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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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— = am~n provided a £ 0
an
When two terms in exponent form are divided the exponents are subtracted, provided they have the same base. The extra provision of a ^ 0 is needed because when a = 0, the denominator an = 0, and a term cannot be divided by zero.
Examples of this rule are
Subtracting the indices.
Exponents can be negative, and this is explained later
38 = 36
32 , 3 . ,.7 , -4
y ^y = >
an = an~n
an = a° = 1
Since an an = 1, provided an ^0
This last result is quoted as the following rule.
Rule 4:
a° = 1 provided a ^ 0
The extra provision of a ^ 0 is needed, because 0° is undefined. Examples of this rule are
10° = 1 (-4)° = 1 (a + b)° = 1 Provided (a + b) ^ 0
Rule 5:
(amy
According to this rule, when a term in exponent form is itself expressed to an exponent, then the exponents are multiplied.
Examples of this rule are
(54)3 = 512 (a4)3 = a12
Note. Confusion may arise, because sometimes we add exponents and at other times we multiply exponents, and the examples above make it clear when to add and when to multiply.
EXPONENT 191
Rule 6:
a~n = — an
This rule enables negative exponents to be expressed as positive exponents. Examples of this rule are
-9 1
6 “62
1
“ 36 3 x 42
3
4~2
48
Rule 7:
3 _2 _ 3
4^ Ay2
ra=am
This rule expresses the square root of a term as an exponent. It also expresses surds as exponents.
Examples of this rule are
91/2 = a/9 = 3
= (x16)m
= x8 When taking the square root of a term in
exponent form the exponent is halved
The bases of some numbers can be changed, and this is explained in the following example.
Example. Simplify the expression (2n x 42n)/8n.
Solution. Each of the bases 4 and 8 can be expressed in base 2:
2» x 42» 2n x (22)n
(2 3)"
2n x 22t' 23n 23n
23n
2=4 and 2=8 Using Rule 5 Using Rule 2
192 EXPONENTIAL CURVE
= 23n~3n Using Rule 3
= 2°
= 1 Using Rule 4
Calculators are programmed to work out exponents; check with your calculator handbook.
Reference: Square Root.
EXPONENTIAL CURVE
The exponential curve is sometimes called a growth curve, because it often models the way populations grow. This is a curve whose equation is y = ax, where a is any number greater than zero. For example, y = 2X is the equation of the exponential curve in figure a. The curve has a horizontal asymptote that is the x-axis.
The curve crosses the y-axis at y = 1, which is true for all curves of the type y = ax. A characteristic of this type of curve is that when a > 1 the curve becomes very steep. The function y = 2X is an increasing function for all values of x, which means its gradient is always positive.
When a is a fraction, say a = |, the graph of y = (|)x is as shown in figure b. This is also the graph of y = 2~x. This kind of curve is used to model radioactive decay. The function y = 2~x is a decreasing function for all values of x, which means its gradient is always negative.
EXPONENTIAL DECAY 193
References: Asymptote, Exponential Decay, Gradient, Graphs, Increasing Function, Logarithmic Curve.
EXPONENTIAL DECAY
The mass of a radioactive element decays over a period of time according to an exponential rule. The time taken to decay to half of its original mass is called the half-life of the element. The mass m of an iodine element has a fast rate of decay, as shown in the following table. The original mass of the element is 80 grams and in 8 days it decays to 40 grams, which is half its mass, so the element’s half-life is 8 days. The graph of this function is an exponential curve, as shown in figure a.
(a)
Time in days (f) 0 8 16 24 32
Mass of iodine in grams (m) 80 40 20 10 5
Exponential Growth The number n of cells of yeast, used to make bread, doubles every hour, and the data showing how the cells multiply is contained in the following table. The exponential graph showing the growth of this population is shown in figure b.
(b)
Time in hours (f) 0 12 3 4
Number of cells (n) 1 2 4 8 16
References: Graphs, Exponential Curve, Logarithmic Curve.
194 EXTERIOR ANGLE OF A POLYGON
EXTERIOR ANGLE OF A CYCLIC QUADRILATERAL
Reference: Cyclic Quadrilateral.
EXTERIOR ANGLE OF A POLYGON
This is the angle between one side of a polygon and the extension of the next side. In the pentagon in figure a it is the shaded angle. A pentagon has five exterior angles.
The geometry theorem about the exterior angles of a polygon states:
The sum of all the exterior angles of a polygon is 360°.
This theorem is true for all polygons whether it has 3 sides, 5 sides, or 100 sides.
Example. One of the exterior angles of a regular polygon is 40°. How many sides has it?
Solution. Two sides of the polygon are drawn in figure b. Each exterior angle is 40°. Dividing 360° by 40° will give the number of sides of the polygon:
360
Number of sides = ——
40
The number of sides of the polygon is 9, which means it is a nonagon.
References: Adjacent Angles, Angle, Nonagon, Pentagon, Polygon.
EXTRAPOLATION
195
EXTERIOR ANGLE OF A TRIANGLE
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