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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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Example. Work out “3 - +5.
Solution. Write
“3 — +5 = “3 + “5 Change subtraction into addition, and replace +5 by its
opposite, which is “5
= “8 Doing addition in the usual way
Multiplying Two Integers There are two rules to remember for multiplying two integers together:
1. If the signs of the two numbers are the same, either both + or both —, the answer is always positive.
2. If the signs of the two numbers are different, say one is + and one is —, the answer is always negative.
Examples are
_3 x “4 = +12 The signs of the two numbers are the same, both negative
-5x+3 = -15 The signs of the two numbers are different, one is negative
and the other is positive
If there are more than two numbers being multiplied, multiply them two at a time.
INTERIOR ANGLES 257
Dividing Two Integers The rales for obtaining the sign of the answer are exactly the same as those for obtaining the sign when multiplying two integers. An example is
+6 -f ~4 = -1.5 The signs of the two numbers are different, and 6-^4= 1.5
It is a straightforward process to use the calculator to add, subtract, multiply, and divide integers.
References: Number Line, Opposite Integers, Pythagoras’ Theorem.
INTERCEPT FORM OF A STRAIGHT LINE
Reference: Gradient-Intercept Form
INTERCEPT
An intercept is a point on the x-axis or y-axis where the graph of a straight line or a curve intersects the axis. An intercept can also be the value of the coordinate at that point. More information is given in the entries in the references, especially Graphs.
References: Coordinates, Equations, Gradient-Intercept Form, Graphs.
INTEREST
References: Compound Interest, Simple Interest.
INTERIOR ANGLES
Polygons have interior angles and exterior angles. When we talk about the angles of a polygon we usually mean the interior angles. The figure shown is a pentagon, which has five sides and five angles. The five angles, which are drawn shaded, are all interior angles, because they are inside the polygon. The sum of these five interior angles of the pentagon is 540°.
References: Exterior Angle of a Polygon, Polygon.
258 INTERQUARTILE RANGE
INTERPOLATION
Reference: Extrapolation.
INTERQUARTILE RANGE
Under this entry we will study the following: median, quartiles (upper and lower), interquartile range, and range.
One set of data is used to explain the different terms. The data we use are the set of test marks, out of 10, scored by Nathan’s class of 25 students in a mathematics test. The processes for dealing with a large amount of data are explained in the entry Cumulative Frequency Graph.
The set of 25 marks in the math test, ranked in order of size, is
1,1,2, 2,2, 2, 3, 3, 3,4,4,4, 5, 5,5, 6, 6,6,7,7,7, 8, 8, 8, 9
Med ian When the marks have been arranged in order of size the median is the middle mark. There are 25 marks, which is an odd number. To find the middle mark, we add 1 to 25 to get 26. Divide 26 by 2 to get 13. The median is the 13th mark, which can be counted from either end of the list. In this example the median mark is 5 (see figure a).
9
8
8
8
7
7
UQ=7[7
6
6
6
5
5
5___
4
4
4
3
3
T\ LQ=3 2 2 2 2 1
1
(a)
Lower Quartile (LQ) This is the middle mark of the bottom half of marks. When we are finding the upper and lower quartiles of an odd number of marks the median mark is included in both the bottom half of the list and also in the top half. There are now 13 marks in the bottom half, which is an odd number. To find the middle mark, we add 1 to 13 to get 14. Divide 14 by 2 = 7. The lower quartile is the 7th mark counted from the bottom of the list. The lower quartile mark is LQ = 3.
INTERQUARTILE RANGE 259
Upper Quartile (UQ) This is the middle mark of the top half of marks. Using the same process as in finding the LQ gives the upper quartile as the 7th mark counted from the top of the list. The upper quartile mark is UQ = 7.
Interquartile Range (IQR) This is the difference between the UQ and the LQ,
IQR = UQ - LQ IQR = 7 - 3 = 4
The interquartile range is 4.
The Range This is the difference between the highest mark (9) and the lowest mark (1),
Range = 9—1 = 8
The range is 8.
It is worth considering how the above calculation would have been affected had there been an even number of students in Nathan’s class (say 26 students) instead of 25. The process is slightly different (see figure b). Suppose the 26 marks scored in the test are
1,1,2, 2,2, 2, 3, 3, 3,4,4,4,4, 5, 5, 5, 6,6, 6,7,7,7, 8, 8, 9, 9
7
7
T\ UQ=7
6
6
6
5
5
Lsn
Median=4.5
4
4
4
3
3
3] LQ=3 2 2 2 2 1 1
(b)
260 INTERSECTING CHORDS
Median There is an even number of marks, so there will be two middle marks, and we have to add the two marks together and divide by 2 to get the median. The two middle marks are found by dividing 26 by 2 to give 13, and counting the 13th and 14th marks from either end of the list. The two middle marks are 4 and 5, so the median will be their mean, which is 4.5.
Q uart iles The median is obtained from the two marks 4 and 5. The lower one, which is 4, is included in the bottom half, and the higher one, which is 5, is included in the top half. Using the same procedures as before, we find that the lower quartile mark is LQ = 3 and the upper quartile mark is UQ = 7.
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