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!AWApr95/Goodies/Calc/ManualTxt
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File contents
Reverse Polish Calculator - By Circle Software - � Jan 1995
===========================================================
This program is shareware.
Please read the file Licence in this directory.
Introduction.
============
The Reverse Polish system is different from the way most calculators
work, and although this is simple, you should not try to use it without
first reading these notes.
The nature and origin of Reverse Polish notation is explained at the
end of this file.
Note that the !Help facility is fully implemented. Select Help from the
iconbar menu.
This manual is divided into sections.
Conventions: Describes the nomenclature used in the text.
A Quick Start: Gives simple instructions without getting technical.
How to use RPCalc: Explains the finer points of using the program.
Reverse Polish: Describes the origin and nature of Reverse Polish.
The Functions: Describes the use af all calculator functions.
More Complex Examples.
To find any function, search for the button as
[name] e.g. [+] or [Sine] etc.
N.B. To correctly display some button legends, your
text editor should be set to display standard
system font. e.g. [�] should appear as [down arrow].
Conventions
===========
Throughout this manual, references to button icons on the calculator
are shown between square brackets, e.g. [Enter].
The terms SELECT and ADJUST refer to the first and third mouse buttons,
as normally defined, and may be used to prefix a function button.
When not specifically stated, you should always use the SELECT button,
as in most cases, ADJUST will cause an alternative action.
A Quick Start
=============
First, see how easy it is to use by doing a simple addition calculation,
before going into details. Press the following keys in order -
[3] [Enter] [4] [+]
The value shown in the lower display is the result of adding 3 + 4, (7).
To multiply the result by 5, press -
[5] [x]
And we get 35.00 as expected. Its as simple as that!
Using this method you can always see exactly what values are about to be
operated on, unlike ordinary calculators.
You can also now see why it is called 'Reverse' Polish. On this calculator
you press these buttons in the reverse order, compared to a normal one.
However, this simple example may illustrate how to use the machine, but
does not show the real power of the system. For that, read on.
How to use RPCalc.
=================
The Displays.
The calculator has four display registers rather than the usual one.
The lower register, labelled 'X', is equivalent to the single display
found on ordinary calculators. Above this, the Y register is used to
hold a second value to be operated on, while the remainder will
display intermediate calculation results, where appropriate.
These registers constitute a 'stack', where numbers are placed until
needed.
When you click on the [+] operator button, for example, the value
displayed in the X register will be added to the value in Y, and the
result displayed in X.
A few functions produce two results, and in these cases the second
result will be displayed in the Y register.
Entering Values.
You enter values into the X register in the normal way, by clicking
on the number buttons, or by using the keyboard number pad. (In the
later case, the calculator must have the input focus. If not, click
in the calculator background, or the iconbar icon.)
To get a value into the Y register, first enter it into X, as above,
and then use the [Enter] key. This will move the value in the X
register up to the Y, and clear X ready for a new value.
When [Enter] is used, any values in higher registers will also be moved
up, and any value in the top register will be lost. This should rarely
happen.
Adding Two Numbers.
Thus, to add two values, enter the first value into X, click [Enter] to
push this up to Y, enter the second number into X, then click [+].
This simple example does not show the advantages of Reverse Polish over
the more conventional calculator, so let's try something more complex.
Evaluating an Expression.
To evaluate the expression ( 2 + 4 ) * ( 7 - 3 ) follow the button
clicks in the first column below, and check the actions listed.
In particular note the display contents as given in the last column.
Button Action Stack
====== ================================== ========
X Y
[2] '2' appears in X. 2
[Enter] The 2 moves up to Y, and X clears. 2
[4] '4' appears in X. 4 2
[+] '4' and '2' are unstacked, and 6 placed in X 6
[7] '7' appears in X, the 6 moves up to Y 7 6
[Enter] The 7 moves up to Y, and X clears. 7 6
[3] '3' appears in X 3 7 6
[-] '3' and '7' are unstacked, and 4 placed in X 4 6
[*] '4' and '6' are unstacked, and 24 placed in X 24
Thus X contains the final result, which we can use in further
calculations if required. Note that it is not necessary to write down,
remember, or re-enter either intermediate result 6 or 4, as these
remain on the stack until needed. This calculation required 3 of
the 4 registers provided, which you will find sufficient for the most
complex calculations.
Another Example
Suppose you needed to calculate the VAT due and the total price of
an item costing �123.50 We will use a property of the [Enter] button
invoked by clicking with the ADJUST button. When you do this, the value
in X will be moved up as before, but X will not be cleared, so its value
may be used again.
Press the buttons shown in the first column, as before -
( Remember to use the ADJUST button for [Enter] ).
Buttons Action Stack
======= ================================== =============
X Y
[1][2][3][.][5] 123.5 appears in X 123.50
[Enter] Use the ADJUST mouse button 123.50 123.50
[1][7][.][5] Stack rises & 17.5 appears in X 17.5 123.50 123.50
[%] X % of Y is computed in X 21.61 123.50
(This is the VAT due on 123.50)
[+] 21.61 + 123.50 is computed in X 145.11
(This is the total due.)
Note how the final total was computed with a single key stroke, without
the need to re-enter any values at all. This was because we had duplicated
the original 123.5 by clicking [Enter] with the ADJUST button.
The Storage Registers
In addition to the stack, the value in X may be stored in any of 10 special
registers numbered from 0 to 9. To do this click on [Store] followed by the
single digit indicating the required store number. To retrieve the stored
number click on 'Store' with the Adjust button, followed by the store number.
Try this -
Buttons Action Stack
======= ================================== =============
X Y
[pi] 3.14 appears in X 3.14
[Store] The store button stays 'in'. 3.14
[7] The value is stored. 3.14
[Clear] The X register is cleared 0.00
Now prove the value is stored by (Use the ADJUST button) -
[Store] Use the Adjust button to 'fetch' 0.00
[7] Value stored in store 7 is shown 3.14
In the VAT example above, we could have stored the VAT rate, 17.5, in one
of these storage registers, and used it in repeated VAT calculations.
The Display Format
In the above example, the value of 'pi' was shown as 3.14 because the
calculator is initially set to display only 2 decimal places. The full
value of pi (to about 15 significant figures) is of course used internally,
and may be displayed by increasing the decimal places using the adjuster
buttons at the top right of the calculator.
Other display modes are also available.
!Help
RPCalc supports the Acorn !help facility, which may be used to explore all
the functions on the calculator. To invoke this, select 'Help' from
the iconbar menu, and move the pointer over the function buttons provided.
You will find that most of the buttons provide two separate functions. The
function displayed on the button is invoked using the SELECT mouse button,
while another, often the inverse function, is invoked using the ADJUST
button.
Reverse Polish
==============
Reverse Polish is a notation system for mathematical expressions
invented by a Polish gentleman by the name of Lukasiewicz.
Unfortunately for him, nobody could pronounce his name so the
notation became known as Reverse Polish, and his name is all but
forgotten.
It is possible that even the notation itself would have been
forgotten were it not for the arrival of computers, where it has
become probably one of the most often used systems, not for
writing expressions, but for storing them for later computation,
as for example in compiled code.
The Forth programming language is based upon the Reverse Polish
system, and Postscript also works this way.
The tag 'reverse' simply refers to the fact that operators, such
as '+' and '-' are written after the values on which they operate
rather than before, as in more conventional notation.
The advantage of the notation is that it permits an expression
to be written without the use of parentheses, while remaining
totally un-ambiguous. For example, the expression -
( 2 + 4 ) * ( 7 - 3 )
would be written, or stored as -
2 4 + 7 3 - *
To evaluate this, the expression is scanned from left to right, and
any values found removed and placed aside on a stack (i.e. a heap or
pile). When an operator is found, the required number of values are
retrieved from the stack (last on, first off), the operation carried
out on them, and the result put back on the stack. This system is
continued to the end of the expression.
Thus in the above case, 2 and then 4 are placed on the stack before
the '+' operator is found. These values are then retrieved and the
addition carried out. The result, 6, is then placed back on the
stack.
Then 7 and 3 are found in turn and placed on top of the 6. At this
point a '-' operator is found, so the last two values, 7 and 3, are
retrieved and the result of the subtraction, 4, is placed back on
the stack, as before.
At this point the stack now has a 4 on top of 6, so when the last
operator is found, the '*', these two numbers are multiplied together
and the result, now the final answer, 24, is placed back on the
stack.
Notice that whenever an operator is found, only the required number of
values are unstacked, so that the system works equally well for unary
operators such as x! or sin(x), as for binary ('+' '/' etc) or other
operators.
The system may be extended to include multi-parameter functions, such
as function( a, b, c ). Which is simply written -
a b c function
It is also possible to include programming constructs, such as
if-then-else, repeat, etc, as is done in Forth and Postscript.
Note also that the notation does not require any knowledge of operator
priority. Operators are always executed when found, using however
many values are needed.
The Functions.
=============
This section gives full descriptions of each available function,
for each calculator button.
In all cases, X, or X and Y where appropriate, are unstacked, and
the result placed in X, unless otherwise stared.
Use of the SELECT mouse button is assumed where not stated.
The stack manipulations.
[Enter] Causes the stack to rise.
SELECT Clears the X register
ADJUST Preserves the X register.
[�] SELECT Rotates the stack downwards.
ADJUST Rotates the stack upwards.
In both cases end values are wrapped around.
[X� Y�] Swaps the X and Y register values.
The stack does not rise.
Simple Arithmetic.
[-] Subtract. Computes Y - X
[+] Add. Computes Y + X
[�] Multiply. Computes Y * X
[�] Divide. Computes Y � X
Functions of X.
[x!] Factorial. Computes x * (x-1) * (x-2) * (x-3) ... * 2 * 1
e.g. to compute factorial 10, key in -
[10] [x!] result 3628800
[1/x] Reciprocal. Computes 1 � x
e.g. to compute 1 / e, key in -
[e] [1/x] result 0.37
[x^y] Powers. ( Actual legend not reproducible here)
SELECT Computes X the the power of Y
ADJUST Computes Y the the power of X
[x�] Squares.
SELECT Computes x * x
ADJUST Computes the square root of x
Miscellaneous Functions.
[�] Changes the sign of X. i.e. -x becomes x, and x becomes -x
[Int] Integer.
SELECT Computes the integer part of x, removing fractions.
ADJUST Computes the fractional part of x.
[Abs] Absolute Computes |x| i.e. Removes any -ve sign.
Logarithms.
[Log] SELECT Computes the log to base 10.
ADJUST Computes the anti-log to base 10.
[Ln] SELECT Computes the log to base E.
ADJUST Computes the anti-log to base E.
Trig Functions.
[Sin] SELECT Computes sine(x)
ADJUST Computes angle whose sine is x
[Cos] SELECT Computes cos(x)
ADJUST Computes angle whose cosine is x
[Tan] SELECT Computes tan(x)
ADJUST Computes angle whose tangent is x
Hyperbolic Functions.
[Sinh] SELECT Computes sinh(x)
ADJUST Computes inverse of sinh(x)
[Cosh] SELECT Computes cosh(x)
ADJUST Computes inverse of cosh(x)
[Tanh] SELECT Computes tanh(x)
ADJUST Computes inverse of tanh(x)
Other Functions.
[Polar] This function produces two result values.
SELECT Computes polar from cartesian co-ordinates.
X = Sqrt( x*x + y*y )
Y = angle whose tangent is Y � X
e.g. Given a 3, 4, 5 triangle, key in -
[3] [Enter] [4] [Polar] results in -
X = 5, the (hypotenuse)
Y = 36.87�
ADJUST Computes cartesian from polar co-ordinates.
X = x * cos(y)
Y = x * sin(y)
[H.MS] Hours, Minutes and Seconds Conversion.
The display should be set to Fixed point, 4 decimal places
to use this function.
SELECT Computes hrs, mins, secs from decimal hrs.
i.e. result X = H.MMSS where MM == Minutes, SS == Seconds.
e.g. Given 12.56 hrs, key in -
[12.56] [H.MMSS] results in 12.3336
i.e. 12 hrs 33 mins 36 secs.
ADJUST Computes the inverse, converting hrs, mins, secs to hrs.
e.g. Map reference 12� 18', key in -
[12.18] ADJUST [H.MMSS] results in 12.30�
[yCx] Combinations and Permutations.
SELECT Computes the number of possible combinations of Y objects
taken X at a time, where order is unimportant.
e.g. To compute the number of ways to select 8 objects from a
total of 12, key in -
[12] [Enter] [8] [yCx], result 495.
N.B. This is how to compute (so called) permutations on your
Pools coupon. These are really Combinations, as order is
not important.
ADJUST Computes the number of possible permutations of Y objects
taken X at a time, where order IS important.
e.g. Taking the above example, key in -
[12] [Enter] [8] ADJUST [yCx], result 19,958,400
More Complex Examples.
=====================
1. A laser range finder gives the distance to the top of a building as
97.66 meters at an elevation of 12� 25'. Compute the height of the
building, and its horizontal distance.
We must first convert the angle to decimal degrees, then enter the
angular distance and convert to cartesian co-ordinates.
key in - [12.25] ADJUST [H.MS] [97.66] ADJUST [POLAR]
Results are X = 95.38, the horizontal distance.
Y = 21.00, the vertical height of the building.
2. Compute the roots of the equation: 3x� - 9x + 6 = 0
Given the equation -b � �(b� - 4ac) i.e. a = 3, b = -9, c = 6
----------------
2a
This example demonstrates the use of the internal storage registers
as well as some stack manipulation, all of which eliminates the
need to remember intermediate results, and the necessity to enter
the same value more than once.
The best way of tackling a problem like this is to start in the
middle, with the most complex part, like the inner root function.
Buttons Action Stack
======= ====================== ======================
[9] value -b 9
[Store] [1] We will need this later 0
[x�] compute b� 81
[4] enter 4 4 81
[Enter] push the stack up 4 81
[3] value a 3 4 81
[Store] [2] We will need this again 3 4 81
[x] compute 4a 12 81
[6] value c 6 12 81
[x] compute 4ac 72 81
[-] compute b� - 4ac 9
Adj [x�] compute sqr root. 3
Adj [Enter] duplicate value 3 3
Adj [Store] [1] recall -b 9 3 3
[+] compute -b + (b� - 4ac) 12 3
Adj [Store] [2] recall a 3 12 3
[2] value 2 2 3 12 3
[x] compute 2a 6 12 3
[�] First result = 2 2 3
[�] loose result 3
Adj [Store] [1] recall -b 9 3
[x� y�] swap values over 3 9
[-] compute -b - (b� - 4ac) 6
Adj [Store] [2] recall a 3 6
[2] value 2 2 3 6
[x] compute 2a 6 6
[�] Second result = 1 1
Hence, the roots are x = 1 and x = 2.
There may be shorter ways of doing this. Note that we stored -b and a
values for later use. In this simple contrived example this may have
been pointless, as they could easily be re-entered. However, in a real
example these values may have had 9 or more digits, so that storing
them would be a real time saver.
If the inner function had been negative indicating complex roots, we
would have had to proceed slightly differently.
End.
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00000010 61 6c 63 75 6c 61 74 6f 72 20 2d 20 42 79 20 43 |alculator - By C|
00000020 69 72 63 6c 65 20 53 6f 66 74 77 61 72 65 20 2d |ircle Software -|
00000030 20 a9 20 4a 61 6e 20 31 39 39 35 0a 3d 3d 3d 3d | . Jan 1995.====|
00000040 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d |================|
*
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00003320 0a 20 20 20 20 20 20 20 41 44 4a 55 53 54 20 20 |. ADJUST |
00003330 20 43 6f 6d 70 75 74 65 73 20 74 68 65 20 73 71 | Computes the sq|
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00003370 20 20 43 68 61 6e 67 65 73 20 74 68 65 20 73 69 | Changes the si|
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000033c0 20 20 20 20 20 20 53 45 4c 45 43 54 20 20 20 43 | SELECT C|
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00003410 20 20 20 43 6f 6d 70 75 74 65 73 20 74 68 65 20 | Computes the |
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000034a0 20 62 61 73 65 20 31 30 2e 0a 20 20 20 20 20 20 | base 10.. |
000034b0 20 41 44 4a 55 53 54 20 20 43 6f 6d 70 75 74 65 | ADJUST Compute|
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000034d0 6f 20 62 61 73 65 20 31 30 2e 0a 0a 5b 4c 6e 5d |o base 10...[Ln]|
000034e0 20 20 20 53 45 4c 45 43 54 20 20 43 6f 6d 70 75 | SELECT Compu|
000034f0 74 65 73 20 74 68 65 20 6c 6f 67 20 74 6f 20 62 |tes the log to b|
00003500 61 73 65 20 45 2e 0a 20 20 20 20 20 20 20 41 44 |ase E.. AD|
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00003540 6e 63 74 69 6f 6e 73 2e 0a 0a 5b 53 69 6e 5d 20 |nctions...[Sin] |
00003550 20 53 45 4c 45 43 54 20 20 43 6f 6d 70 75 74 65 | SELECT Compute|
00003560 73 20 73 69 6e 65 28 78 29 0a 20 20 20 20 20 20 |s sine(x). |
00003570 20 41 44 4a 55 53 54 20 20 43 6f 6d 70 75 74 65 | ADJUST Compute|
00003580 73 20 61 6e 67 6c 65 20 77 68 6f 73 65 20 73 69 |s angle whose si|
00003590 6e 65 20 69 73 20 78 0a 0a 5b 43 6f 73 5d 20 20 |ne is x..[Cos] |
000035a0 53 45 4c 45 43 54 20 20 43 6f 6d 70 75 74 65 73 |SELECT Computes|
000035b0 20 63 6f 73 28 78 29 0a 20 20 20 20 20 20 20 41 | cos(x). A|
000035c0 44 4a 55 53 54 20 20 43 6f 6d 70 75 74 65 73 20 |DJUST Computes |
000035d0 61 6e 67 6c 65 20 77 68 6f 73 65 20 63 6f 73 69 |angle whose cosi|
000035e0 6e 65 20 69 73 20 78 0a 0a 5b 54 61 6e 5d 20 20 |ne is x..[Tan] |
000035f0 53 45 4c 45 43 54 20 20 43 6f 6d 70 75 74 65 73 |SELECT Computes|
00003600 20 74 61 6e 28 78 29 0a 20 20 20 20 20 20 20 41 | tan(x). A|
00003610 44 4a 55 53 54 20 20 43 6f 6d 70 75 74 65 73 20 |DJUST Computes |
00003620 61 6e 67 6c 65 20 77 68 6f 73 65 20 74 61 6e 67 |angle whose tang|
00003630 65 6e 74 20 69 73 20 78 0a 0a 0a 48 79 70 65 72 |ent is x...Hyper|
00003640 62 6f 6c 69 63 20 46 75 6e 63 74 69 6f 6e 73 2e |bolic Functions.|
00003650 0a 0a 5b 53 69 6e 68 5d 20 53 45 4c 45 43 54 20 |..[Sinh] SELECT |
00003660 20 43 6f 6d 70 75 74 65 73 20 73 69 6e 68 28 78 | Computes sinh(x|
00003670 29 0a 20 20 20 20 20 20 20 41 44 4a 55 53 54 20 |). ADJUST |
00003680 20 43 6f 6d 70 75 74 65 73 20 69 6e 76 65 72 73 | Computes invers|
00003690 65 20 6f 66 20 73 69 6e 68 28 78 29 0a 0a 5b 43 |e of sinh(x)..[C|
000036a0 6f 73 68 5d 20 53 45 4c 45 43 54 20 20 43 6f 6d |osh] SELECT Com|
000036b0 70 75 74 65 73 20 63 6f 73 68 28 78 29 0a 20 20 |putes cosh(x). |
000036c0 20 20 20 20 20 41 44 4a 55 53 54 20 20 43 6f 6d | ADJUST Com|
000036d0 70 75 74 65 73 20 69 6e 76 65 72 73 65 20 6f 66 |putes inverse of|
000036e0 20 63 6f 73 68 28 78 29 0a 0a 5b 54 61 6e 68 5d | cosh(x)..[Tanh]|
000036f0 20 53 45 4c 45 43 54 20 20 43 6f 6d 70 75 74 65 | SELECT Compute|
00003700 73 20 74 61 6e 68 28 78 29 0a 20 20 20 20 20 20 |s tanh(x). |
00003710 20 41 44 4a 55 53 54 20 20 43 6f 6d 70 75 74 65 | ADJUST Compute|
00003720 73 20 69 6e 76 65 72 73 65 20 6f 66 20 74 61 6e |s inverse of tan|
00003730 68 28 78 29 0a 0a 0a 4f 74 68 65 72 20 46 75 6e |h(x)...Other Fun|
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000045b0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 63 6f | co|
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000045e0 20 20 20 5b 2d 5d 20 20 20 20 20 20 20 20 20 20 | [-] |
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00004740 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00004750 20 20 20 32 20 20 20 33 20 20 31 32 20 20 20 33 | 2 3 12 3|
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000047a0 f7 5d 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |.] |
000047b0 46 69 72 73 74 20 72 65 73 75 6c 74 20 3d 20 32 |First result = 2|
000047c0 20 20 20 20 20 20 20 20 20 20 20 32 20 20 20 33 | 2 3|
000047d0 0a 20 20 20 20 5b 8a 5d 20 20 20 20 20 20 20 20 |. [.] |
000047e0 20 20 20 20 20 20 6c 6f 6f 73 65 20 72 65 73 75 | loose resu|
000047f0 6c 74 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |lt |
00004800 20 33 0a 20 20 20 20 41 64 6a 20 5b 53 74 6f 72 | 3. Adj [Stor|
00004810 65 5d 20 5b 31 5d 20 20 72 65 63 61 6c 6c 20 2d |e] [1] recall -|
00004820 62 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |b |
00004830 20 20 20 39 20 20 20 33 0a 20 20 20 20 5b 78 8b | 9 3. [x.|
00004840 20 79 8a 5d 20 20 20 20 20 20 20 20 20 20 73 77 | y.] sw|
00004850 61 70 20 76 61 6c 75 65 73 20 6f 76 65 72 20 20 |ap values over |
00004860 20 20 20 20 20 20 20 20 20 33 20 20 20 39 0a 20 | 3 9. |
00004870 20 20 20 5b 2d 5d 20 20 20 20 20 20 20 20 20 20 | [-] |
00004880 20 20 20 20 63 6f 6d 70 75 74 65 20 2d 62 20 2d | compute -b -|
00004890 20 28 62 b2 20 2d 20 34 61 63 29 20 20 20 20 36 | (b. - 4ac) 6|
000048a0 0a 20 20 20 20 41 64 6a 20 5b 53 74 6f 72 65 5d |. Adj [Store]|
000048b0 20 5b 32 5d 20 20 72 65 63 61 6c 6c 20 61 20 20 | [2] recall a |
000048c0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
000048d0 20 33 20 20 20 36 0a 20 20 20 20 5b 32 5d 20 20 | 3 6. [2] |
000048e0 20 20 20 20 20 20 20 20 20 20 20 20 76 61 6c 75 | valu|
000048f0 65 20 32 20 20 20 20 20 20 20 20 20 20 20 20 20 |e 2 |
00004900 20 20 20 20 20 20 20 32 20 20 20 33 20 20 20 36 | 2 3 6|
00004910 0a 20 20 20 20 5b 78 5d 20 20 20 20 20 20 20 20 |. [x] |
00004920 20 20 20 20 20 20 63 6f 6d 70 75 74 65 20 32 61 | compute 2a|
00004930 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00004940 20 36 20 20 20 36 0a 20 20 20 20 5b f7 5d 20 20 | 6 6. [.] |
00004950 20 20 20 20 20 20 20 20 20 20 20 20 53 65 63 6f | Seco|
00004960 6e 64 20 72 65 73 75 6c 74 20 3d 20 31 20 20 20 |nd result = 1 |
00004970 20 20 20 20 20 20 20 31 0a 0a 48 65 6e 63 65 2c | 1..Hence,|
00004980 20 74 68 65 20 72 6f 6f 74 73 20 61 72 65 20 78 | the roots are x|
00004990 20 3d 20 31 20 61 6e 64 20 78 20 3d 20 32 2e 0a | = 1 and x = 2..|
000049a0 0a 54 68 65 72 65 20 6d 61 79 20 62 65 20 73 68 |.There may be sh|
000049b0 6f 72 74 65 72 20 77 61 79 73 20 6f 66 20 64 6f |orter ways of do|
000049c0 69 6e 67 20 74 68 69 73 2e 20 4e 6f 74 65 20 74 |ing this. Note t|
000049d0 68 61 74 20 77 65 20 73 74 6f 72 65 64 20 2d 62 |hat we stored -b|
000049e0 20 61 6e 64 20 61 20 0a 76 61 6c 75 65 73 20 66 | and a .values f|
000049f0 6f 72 20 6c 61 74 65 72 20 75 73 65 2e 20 49 6e |or later use. In|
00004a00 20 74 68 69 73 20 73 69 6d 70 6c 65 20 63 6f 6e | this simple con|
00004a10 74 72 69 76 65 64 20 65 78 61 6d 70 6c 65 20 74 |trived example t|
00004a20 68 69 73 20 6d 61 79 20 68 61 76 65 0a 62 65 65 |his may have.bee|
00004a30 6e 20 70 6f 69 6e 74 6c 65 73 73 2c 20 61 73 20 |n pointless, as |
00004a40 74 68 65 79 20 63 6f 75 6c 64 20 65 61 73 69 6c |they could easil|
00004a50 79 20 62 65 20 72 65 2d 65 6e 74 65 72 65 64 2e |y be re-entered.|
00004a60 20 48 6f 77 65 76 65 72 2c 20 69 6e 20 61 20 72 | However, in a r|
00004a70 65 61 6c 0a 65 78 61 6d 70 6c 65 20 74 68 65 73 |eal.example thes|
00004a80 65 20 76 61 6c 75 65 73 20 6d 61 79 20 68 61 76 |e values may hav|
00004a90 65 20 68 61 64 20 39 20 6f 72 20 6d 6f 72 65 20 |e had 9 or more |
00004aa0 64 69 67 69 74 73 2c 20 73 6f 20 74 68 61 74 20 |digits, so that |
00004ab0 73 74 6f 72 69 6e 67 0a 74 68 65 6d 20 77 6f 75 |storing.them wou|
00004ac0 6c 64 20 62 65 20 61 20 72 65 61 6c 20 74 69 6d |ld be a real tim|
00004ad0 65 20 73 61 76 65 72 2e 0a 0a 49 66 20 74 68 65 |e saver...If the|
00004ae0 20 69 6e 6e 65 72 20 66 75 6e 63 74 69 6f 6e 20 | inner function |
00004af0 68 61 64 20 62 65 65 6e 20 6e 65 67 61 74 69 76 |had been negativ|
00004b00 65 20 69 6e 64 69 63 61 74 69 6e 67 20 63 6f 6d |e indicating com|
00004b10 70 6c 65 78 20 72 6f 6f 74 73 2c 20 77 65 0a 77 |plex roots, we.w|
00004b20 6f 75 6c 64 20 68 61 76 65 20 68 61 64 20 74 6f |ould have had to|
00004b30 20 70 72 6f 63 65 65 64 20 73 6c 69 67 68 74 6c | proceed slightl|
00004b40 79 20 64 69 66 66 65 72 65 6e 74 6c 79 2e 0a 0a |y differently...|
00004b50 0a 0a 45 6e 64 2e 0a 0a |..End...|
00004b58 
.