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Part 1

Expressions

arithmetic, dfns, order of execution

  1. Without using a computer, evaluate the following expressions. Then, use an APL interpreter to check your answers.

    1. 3×2+4
    2. 4÷2+6-2
    3. 3+2 2×2
    4. 4+2 6×2
    5. (3+(6×(2+3)))
    6. (3+(6×2)+3)
    7. 3+6×(2+3)
    8. ((3+6)×2+3)
    9. ((3+6)×3)+3
    10. ((3+6×2)+3)
    11. 2×7-6+3
    12. (2×7)-6+3

  2. The average daily temperatures, in degrees Celcius, for 7 days are stored in a variable t_allweek.

    t_allweek ← 11.7 8.6 9.7 14.2 6.7 11.8 9.2

    Use APL to compute the follwing:

    1. The highest daily temperature
    2. The lowest daily temperature
    3. The range of (difference between the largest and the smallest) temperatures
    4. Each temperature rounded to the nearest whole number

  3. Rewrite the following expressions so that they do not use parentheses.

    1. (÷a)×b
    2. (÷a)÷b
    3. (a+b)-5
    4. (a+b)+5

Simple Functions

The following problems can be solved with single-line dfns.

  1. Eggs

    A recipe serving 4 people uses 3 eggs. Write the function Eggs which computes the number of eggs which need cracking to serve ⍵ people. Using a fraction of an egg requires that a whole egg be cracked.

          Eggs 4
3

      Eggs 100
75

      Eggs ⍳12
1 2 3 3 4 5 6 6 7 8 9 9

  2. The formula to convert temperature from Celsius (\(T_C\)) to Fahrenheit (\(T_F\)) in traditional mathematical notation is as follows:

    \[T_F = {32 + {{9}\over{5}}\times {T_C}}\]

    Write the function CtoF to convert temperatures from Celcius to Farenheit.

          CtoF 11.3 23 0 16 ¯10 38
52.34 73.4 32 60.8 14 100.4

Generating Sequences

  1. A Mathematical Notation

    Use APL to evaluate the following

    1. \(\prod_{n=1}^{12} n\) (multiply together the first twelve integers)

    2. \(\sum_{n=1}^{17}n^2\) (add together the first seventeen squared integers)

    3. \(\sum_{n=1}^{100}2n\) (add together the first one hundred positive even integers)

    4. \(\sum_{n=1}^{100}2n-1\) (add together the first one hundred odd integers)

    5. In TMN, the following expression is equal to 0, why does the following return 70 in APL? 

            84 - 12 - 1 - 13 - 28 - 9 - 6 - 15  
70

  2. Pyramid Schemes

    1. Sugar cubes are stacked in an arrangement as shown by Figure 1.

      Stacked sugar cubes

      Figure 1. Stacked sugar cubes

      This stack has 4 layers and a total of 30 cubes. How many cubes are there in a similar stack with 467 layers?

    2. Now consider the stack in Figure 2.

      Differently stacked sugar cubes

      Figure 2. Differently stacked sugar cubes

      The arrangement in Figure 2 has 4 layers and 84 cubes. How many cubes are there in a similar stack with 812 layers?

    3. Now look at Figure 3.

      This is just a waste of sugar cubes by now...

      Figure 3. This is just a waste of sugar cubes by now...

      The stack in Figure 3 has 3 "layers" and 36 cubes in total. How many cubes are there in a similar stack with 68 "layers"?

  3. Write a function To which returns integers from ⍺ to ⍵ inclusive.

          3 To 3
3
      3 To 4
3 4
      1 To 7
1 2 3 4 5 6 7
      ¯3 To 5
¯3 ¯2 ¯1 0 1 2 3 4 5

    BONUS: What if ⍺>⍵?

          3 To 5
3 4 5
      5 To 3
5 4 3
      5 To ¯2
5 4 3 2 1 0 ¯1 ¯2