MATH 4
INTERMEDIATE ALGEBRA
SUMMER TERM  2001

REAL NUMBER SYSTEM

Naturals, Fractions, Decimals, Rationals, Square Roots, Irrationals, Reals, Formulas, Scientific Notation, Graphing

Integers - The set of numbers known as integers could be seen as a solution to the problems, which arise in the natural number set. It is about second grade when the concept of integers is generally introduced. One could look at it from the perspective of a child who, at lunch, borrows a sandwich from another second grader. The next week the child is asked to repay his debt, and although both children have an intuitive sense of what this means, neither can express the debt on paper. This is where the integers are introduced. Integers encompass all numbers that are included in the natural number set, but also encompass all of the negative whole numbers as well. Thus numbers such as 0, 1, 2, 3 are in the set of integers, but now -1, -2, -3 are also included in the set. In fact, for every positive number in the set of integers there is also a number equal in value, but opposite in sign. Integers, like natural numbers, do not include fraction such as 1/4. The formal definition of integers could be stated as:

I= {. . ., -4, -3,-2, -1, 0, 1, 2, 3, 4, . . . }

The collection of all such whole numbers, positive, negative or zero.

The set of integers has certain physical features to which it adheres.

  1. Like the natural numbers;
    1. one number from the set is always greater than or equal to another number from the set.
    2. the sum of any two integers will always be another integer.
    3. the product of any two integers will always be another integer.
    4. the set of integers does not always fulfill the division function, for three divided by five does not have an answer from within the set.

    2.  

  2. However, unlike the natural numbers;
    1. subtraction does hold for the integers, such that any integer subtracted from another integer will yield another integer.
    2. an integer cannot always be raised to the power of another integer, since

      2-2 = 1/4

      which is clearly not an integer.

Complete the following problems on a separate sheet of paper. Be sure to show all of the work necessary to complete the problems. (Calculators may be used to check the answer, but should not be used to solve the problem originally.)

Addition

  1. Find the sum of -28, 23, -19, -45 and 62.

    Step 1 - Add the positive numbers, and then add the negative numbers.

    MA2_1.jpg (3203 bytes)

    Step 2 - Subtract the smaller valued number from the larger valued number.

    MA2_2.jpg (1708 bytes)

    Step 3 - Give the answer in Step 2 the sign of the larger valued number.

    The answer is -7.

     

    HOMEWORK PROBLEMS :

  2. Find the sum of - 93, 4637, 201 and - 2745.
  3. Find the sum of 239, - 75423, 1096, - 387 and 37620.
  4. Find the sum of 1469, - 4732, 5832 - 5692, 12073 and -788.
  5. Find the sum of - 82723, -19545, 32961, -9074, 75814 and - 3984.

 

Subtraction

  1. Subtract - 219 from 1732.

    Step 1 - Write the problem using only symbols, not words.

    1732 - (- 219)

    2.  

     

    Step 2 - Change the subtraction symbol to an addition symbol and change the sign of the number being subtracted.

    1732 + (+219)

    Step 3 - Add the resulting integers.

    The answer is 1951.

    HOMEWORK PROBLEMS :

  2. Subtract 7539 from - 61358.
  3. Subtract - 78482 from 29045.
  4. Subtract 9597632 from - 20462384.
  5. Subtract - 2016832 from - 9016327.

Multiplication

  1. Multiply 29 times -732.

    Step 1 - Multiply the two numbers disregarding their signs.

    MA1_3.jpg (3610 bytes)

    Step 2 - If both numbers have the same sign, the answer is positive.

    Step 3 - If both numbers do not have the same sign, the answer is negative.

    The answer is -21228.

     

    HOMEWORK PROBLEMS :

  2. Multiply -329 times -752.
  3. Multiply -1782 times 2746.
  4. Multiply 4183 times -7639.
  5. Multiply 27406 times 478321.

 

Division

  1. Divide -1748 by 23.

    Step 1 - Divide the two numbers disregarding their signs.

    MA2_3.jpg (2745 bytes)

    Step 2 - If both numbers have the same sign, the answer is positive.

    Step 3 - If both numbers do not have the same sign, the answer is negative.

    The answer is -76.

    2. HOMEWORK PROBLEMS:

  2. Divide 330595 by -37.
  3. Divide -984753 by -231.
  4. Divide -2732478 by 714.
  5. Divide 12686362 by 5317.

RULE OF SIGNS

+ +  =  +

+ -  =  -

- +  =  -

- -  =  +

 

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