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Solution. The factors of 8 are 1, 2, and 4, not including itself. 1 + 2 + 4 = 7, so 8 is not a perfect number since it is not the sum of all its factors.
The followers of Pythagoras (who lived about 450 BC) thought that perfect numbers had mystical properties. For example, God created the world in 6 days, and the number 6 is a perfect number. Until the year 1952 there were only 12 known perfect numbers and all of them were even numbers. No one is quite sure whether or not a perfect number can be an odd number. The first three perfect numbers are 6, 28, and 496. At present only 24 perfect numbers have been discovered.
References: Factor, Pythagoras’ Theorem, Whole Numbers.
Reference: Completing the Square.
The perimeter of a closed curve or of a polygon is either the boundary line of the closed curve or polygon, or it can mean the total length of the boundary line. In this entry we will discuss the total length of the boundary line. The perimeter of the rectangle in figure a, drawn on a 1-cm grid, is found by adding together the lengths of the four sides, which will be the total length of the boundary line:
Perimeter =4+3+4+3 = 14
The perimeter of the rectangle is 14 cm.
An alternative method for finding the perimeter of the rectangle is to add together 4 and 3 cm and double the answer.
For curved shapes like the circle it is difficult to find the perimeter by measuring the length of the boundary line, so we use a formula for the perimeter. The perimeter of a circle is usually called the circumference. To find the perimeters of well-known shapes, such as the circle, the parallelogram, the trapezium, the triangle, etc., see the respective entries.
References: Area, Diameter, Pi, Polygon, Radius, Rectangle.
These are also called arrangements. Reference: Combinations.
References: Locus, Mediator.
Two straight lines are perpendicular if they are at right angles to each other, which means that the angle between them is 90°. For example, If two lines are perpendicular and one of them is horizontal, then the other line is vertical.
The symbol for pi is tt, which is a letter of the Greek alphabet. The value of it is available in a calculator as tt = 3.141592653... and cannot be written down as an exact value, tt is an irrational number, which means it cannot be written as a fraction, and can never be expressed as a recurring decimal. Sometimes the fraction 22/7 is used as a value for tt, but when expressed as a decimal is only accurate to two decimal places. Certain sentences or verses can be used to help remember its value. For example, the value of tt to seven decimal places can be remembered according to the number of letters in the respective words of the following sentence:
MAY I HAVE A LARGE CONTAINER OF COFFEE?
3 1 4 1 5 9 2 6
The origin of tt can be explained as follows: Suppose you obtain a circular object, say a dinner plate, and measure its circumference (C) and its diameter (D) using a length of string and a ruler. The definition of tt is
it = —
You can calculate an approximate value for tt using your dinner plate measurements by dividing the circumference C by the diameter D. Using these measurements, provided they are fairly accurate, you can expect your value for tt to be accurate to one decimal place. In Biblical times the ancient Hebrews believed that the value for tt was equal to 3 (see 2 Chronicles, Chapter 4, Verse 2). An ancient Chinese ratio for tt was 355/113, which is accurate to six decimal places when expressed as a decimal.
Throughout history mathematicians have striven to improve on the accuracy of tt, and various formulas have been developed to achieve this. By 300 AD, the mathematician Ptolemy had calculated tt to be 3.1416, which is correct to four decimal places. Over 300 years ago a formula for tt was known to be given by the following the infinite series, although it converges too slowly to be of practical use:
TT 1 1 1 1 1
4 _ 1 _ 3 + 5 _ 7 + 9
Using computers, mathematicians have calculated tt to over one billion decimal places. We use tt to find, where applicable, the perimeters, areas, and volumes of shapes like the circle, the sphere, the cylinder, and the cone. Explanations are given under the respective entries.
References: Area, Irrational Numbers, Perimeter, Volume.
A pictogram is one of the statistical graphs used for displaying discrete data. The graph is made up of a series of pictures, and each picture represents a certain quantity of data.
Example. Pat is the top salesperson for “We Sell,” a business that sells Real Estate. The number of houses sold per year by each member of the “We Sell” team is shown in the table. Draw a pictogram to represent these sales.
Salesperson Number of Houses Sold