Archive for November, 2009

Function of a variable

A function is a rule that associates, with each value of a variable x in a certain set, exactly one value of another variable y. The variable y is then called the dependent variable, and x is called the independent variable. The set from which the values of x can be chosen is called the domain of the function. The set of all the corresponding values of y is called the range of the function. Examples are below:

If f(x) = x3 – 4x + 2, then
f(1) = 13 – 4(1) + 2 = 1 – 4 + 2 = -1
f(-2) = -23 – 4 (-2) + 2 = -8 + 8 + 2 = 2
f(a) = a3 – 4a + 2

The function f(x) = 18x – 3x2 is defined for every number x; that is, without exception, 18x – 3x2 is a real number whenever x is a real number. Thus, the domain of the function is the set of all real numbers.

The area A of a certain rectangle, one of whose sides has length x, is given by A = 18x – 3x2 is a real number whenever x is a real number. Thus, the domain of the function is the set of all real numbers.

The area A of a certain rectangle, one of whose sides has length x, is given by A = 18x – 3x2. Here, both x and A must be positive. By completing the square, we obtain A = -3(x -3)2 + 27. In order to have A > 0, we must have 3(x-3)2 < 27. In order to have A > 0, we must have 3(x-3)2 < 27, which limits x to value below 6; hence, 0 < x < 6. Thus, the function determining A has the open interval (0,6) as its domain.

Introduction of Hindu-Arabic & Roman numerals

At first, the ancients developed names for the numbers. They spoke of having one sheep, two sheep, etc. But you can see how difficult it would be to add or subtract columns of numbers expressed only in words. Thus we learn that arithmetic computation did not begin until man came to use symbols for numbers. The kinds of symbols used for numbers went through various changes starting with the simple vertical mark of ancient Mesopotamia, progressing to the combination of the Egyptians, the familiar numerals of the Romans, and finally to our present figures.

We are indebted to the Arabs for our present method of writing numbers. For this reason, the numerals 0 through 9, the ingredients for any number combinations we wish to write, were called Arabic numbers for a long time. But more recently historians have discovered that the system of writing numbers now used by civilized people throughout the world was originated by the Hindus in India. The Arabs learned the system from the Hindus and are credited with having brought it to Europe soon after the conquest of Spain in the eighth century AD. For this reason, we now property call it the Hindu-Arabic system of numerals.

Reading and writing Roman numerals

An early system of writing numbers is the Roman system. It is generally agreed that it is of little practical value in today’s world of advanced mathematics.

Because you will still see Roman numerals used in recording dates, in books, as numbers of a clock face, and in other places, it is worth taking a little time to learn how to read them.

The Roman number system is based on seven letters, all of which are assigned specific values. They are:

I=1, V=5, X=10, L=50, C=100, D=500, M=1000

Here are a few rules to help you read Roman numerals:

Rule 1: When a letter is repeated, its value is repeated.
eg, I=1, II=2, III=3, XX=20, CCC=300

Rule 2: When a letter follows a letter of greater value, its value is added to the greater value.
eg, VI=6, XV=15, LX=60, DC=600

In these examples, observe that the smaller value I after V means add 1 to the 5 to give 6. In the same way, the V following the X means add 5 to 10 which equals 15. Similarly, LX represents 10 added to 50 to give 60. To write 70, merely add XX after the L to give LXX. In like manner, to write 800, add CC after DC to give DCCC.


Some basic ideas of arithmetic

There are four basic operations in mathematics: addition, subtraction, multiplication, and division. Often when we talk about a collection of numbers, such as the numbers 1, 2, and 3, we use the word set. We could use set notation with braces, [ ], to list the number: [1, 2, 3]. The set of even numbers could be written as [2, 4, 6, 8, 10, …], and the set of odd numbers as [1, 3, 5, 7,…]. (The three dots indicate that the numbers continue indefinitely. In any collection of numbers ending in dots, there is no largest number.)

Here, we deal with two sets of numbers: the counting numbers [1, 2, 3, 4,…] and the whole numbers [0, 1, 2, 3,…]. The whole numbers are just the counting numbers plus zero. When we count, we start with 1. When we answer the question “How many?” we need zero as a possible answer.

Symbols are necessary to make mathematical statements complete. For example, we use symbols for addition (+) and multiplication (X).

= as in 8 + 3 = 11
8 plus 3 equals 11

< as in 3 < 8
3 is less than 8

> as in 8 > 3
8 is greater than 3

Notice that the symbols for less than and greater than are always open toward the larger number. When statements are not true, we put a slash through the symbol:

6 + 3 =/ 11
6 + 3 does not equal 11

5 >/ 7
5 is not greater than 7

9 </ 6
9 is not less than 6

Numerals are symbols for numbers, which are abstract ideas. For example, a fisherman 8000 years ago might record that he caught ||| fish. We could write 3 for the amount ||| and 3 are the symbols for the same numbers. Our number symbols are called arabic numerals.

Digits are the number symbols (numerals) 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 in our number system. Numbers are written as combinations of any of these ten digits.

A whole number is written as a string of digits, 7 is a one-digit number; 32 is a two-digit number with 3 as the first digit and 2 as the second digit; 487 is a three-digit number with 4 as the first digit, 8 as the second digit, and 7 as the third digit.


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