1.01  Number Systems, Real Numbers, Operations

 from Basic Algebra: One Step at a Time © 2002

P. 1-6

In any study of mathematics, there must be number systems and operations (such as addition, subtraction, multiplication, and division).  It seems only "natural" to begin this study with the natural number system.  The natural numbers, also known as the counting numbers, is the set of numbers that is used for counting {1, 2, 3, 4 . . .} (three dots mean "and so on").  When the number 0 is included, this set {0, 1, 2, 3, 4 . . .} is the set of whole numbers.  With the advent of credit, it became necessary to have negative numbers.  The set of integers is defined to be the set of all whole numbers and their negatives:    {. . . -4, -3, -2, -1, 0, 1, 2, 3, 4 . . .} .

 You probably also noticed that each set of numbers so far is developed or built upon the previous set of numbers.  This makes each previous set of numbers a subset (i.e., a set contained within a set) of each succeeding set of numbers.  Illustrations that can be used to demonstrate this are called Venn Diagrams, named after the mathematician John Venn ( circa 1880).

Having built up to the set of rational numbers, there are some numbers that are not rational--that is, these numbers cannot be expressed as a ratio of integers.  For example, the solution to the equation  x² = 2 is  ± 2.  Another example of a number that cannot be expressed as a ratio of integers is the number  , whose value is approximately (but not exactly!) 22/7 or 3.14.  It can be proven that the actual value of   will never terminate, and it will never repeat a pattern.  The set of all numbers like  2, - 5, and   that never terminate and never repeat a pattern form the set of irrational numbers.

These two sets, the rational and the irrational numbers, are said to be disjoint sets, which means that they have no members in common.  The intersection of the sets (i.e., what is common to both sets) is the empty set, denoted  Фor { }.  Notice that f is the Greek letter “phi” (as in the Phi Theta Kappa Honor Society), and it is NOT a zero with a slash!!  Also notice that  {Ф} is not an acceptable notation for the empty set--this is not empty because it has a "Ф " in it!  The set of all rational and irrational numbers combined (the union of the two sets) is the real number system R.  [Note:  in math, this is the real thing!]

REAL NUMBERS:  All of the above!!

Now that you have all of these numbers, what can you do with them? Well, you can take them two at a time and add them, subtract them, multiply them, or divide them.  These are called arithmetic operations.  Since these operations combine two numbers at a time, they are called binary operations.   Addition uses the “+” sign,  subtraction the “-” sign.  For multiplication, the “x,” “×,” or “ ” sign is used.  Notice that either the cross sign or the dot means multiplication.

Example 1.     6 × 5 = 30, or 6 · 5 = 30.  Moreover, if no sign is given between the two numbers, then the operation is still multiplication, although parentheses are necessary to ensure that “(6)(5)” = 30 doesn’t look like “65."  All of the following mean the same:

                       6 × 5 = 30                  (6)(5) = 30                   (6) · (5) = 30          (6) × (5) = 30

                      6 ·  5 = 30                    6 (5) =  30                     6 · (5) = 30             6 × (5) = 30

                                                           (6) 5 = 30                    (6) · 5   = 30            (6) × 5 = 30

In algebra, there is a tendency to avoid use of the “×” or “x” signs for multiplication, since the letter “x” is usually reserved to be used as an unknown.  (See Section 1.05 on variables.)    

Another operation used in math as a shorthand notation is called “raising to a power.” 

Example 2.    Suppose you wanted to take 2 times itself five times: that is, 2×2×2×2×2 or 2·2·2·2·2.  Instead of writing out all those two’s, you can write 2⁵, where the 5 is a superscript.  The 5 is called an exponent, and 2⁵ is read “two raised to the fifth power.”  The value of 2⁵ is 32.   

Example 3.   3⁴ (read “three raised to the fourth power”) means 3·3·3·3, which equals 81.  In the case of raising to the second power, we call it “squaring,” while raising to the third power is called “cubing.”

Example 4.     3²  means   3 · 3  = 9   (“three raised to the second power” or “three squared.”)

Example 5.     3³  means 3 · 3 · 3 = 27 (“three raised to the third power” or “three cubed.”)

As there are different symbols to indicate the operation of multiplication, there are also different symbols for division. 

Example 6.  To indicate 12 divided by 4 (or 4 divided into 12), you may write

                        12 ÷ 4 = 3,        ,       12/4 = 3,     

EXERCISES.  Perform the indicated operations.

1.  7 · 8                       2.  6×9                       3.  7(9)                       4.  (8)(9)

5.  (8) · (6)                  6.  (7)×(6)                 7.   4²                            8.   5²

9.    2³                       10.   4³                         11.   5³                        12.   6³

13.    2⁴                       14.   4⁴                        15.   5⁴                      16.   6⁴

17.   10²                      18.  10³                       19.  10⁴                     20.  10⁵

21.   36 ÷ 4                  22.                         23.  72/8                   24. 

 25.                         26.  72 ÷ 6                   27.                  28.   92/4            

In the previous division exercises, you may have noticed that the answers all came out even.  However, math, like life, doesn’t always come out even.  Remember that every fraction can be expressed as a decimal by simply dividing the denominator into the numerator.  If you continue the division process far enough, there will either be no remainder (a terminating decimal) or a pattern will repeat (a repeating decimal)--guaranteed!  When the decimal repeats a pattern, the number(s) in the repeating pattern may be indicated with a bar over the number.  As an example, 0.3333 . . .  = 0.3  . 

Example 7.     Do you think there is a difference between the decimals 0.34 and 0.34  ? 

Answer:         Yes!  There is a difference.

                       In the decimal 0. 34, the 34 repeats, so you have 0.34343434 . . .

                       In the decimal 0.34, only the 4 repeats, so you have 0.344444 . . .

A calculator is a good way to convert fractions to decimals, by simply dividing the numerator by the denominator.  In most cases, it will be obvious whether the decimal is terminating or repeating.  Consider the following examples.

Example 8.        means    5 ÷ 4    or           which is 1.25    (a terminating decimal!)

Example 9.        means    1 ÷ 3    or  which is 0.3333. . .   or    0.3

Example 10.   means    6 ÷ 11   or  which is 0.545454 . . .    or 0. 54

Example 11.     means  3 ÷ 7   or  which is 0.428571428571 . . .   or 0.428571

Example 12.     means   1 ÷ 16   or  which is 0.0625  (a terminating decimal!)

EXERCISES.   Divide the fractions (by calculator and by hand) to express as terminating or repeating decimals.

29.                                    30.                           31.                                    32.    

33.                                   34.                         35.                                36. 

37.                                   38.                         39.                                40.    

ANSWERS 1.01

 p. 1 - 6:           

  1. 56; 2. 54; 3. 63; 4. 72; 5. 48; 6. 42; 7. 16; 8. 25;  9. 8;  10. 64; 11. 125;  12. 216;  13. 16;  14. 256;    15. 625;  16. 1296;  17. 100;  18. 1000;  19. 10,000;  20. 100,000;  21. 9;  22. 8;  23. 9; 24. 4; 25. 18; 26. 12;  27. 6;  28. 23;  29. 0.75;  30. 0.6;  31. 0.2;  32. 0.18;  33. 1.625;  34. 4.3;  35. 0.3125; 36. 0.15; 37. 0.3863; 38. 0.425; 39. 0.3571428; 40. 1.190476.