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References: Distance, Height.
This is the difference between one quantity and another quantity. One of the quantities is usually a fixed quantity or a standard quantity.
One of the measures of intelligence is intelligence quotient, which is commonly referred to as IQ. A super brain has an IQ of 140. If Walton has an IQ of 128, the deviation of Walton’s brain from a super brain is
Reference: Standard Deviation.
The diagonal of a polygon is a line joining any two vertices, provided that the two vertices are not adjacent to each other on the polygon. For example, in the pentagon ABCDE in figure a, the line AD is a diagonal, and so is BD. The line AE is not a diagonal of the pentagon, because A and E are adjacent vertices on the pentagon. The pentagon has five diagonals, and they are drawn dashed in the figure.
Deviation = 140 — 128
If a polygon has n sides, the formula for the number of its diagonals is d = ft (ft — l)/2, where d is the number of diagonals.
The diagonal of a polyhedron is a line joining any two vertices, provided that the two vertices are not in the same face of the polyhedron. For example, in the cuboid ABCDEFGH in figure b, the line AG is a diagonal of the cuboid, because the vertices A and G are not in the same face of the cuboid. AF is not a diagonal of the cuboid, because the vertices A and F are in the same face of the cuboid, which is the face ABFE. Similarly, AH is not a diagonal. The other three diagonals are BH, CE, and DF.
References: Face, Pentagon, Polygon, Polyhedron, Vertex.
Dice is the plural of the word die. A die is a small cube, usually made of plastic, with its faces marked with dots representing the numbers 1, 2, 3,4, 5, and 6 (see figure a). Since a die is used in many board games and games of chance, it provides intriguing questions in probability. The numbers on its faces are arranged so that 1 is on the opposite face to 6, 2 is opposite to 5, and 3 is opposite to 4. They are arranged so that the numbers on opposite faces add to 7.
8 9 10 11 12
7 8 9 10 11
6 7 8 9 10
5 6 7 8 9
4 5 6 7 8
3 4 5 6 7
2 3 4 5 6
Example. Two dice, one red and one blue, are rolled together onto a table. The score is obtained by adding the number facing up on the red die to the number facing up on the blue die. What is the probability that the score will be 9? What is the most likely score?
Solution. This problem can be analyzed using a table of all possible outcomes as shown in figure b. When the blue die shows a 2 and the red die shows 4, the score is 6. The total number of ways the dice can score is 36, and this is called the total number
156 DIFFERENCE TABLES
of outcomes. The number of ways the dice can score 9 is 4 ways. This is called the number of favorable outcomes. Write
Number of ways of scoring 9
Probability(scoring 9) = —— -----------------------
Total number of outcomes
Probability(scoring 9) = - Canceling the fraction 9
The most likely score is 7, which occurs 6 times, as seen in the table.
References: Outcome, Probability of an Event, Tree Diagram.
DIFFERENCE OF TWO SQUARES
These tables are used to find a formula (or general term) for a sequence of numbers. We shall examine only two types of difference tables, the first relating to a linear formula and the second relating to a quadratic formula. They are explained in the following examples.
Example 1. Pat has $5 and saves $4 each week. Write down how much she will have saved after each week for the first 5 weeks, and find a formula for this sequence of numbers.
Solution. The sequence of money saved is written down, and the differences between the terms of the sequence are listed in a table, as follows:
0 weeks 1 st week 2nd week 3rd week 4 th week 5th week
Savings in $ Difference 5 4 9 4 13 4 17 4 21 4 25
The differences in the table are between one week’s savings and the next. When each difference is the same, as in this sequence of numbers, the formula for the sequence is a linear formula. This formula is expressed in terms of n, the number of weeks, and S, the accumulated savings. The linear formula, in general terms, is S = an + b, where a and b are constants which need to be found. The value of the constant a is always equal to the value of the differences in the table, which in this example is 4. Therefore
DIFFERENCE TABLES 157
a = 4, and so S = An + &. The value of & is found by substituting a value for 5 and a corresponding value for n into the formula 5 = 4n + b. For instance, when n = 0 and S = 5 we write
5 = 4n + b
5 = 4 x 0 + & Substituting S = 5 and n = 0 b = 5 Solving the equation
The formula is 5 = 4n + 5, where n is the number of weeks and 5 is the money saved in $.
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Example 2. If we arrange dots in the shape of triangles we obtain the set of triangle numbers, and the first four are shown in the figure. The next two triangle numbers are 15 and 21. The first five triangle numbers are entered in a table as follows that shows first- and second-order differences: