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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
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2, 2, 3, 2, 1, 3, 4, 3, 5, 3, 2, 2, 2, 1, 2, 4, 6, 3, 4, 2, 2, 1, 1, 2, 3, 2, 1, 5, 4, 3
To make these data more manageable, they are arranged as a frequency distribution in the form of a table, and then displayed as a bar graph (see figure):
Gaps are left between the columns of the bar graph, because the data are discrete.
Number of children Frequency
1 2 3 4 5 6
5 11 7 4 2 1
10
Number of Children in Families
8
Frequency 6 4
2
0
n
2 3 4 5 6
Number of children
References: Bar Graph, Data, Discrete, Frequency Distribution, Statistics.
DISTRIBUTIVE LAW
References: Addend, Associative Law, Commutative Law.
162 DIVISIBILITY TESTS
DIVIDEND
The dividend is the number that is being divided when we divide one number by another. For example, Joanne has 7 Easter eggs for her 3 children. She divides 7 Easter eggs by 3 children to get an answer of 2 eggs for each child and 1 egg left over. In this example 7 is the dividend, 3 is the divisor, 2 is the quotient, and 1 is the remainder. Who gets the egg left over? Dad of course!
The division of 7 by 3 to get an answer of 2 and leave a remainder 1 can be written as a division identity:
7 =3 x 2+1
Expressed in words, the division identity is
Dividend = divisor x quotient + remainder
If the remainder is zero, the dividend is exactly divisible by the divisor. This would be the case if Joanne had 6 Easter eggs instead of 7, and we would say that 6 is divisible by 3, because the remainder is zero.
DIVISIBILITY TESTS
These are short cuts to see if one counting number is exactly divisible by another counting number, without actually doing the division. They provide a check to see if one number is a factor or multiple of another number without actually doing the division process. Divisibility tests were more useful some years ago before calculators were freely available, but they still provide a quick test for divisibility.
Test 1. A number is divisible by 10 if the number ends in 0.
Example 1. The number 650 is divisible by 10, since it ends in 0, but 507 is not, since it ends in 7.
Test 2. A number is divisible by 5 if the number ends in 0 or ends in 5.
Example 2. The numbers 105 and 790 are both divisible by 5.
Test 3. A number is divisible by 2 if the number ends in 0 or 2,4, 6, 8. This means it is an even number.
Example 3. The number 738 is divisible by 2, since it ends in 8.
Test 4. A number is divisible by 3 if the sum of its digits is divisible by 3.
Example 4. The number 2673 is divisible by 3, because the sum of its digits is 2 + 6 + 7 + 3 = 18, and 18 is divisible by 3.
DIVISION IDENTITY 163
Test 5. A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
Example 5. The number 3524 is divisible by 4, because its last two digits form the number 24, which is divisible by 4.
Test 6. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
Example 6. The number 933,248 is divisible by 8, because its last three digits form the number 248, which is divisible by 8. Check: 248 -r 8 = 31.
Test 7. A number is divisible by 6 if its last digit is even and the sum of its digits is divisible by 3.
Example 7. The number 15,408 is divisible by 6 because its last digit, which is 8, is even, and the sum of its digits, which isl+5 + 4 + 0 + 8 = 18, is divisible by 6.
Test 8. A number is divisible by 9 if the sum of its digits is divisible by 9.
Example 8. The number 67,320 is divisible by 9, because the sum of its digits, which is6 + 7 + 3 + 2 + 0 = 18, is divisible by 9.
Test 9. A number is divisible by 11 if the difference and sum alternately of its digits, starting with the last digit and working in order to the first digit, make a number that is either 0 or a number divisible by 11.
Example 9. The number 2783 is divisible by 11, because 3 — 8 + 7 — 2 = 0.
Example 10. The number 10,857 is divisible by 11, because 7 — 5 + 8 — 0 + 1 = 11.
The calculator can be used for a divisibility test.
References: Digit, Even, Factor, Integer, Multiple.
DIVISIBLE
Reference: Dividend.
DIVISION IDENTITY
Reference: Dividend.
164 DODECAGON
DIVISOR
Reference: Dividend.
DODECAGON
A dodecagon is a polygon that has 12 sides. To find the angle sum of a dodecagon, search under the entry Polygon. A regular dodecagon is shown in figure a, and each interior angle is equal to 150°.
The dodecagon has 12 axes of symmetry, and its order of rotational symmetry is also 12. The regular dodecagon tessellates with two equilateral triangles and one square; see the entry Tessellations.
The regular dodecagon is made up of 12 isosceles triangles that are all the same size (see figure b).
References: Equilateral Triangle, Isosceles Triangle, Polygon, Regular Polygon, Symmetry, Tessellations.
DOT PLOT 165
DODECAHEDRON
This is a regular polyhedron (solid) which has 12 congruent faces and each face is a regular pentagon (see figure). The regular dodecahedron is one of the Platonic solids.
References: Net, Pentagon, Platonic Solids, Polyhedron.
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