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The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2
Download (direct link): theatozofmath2002.pdf
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References: Integers, Number Line.
The set of integers is {..., — 3, — 2, — 1, 0, 1, 2,3,...}. Negative integers are all those integers to the left of zero, which are {..., — 3, — 2, — 1}. Similarly, positive integers are all those integers to the right of zero, which are {1, 2, 3,...}. The set of counting numbers is the same as the positive integers. Zero is neither positive nor negative.
References: Integers, Negative.
A net is a two-dimensional drawing which can be cut out and folded up to make a solid shape. The various nets of an open cube are investigated under Pentominoes. The nets of the solid shapes, such as cube, cuboid, cylinder, cone, tetrahedron, etc., are illustrated under their own entries.
The study of networks, also known as graph theory, is part of a branch of mathematics called topology. A network is made up of a set of points that are joined by lines (see figure a). The lines joining the points are called arcs, and the points are called nodes. The lines do not have to be straight. The enclosed spaces in the network are called regions.
If the number of nodes in a network is denoted by N, the number of arcs by A, and the number of regions by R, then in the network in figure a, N = 5 nodes, A = 8 arcs, and R = 5 regions. We always count the outside of the network as a region. The data fit Euler’s formula for networks, which states that
N + R = A + 2
We can check that the network drawn in the figure does fit Euler’s formula:
5 + 5 = 8 + 2 Substituting into Euler’s formula iV = 5, A = 8, R = 5 Which is true.
A node is a point where arcs meet, or where an arc starts. The order of a node is the number of arcs leaving it. In figure b, the number at the side of each node is its order.
In addition to Euler’s formula, another interesting fact about networks is that for a network to exist it must have an even number of odd nodes. The network shown in figure c has two odd nodes, which means that this network complies with the rale, because it has an even number of odd nodes.
A network is traversable if it can be drawn without taking the pencil from the paper and traveling over every arc just once. You can pass through a node any number of times, but you cannot redraw a line. Euler discovered the following rale for a network to be traversable, whether you return to your starting point or not:
To be traversable, a network must have either two odd nodes, or it must have no odd nodes. It may have any number of even nodes (see figure d).
No odd nodes,
Two odd nodes,
Four odd nodes,
therefore traversable therefore traversable therefore not traversable
References: Euler’s Formula, Königsberg Bridge Problem.
References: Königsberg Bridge Problem, Networks.
Reference: Data.
A nonagon is a polygon with nine sides. Refer to the entry Polygon for more information on angle sum. Figure a shows three nonagons; the last one is a regular nonagon.
A regular nonagon has nine sides of equal length, and each of its nine angles is of equal size and equal to 140° (see figure b). The regular nonagon is made up of nine isosceles triangles, and each angle at the center of the nonagon is 40°, since there are 360° in a full turn.
References: Polygon, Regular Polygon.
In order to describe a normal, it is first necessary to know what a tangent is. A tangent to a curve is a straight line that touches the curve at a point and has the same gradient, or direction, as the curve at that point. Examples of tangents to curves are drawn in figure a, where the point of contact of the tangent with the curve is indicated by a dot. The drawing on the far right is an example where the tangent to the curve is at a point of inflection. At a point of inflection, the tangent changes from one side of the curve to the other.
We can now explain the term normal to a curve in relation to a tangent to a curve. A normal to a curve is a straight line that is at right angles to the tangent to the curve, and passes through the same point of contact with the curve as the tangent. In figure b, the tangents are drawn dashed and the normals are at right angles to the tangents. The normal to a circle passes through the center of the circle.
Gradients of Tangents and Normals If a tangent and normal are drawn to a curve at the same point, the product of their gradients is — 1. This rule is true for any two lines in the same plane that are at right angles. If the gradient of the tangent is m\ and the gradient of the normal is m2, then
If two lines with gradients m\ and m2 are perpendicular, then the quick way to obtain m2 is to turn mi upside down and change its sign.
So far we have discussed the normal to a curve, but there can also be a normal to a plane. A normal to a plane is a straight line that is at right angles to the plane.
References: Gradient; Inflection, Point of; Plane; Slope; Tangent.
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