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Numbers and Their Application to Math and Science

Homework for Numbers Lesson 20

Mark the boxes corresponding to which classification of numbers the given number belongs. Some simplification may be necessary.

Note, the following notation is used for column headings: N-the Natural Numbers: [0], 1, 2, 3, ...; Z-the Integers: the natural numbers, 0, and their "opposites"; Digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9); Even (remainder is 0 when divided by 2); Q-the rational numbers (expressible as ratio of two integers); R/Q-the irrational numbers (reals minus rationals); R-the real numbers (all numbers on a number line); R+-the positive reals (x>0); R--the negative reals (x<0); R/A-transcendental numbers (reals minus algebraics); Imaginary-complex numbers with zero real part. In a final column are given a tally of check-marks expected for each row. Once that is correct, please tally each column for grading purposes.

  N     Z   Digit Even   Q     R/Q     R     R+     R-     R/A   Imaginary Tally
5            6
2/3            3
-7            4
           3
(16)           7
(-16)           1
(-15)           1
44           6
           4
1.765           3
-10000           5
-1½           3
-(6)           3
0 *          6+*
1            6
1/9           3
Tally *+          64

*Note: many computer scientists and mathematicians, especially number theorists, use zero indexing and thus consider zero the first natural number. You can safely ignore this box for the tallies.

 

 

 

  1. Carefully define each of the following sets of numbers. Be sure it is clear which words are previously defined or purposefully undefined.
    1. digits
    2. natural numbers
    3. counting numbers
    4. integers
    5. even numbers
    6. rational numbers
    1. real numbers
    2. irrational numbers
    3. transcendental numbers
    4. positive numbers
    5. negative numbers
    6. imaginary numbers

     

     

     

     

     

  2. Give an example of each type of number listed in the problem above.

     

     

     

     

     

  3. Give additional name or names for the set of natural numbers.  

  4. Which set of numbers was likely the first to be discovered?  

  5.  

  6. Explain how the decimal fraction 3.14 represents a rational or an irrational number.  

  7. Do repeating decimal fractions such as 2.666... always represent rational numbers? [Bonus: Is 0.99999.... equal to 1.000...?]  

  8. What real number is neither positive nor negative? [Bonus: What number is both real and imaginary?]
 

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