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4.2.1 ATM physical layer

The ATM physical layer is divided into two sub-layers: the transmission convergence sub-layer and the physical medium sub-layer.

Functions of the transmission convergence sub-layer include generating and receiving cells, and generating and verifying the cyclic redundancy check in the header error control field. For correct interpretation of ATM cells it is important to identify the beginning of a cell. In theory, if ATM cells are transmitted as a continuous stream of bits,
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7.4 Elixirs of the nervous system: neurotrophins

According to Section 7.2 axons obtain an elixir from targets at their synapses.

Confirmation that there is indeed an elixir came from a series of events that reveals how much of science really works. Elmer Bucker, working with Hamburger in the mid-1940s, had removed a limb bud from a chick and replaced it with a tumour from
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2.1 Zero electrical resistance

In this section we shall discuss some of the most important electrical properties of superconductors, with discussion of magnetic properties to follow in the next section.

The most obvious characteristic of a superconductor is the complete disappearance of its electrical resistance below a temperature that is known as its critical temperature. Experiments have been carried out to attempt to detect whether there is any small residual resistance in the superconducting state. A sensitive t
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1.1 Introduction

An interesting analysis of Napoleon's involvement in Spain is provided by Stendhal in A Life of Napoleon, chapters 36 to 43. Stendhal argues that Napoleon's basic error was to see Spain as susceptible to the imposition by the French of the kind of enlightened reforms which had been welcomed elsewhere in Europe. Stendhal particularises, in a way characteristic of Romantic writers, on what he considers a highly distinctive Spanish national character, which in his view explains the hostil
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2.1.3 Concept cards

Another way to tackle unfamiliar words is to start a ‘concept card’ system, using index cards. When you meet a word which seems important, take a new card and write the word at the top, followed by any useful information you have found. File the cards alphabetically and add details as you come across new information. (It is worth getting an index card box anyway, then you can try out various ways of using it to organise your studies.)


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Drawn from Courtly India: The Conley Harris and Howard Truelove Collection
December 6, 2015 - March 27, 2016: This exhibition presents masterful drawings from the royal courts of northern India. Lovingly amassed by artist Conley Harris and architectural designer Howard Truelove, the collection features practice sketches, preparatory drawings, subtly modeled scenes, and lightly colored compositions created between the 1500s and 1800s. With images at different stages of completion, the collection allows for a fascinating look at Indian workshop practice. Alth
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Protestant Split
This second part of the interview to Dr. Brekus refers to the late 19th cnetury and early 20th century and to the impact liberalism had on protestantism. There is also an explanation to the influence of immigration.
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Pratiques innovantes en enseignement - apprentissage, l'exemple d'Innovalangues (Vidéo)

Monica Masperi

Maître de Conférences (Université Grenoble Alpes)

Pratiques innovantes en enseignement - apprentissage, l'exemple d'Innovalangues


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6 Phylogeny and cladistic analysis

In Section 3.3 the point was made that many physiologists consider that desert birds are successful because of their avian physiology, not because of any specific adaptations. While Williams and Tieleman's research on hoopoe larks demonstrated that desert species are capable of flexibility in metabolic rate and evaporative water loss, it suggested that adaptation is important too. The selective advantages of lowered BMR and TEWL for desert birds include reduced energy demand, and lower produc
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