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Throughout most of human history computation has been a tedious task that was often postponed or avoided entirely. It is only in the last generation that the use of inexpensive handheld calculators has transformed the ways that people deal with quantitative data. Today the use and understanding of electronic computation is nearly indispensable for anyone engaged in technical work. There are a variety of inexpensive calculators available for student use. Some even have graphing and/or symbolic capabilities. The two calculators that will be described in these notes are the Casio fx-300W and the Texas Instruments TI-30Xa. They are both priced under twenty dollars, yet both possess enough computational power to handle the problems faced in most technical settings. While other calculator models have similar or even better features for performing the required computations, the reader will be responsible for learning their detailed use. Never throw away the user’s manual!
In order to perform a computation the correct keystrokes must be entered. In order to indicate the sequence of keystrokes the following notation will be used. Digits (0 through 9 plus any decimal point) will be presented in normal typeface. Any additional keystrokes will be enclosed in boxes. For example, to multiply seven times eight , the command sequence will be written as:
7
8
The number 56 will then appear on the display.
There are eight rows (horizontal) and six columns (vertical) of rectangular
keys on the fx-300W. The
key is in the sixth row, fourth column. The
key is in the eighth row, fifth column, or the bottom right-most key.
There are eight rows and five columns of keys on the TI-30Xa. The
key is in the fourth row, fifth column. The
key is in the eighth row, fifth column, or again the bottom right-most key.
The
key forces a computation and generally must be pressed at the end of any calculation.
The display on the Casio fx-300W has two lines. The top line shows the key strokes entered, while the larger second line shows the result of the calculation. In the first row beneath the display are non-rectangular buttons. The REPLAY buttons allow you to recall and/or edit the most recently entered calculation.
Most keys on both calculators have a second function printed above each key. These
secondary functions appear in gold on the Casio fx-300W and in yellow on the TI-30Xa.
To access these functions on the Casio fx-300W first press the
key on the top left of the calculator. To access these functions on the
TI-30Xa first press the
key also on the top left. To store a result on the Casio fx-300W press the
key (fourth row, first column) followed by one of six registers A through F accessed by
selecting a key from the third row. To store a result on the TI-30Xa press the
key followed by one of the three registers 1 through 3 accessed by
pressing the digit. To recall a stored result on either calculator, press the
button followed by the register key. On the Casio fx-300W you can also access registers
by pressing ALPHA from the row beneath the display followed by the
register name (A through F).
Mathematical expressions which involve more than one operation appear ambiguous. For example, is
or
To clarify this question, mathematics has developed the following hierarchy of computations called order of operations:
Perform all operations that appear in grouping symbols first. If grouping symbols are nested, do the innermost first.
Raise all bases to powers in the order encountered moving from left to right.
Perform all multiplications/divisions in the order encountered moving from left to right.
Perform all additions/subtractions in the order encountered moving from left to right.
Here grouping symbols means parentheses ( ), brackets [ ], braces { }, etc. An example of a nested expression is (6 + 2 × (4 + 1)) ¸ 8. The innermost grouping symbol is (4+1) so the result is
| (6 + 2 × (4 + 1)) ¸ 8 | = | (6 + 2 × 5) ¸ 8 |
| = | (6 + 10) ¸ 8 | |
| = | 16 ¸ 8 | |
| = | 2 |
Raising a base to a power (also known as an exponent) means repeated multiplication as in:
6³ = 6 × 6 × 6 = 216
To perform this calculation on the Casiofx-300W enter the following keystrokes:
6
3
The
key is in the first row, sixth column of the Casio.
To perform this same calculation on the TI-30Xa enter the following keystrokes
6
3
The
key is in the second row, fifth column of the TI-30Xa. We will cover exponents in greater
detail in Unit 3: Decimal Fractions.
In the original problem posed above the multiplication of 6 with 2 is performed before the addition of 5. The proper answer is therefore 17. The other interpretation could be achieved by using parentheses:
Order of operations is built into all scientific calculators. That is, if you enter the keystrokes in the correct order, the calculator will automatically perform the correct calculation. On both the Casio fx-300W and the TI-30Xa entering
5
6
2
yields the correct answer of 17.
In many formulas x occurs as a variable, but then confusion with the times sign can
result. To avoid this, alternative symbols for multiplication are used. They are the dot
notation and adjacent parentheses notation, as in
.
The Casio fx-300W does recognize the adjacent parentheses as multiplication, but
the TI-30Xa does does not. On the TI-30Xa the times operation
must be inserted between the parentheses.
Division is also indicated by a variety of notations. For example, the following all mean 34 divided by 17:
In addition to parentheses, brackets and braces, certain symbols act as implied grouping symbols. The most important of these are the fraction bar and the square root symbol. The fraction bar acts to separate the numerator from the denominator. If either or both of the numerator or denominator consist of an expression with operations, these must be performed first before the division indicated by the fraction bar. For example:
To perform this computation on the calculator, parentheses need to be inserted around both numerator and denominator as shown below:
7
3
2
3
The left parenthesis is in the fourth row, third column of the Casio fx-300W and the right parenthesis is the next key over, in the fourth row, fourth column. On the TI-30Xa the left and right parentheses are in the fourth row, the third and fourth columns, respectively. Parentheses are the only grouping symbol the calculator recognizes or uses.
The square root symbol also acts as a grouping symbol. Any calculation inside the square root needs to be completed before the root is taken. For example:
To perform this computation on the calculator parentheses need to be inserted around the expression inside the square root symbol. On the Casio fx-300W enter the following keystrokes:
25
144
On the TI-30Xa enter the keystrokes:
25
144
Note: On "newer" calculators like the Casio fx-300W one enters
expressions the way "they look". I.e., the square root symbol comes first. On
older models like the TI-30Xa some functions like
come after the expressions they are to evaluate. In any case, using parentheses keys when
necessary is a good habit to acquire. Failure to do so usually results in wrong answers!
47 + 271 = __________
19 + 67 + 43 = __________
269 + 151 - 237 = __________
135 + 932 - 612 = __________
45 × 125 = __________
167 × 134 - 21679 = __________
72498 ÷ 129 = __________
68 × 42 ÷ 136 = __________
12 × 9 - 8 × 7 ÷ 4 × 2 = __________
12 × 9 - 8 × 7 ÷ (4 × 2) = __________
4(19 - 12) ÷ 2 - 11 = __________
32 - 3 × 9 + 16 ÷ 4 × 2 = __________
= __________
= __________
= __________
= __________
From a board 10 feet long, a piece 29 inches was cut off. How long is the piece remaining?
Ignore the width of the cut.
Length = __________
In 1993 General Motors had total sales of $133,621,900,000, while Ford Motor Company had
sales of $108,521,000,000. How much more were GM’s sales than Ford’s?
Difference = __________
If 128 copies of a software package cost $5760, what is the cost per copy?
Cost = __________
A carpenter earns $14 per hour plus time and a half for overtime (more than 40 hours in one
week). If she works 58 hours one week, what is her gross pay for that week?
Pay = __________
You order 15 CDs at $8 a CD and 24 cassette tapes at $3 a tape. What is the cost of the order?
Cost = __________
A truck averages 16 miles per gallon and has a 25 gallon gas tank. What is the farthest
distance the truck can travel without stopping for gas?
Distance = __________
A stairway (shown in the diagram below) consists of rises of 6 inches and must reach a
height of 10 feet. How many rises are needed?
Number = __________
A stairway consists of 4 inch rises and treads of 18 inches. If the height of the stairs
is 4 feet, what is the distance taken up by the stairway on the lower floor?
Distance = __________
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© 2002 by Al Lehnen; HTML-ized by Kevin Mirus (kmirus@madison.tec.wi.us or kjmirus@execpc.com). |
| This document was last modified Wednesday, August 21, 2002, 4:10 PM. |