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Unit 1 - Objective 1 - Numbers


Example 1

Natural Numbers are 1, 2, 3, 4 and so forth.
Whole Numbers are 0, 1, 2, 3, 4 and so forth.
Integers include natural numbers, zero, and the opposites of the natural numbers. Examples of integers are ... -4, -3, -2, -1, 0, 1, 2, 3, 4, ...

The negative numbers are called the negative integers and the natural numbers are called positive integers. The number 0 is neither positive nor negative.

The numbers 1, 16, 753, and 7,234 are examples of positive integers.

The numbers -3, -20, and -5324 are examples of negative integers.

Rational Numbers include the integers and all other numbers that can be expressed as the quotient of two integers. A rational number can also be expressed by a decimal number that repeats or terminates.

The numbers 1/2, -4/5, 76/23, -86/87, 8 (because it can be expressed as 8/1) and 0 (even though it is neither positive or negative) are examples of rational numbers. Other numbers such as 0.500, -3.25, 41/11 (which is 3.727272... where 72 repeats) are rational numbers.

Irrational Numbers are numbers that cannot be expressed as the quotients of two integers. An irrational number can also be expressed by a decimal number that does not repeat or terminate.

The Numbers

     

are examples of irrational numbers where:

  • pi is approximately equal to 3.14159...,
  • e is approximately equal to 2.7182...,
  • the square root of 2 is equal to 1.414213..., and
  • the other roots also do not repeat or terminate.

Real Numbers are the rational numbers combined with the irrational numbers. Real Numbers may be represented by points on the number line. The usual line is a horizontal line that has been marked in equally spaced intervals. One of these marks is called the origin and is indicated by the number zero (0). The marks to the right are labled using the positive integers. The negative integers are used to designate the marks to the left of the origin.

The other real numbers are located between the integers.

Absolute Value is the distance of a number from zero. The absolute value of a number is indicated by placing the number between vertical bars, |   |

The absolute value of 5 is 5 and the absolute value of -10 is 10. Other examples are:

 

Inequalities
A number is larger than another if it is located farther to the right on the number line. We say the number farther to the right on the number line is greater than the other number. The symbol > is used to indicate that one number is greater than another. If the first number is to the left of the second number on the number line then the first number is less than the second. We use the symbol <.

2.3 > 1/2 because 2.3 is farther to the right than 1/2

-1 1/2 > -e because -1 1/2 is farther to the right than -e

-3/4 < 1/2 because -3/4 is farther to the left than 1/2

2 < 3 because 2 is farther to the left than 3

reciprocals

Every number, except zero, has a reciprocal. The reciprocal of a number is equal to 1 divided by that number. If the number is a fraction, then the reciprocal is 1 divided by the fraction or 1/fraction. The reciprocal of 5/4 is 1/5/4 or 4/5.


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Copyright 1996 by B. Chambers and P. Lowry. All Rights Reserved.