# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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100 10 Units

V)

Example. Add together 57, 29, and 47 using hundreds, tens, and units columns.

Solution. Arrange the numbers 57, 29, and 47 in columns. We add each column in turn, starting with the units column (see figure b):

100 10 Units

5 7+

2 9

1 4 2 7

1 3 3

13 23

(b)

66 CARTESIAN COORDINATES

â™¦ Units column: 7 + 9 + 7 = 23, which is two 10â€™s and three units. The two 10â€™s are carried to the column of 10â€™s as a 2, and the 3 is written in the units column.

â™¦ Tens column: 5 + 2 + 4 + 2= 13, which is really 130. The 100 is carried to the column of 100â€™s as a 1, and the 30 is written as a 3 in the 10â€™s column.

â™¦ Hundreds column: There is just a 1 in this column, which is the sum for the column.

Reference: Digit.

CARTESIAN COORDINATES

This is a system for fixing the position of a point in a plane according to its distances from two perpendicular number lines, called axes. An example is a good way to explain how Cartesian coordinates work.

A map is drawn of Treasure Island with an East axis and a North axis (see figure a). These axes are numbered at equal intervals. The position of any point on the map is known by stating both its number on the East axis and its number on the North axis. For Quick Sands the East number is 2 and the North number is 4. This reference for Quick Sands is abbreviated to (2, 4), with the East number written before the North number. We say the coordinates of the point Quick Sands is (2, 4) with respect to these axes.

The point where the two axes meet is called the origin, and instead of writing two zeros, we write one zero for the point of intersection.

CARTESIAN COORDINATES 67

RenĂ© Descartes (1596-1650) invented the grid system made up of two axes drawn on a grid of squares, and it bears his name: â€śCartesian.â€ť He called the East axis the x-axis and the North axis the y-axis and included negative numbers on these axes. The two numbers in a bracket that identify a point, like Quick Sand (2, 4), are called ordered pairs, as well as the coordinates of the point. The first number in the bracket, 2, is called the x coordinate, and the second number in the bracket, 4, is called the y coordinate. Plotting a point means marking the position of the point on a set of axes.

Example. A, B, and C are three of the points of a square, which are drawn on the grid shown in figure b. Write down the coordinates of A, B, and C, and find the coordinates of D, the fourth vertex of the square.

Solution. The coordinates are A(0, 3), B(â€”2, â€” 1), C(2, â€”3). The coordinates of D, the fourth vertex of the square, can be found by drawing a straight line through the vertex C parallel and equal in length to the line BA. This is done by counting two squares along and four squares up from C to reach the point D. The coordinates of D are (4, 1). Cartesian coordinates can be used to solve practical problems like the one below.

Example. Amanda operates a ferry service to the mainland. If a passenger has 80 kilograms of luggage, she charges him $16, for a passenger with 40 kilograms of luggage the charge is $12, and for a passenger with no luggage the charge is $8. What does she charge a passenger with 60 kilograms of luggage?

Solution. The problem can be solved using Cartesian coordinates by first writing the information regarding weight of luggage and charges in dollars as ordered pairs. The ordered pairs are (80, 16), (40, 12), and (0, 8). Axes are drawn on a grid (see figure c), with the horizontal axis labeled w for weight and the vertical axis labeled c for charges. The three points are plotted on the axes and joined up with a line, which happens to be straight.

The point on the straight-line graph is found where w = 60 kg, and the value of c at this point is 14. The cost for a passenger with 60 kg of luggage is $14.

68 CENSUS

Alternatively, the equation of the straight-line graph is found to be c = O.lw + 8, using the gradient-intercept method. The value of c is found by substituting w = 60 into this equation.

Reference: Gradient-intercept Method.

CELSIUS

This unit is used as a measure of temperature. It was formerly called centigrade, which means â€ś100 steps.â€ť

Reference: Temperature.

CENSUS

This is an official process of counting all the people in a population, and at the same time gathering other information about them such as their religion, age, gender, income, possessions, etc. Countries conduct a census of their population every few years. When an investigation is done into a sample of a population it is called a survey. A survey may take the form of administering a questionnaire, measuring some quantity, or counting some items. A survey is usually carried out on a sample of a population and the information gathered is used to make predictions and draw conclusions about the whole population. For example, a city council may survey a sample of its residents about where to build a new library, and use the information from the sample to gauge the view of all the city residents.

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