What is RPN?
 RPN stands for Reverse Polish
Notation. RPN was developed in 1920 by the Polish mathematician
Jan Lukasiewicz as a way to write a mathematical expression without using parentheses.
Why use RPN?
 RPN is more efficient for both the
user and the calculator.
 When you use RPN fewer keystrokes
are needed.
 With RPN there is no need to keep
track of parentheses.
 RPN allows the calculator to solve
very complex equations with out the need for lots of temporary
storage, or complex programing.
 Intermediate results are seen at
each step in the calculation which helps spot errors early, and
gives the user a greater understanding of calculations being
performed.
 Despite some users initial
reaction, RPN is more natural because it emulates the way
calculations are performed by hand.
How to use RPN:
(The RPN version described below
applies to the vast majority of HP calculators, there are some early
models that only have a stack of three, and the newer calculators
have very slight differences that are easy to see if you know how to
use the version described bellow)
Algebra and number crunching are two different things. Algebra must
be done before the calculations to crunch the numbers can begin.
Most calculators today use algebraic entry. This form of entry is
fine if you are doing algebra, but if you are simply crunching
numbers, RPN the more efficient and natural way to do it. Algebraic
entry calculators mimic the way algebra is done on paper. RPN entry
calculators mimic the way calculations are done on paper, which
makes it more natural when doing calculations..
First a quick example: multiply 37 by 81.
With a standard algebraic entry calculator, type “37 X 81 =”, and the
calculator provides the answer.
In RPN it works like this:
This may seem odd if you are not used to it, but
consider how you would perform this calculation on paper.
You would right down 37, then 81, and then perform the
multiplication operation. The ENTER key is just a way of telling the
calculator you are done with the first number and are ready to enter
the next number.
At this point you may have noticed that
the number of keystrokes required to perform this calculation are
exactly the same using both methods. For simple calculations this is
true. However, as soon as the calculation gets even slightly more
complex, RPN will require a significantly smaller number of
keystrokes. Before we can start with the more complex calculations we
first need to understand how RPN works.
RPN calculators use a stack to store
the numbers used in the calculations. This stack has 4 positions.
They are generally called X, Y, Z, and T. When you first type in a
number, it puts it pushes the stack up one and stores the number in
the X position:
T gets pushed off the top, Z goes into
T, Y goes into Z, X goes into Y and the new number is stored in X.
When you hit Enter it also pushes the stack up one position, and sets
it so the next number you type in will overwrite what is in X. When
you perform an operation like addition or multiplication it takes the
numbers in X and Y, performs the operation, puts the answer in X, and
moves everything down one. Similarly to how X is copied when you hit
enter, T is copied when the stack gets moved down after an operation
such as described above. For example: lets start with a zeroed stack
and see what happens when we perform the previous calculation.
T: 0.0000
Z: 0.0000
Y: 0.0000
X: 0.0000
You may have noticed the 0.0000 instead
of just 0. The reason for this is, RPN calculators round off the
number you see in the display to 4 decimal places by default. This
does NOT round off the number stored in the calculator. The
calculator stores the number to a higher precision than can be
displayed so round off error won't be a problem in the digits you can
see on the screen. The number of places displayed can be changed from
4 to whatever is desired.
First type
T: 0.0000
Z: 0.0000
Y: 0.0000
X: 37.0000
Puts 37 in X and moves the stack up one.

Then hit
T: 0.0000
Z: 0.0000
Y: 37.0000
X: 37.0000
Moves the stack up and leaves copy in X

Then type
T: 0.0000
Z: 0.0000
Y: 37.0000
X: 81.0000
Overwrites X with 81

Now hit
T: 0.0000
Z: 0.0000
Y: 0.0000
X: 2,997.0000
Puts answer in X, and moves the stack down one.


Y now has the value from Z, Z has the
value from T. T keeps a copy of whatever was there before, this is
kind of the reverse of hitting enter, where X is moved to Y and X
keeps a copy of what was there before we hit enter. It is hard to see
this, because all that is ever in Z and T is 0 in this simple
calculation.
Once we start performing more complex
calculations, Z and T will both be used. Sometimes when finding the
answer to very complex equations you may find you wish the stack were
bigger, however 99.9% of the time 4 is enough to perform the
calculation. In fact, I have never come across an equation I couldn't
solve with a stack of 4.
Lets try a slightly more complex formula to see how more of the stack comes into play.
The state of the stack is shown below for each step in this calculation.
Step 1)
T: 0.0000
Z: 0.0000
Y: 0.0000
X: 61.0000

Step 2)
T: 0.0000
Z: 0.0000
Y: 61.0000
X: 61.0000

Step 3)
T: 0.0000
Z: 0.0000
Y: 61.0000
X: 45.0000

Step 4)
T: 0.0000
Z: 0.0000
Y: 0.0000
X: 106.0000

Step 5)
T: 0.0000
Z: 0.0000
Y: 106.0000
X: 11.0000

Step 6)
T: 0.0000
Z: 106.0000
Y: 11.0000
X: 11.0000

Step 7)
T: 0.0000
Z: 106.0000
Y: 11.0000
X: 23.0000

Step 8)
T: 0.0000
Z: 0.0000
Y: 106.0000
X: 34.0000

Step 9)
T: 0.0000
Z: 0.0000
Y: 0.0000
X: 3.1176





For this calculation the X, Y, and Z
position on the stack were used, but not T. You have may have noticed
that there is no need to clear the value in the display before
starting the next calculation. Suppose we had two separate
calculations to do, 61+45 and 11+23. We would do everything the same
as above except for the last step. At step 4 we had the answer to
61+45. Then to find the answer to 11+23, we didn't need to clear the
106, we just started the next calculation.
Lets try this same problem on a simple
algebraic calculator.
Step 1) 61
Step 2) +
Step 3) 45
Step 4) =
At this point we have to save this
number somewhere. Either on paper or in the calculator memory if it
has any.
Step 5) store answer to 61 + 45
Step 6) clear
Step 7) 11
Step 8) +
Step 9) 23
Step 10) =
Step 11) store answer to 11 + 23
Step 12) recall answer to 61 + 45
Step 13) /
Step 14) recall answer to 11+23
Step 15) =
This example makes it clear that far
more steps are needed to find the solution with an algebraic
calculator than with RPN. With a little more sophisticated
scientific, algebraic calculator, parentheses can be used, and
intermediate answers do not need to be stored. However, this method
will still require more steps than RPN.
At this point only 0 has been in the T
position of the stack, so we have not observed how other numbers that
end up in T are copied every time the stack is shifted down. This
behavior of the T position can come in handy some times. As an
example, if you need to find the first 9 powers of a number, i.e.
X^1, X^2, X^3, . . . X^9 the T position in the stack is very handy.
To do this, type in the value of X and hit ENTER 3 times, then just
keep hitting X till you have x^9, recording the value in X at each
step. Say for example x were 2, bellow are listed all the steps with
the corresponding stack values.
Step 1)
T: 0.0000
Z: 0.0000
Y: 0.0000
X: 2.0000

Step 2)
T: 0.0000
Z: 0.0000
Y: 2.0000
X: 2.0000

Step 3)
T: 0.0000
Z: 2.0000
Y: 2.0000
X: 2.0000

Step 4)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 2.0000

Step 5)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 4.0000

Step 6)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 8.0000

Step 7)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 16.0000

Step 8)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 32.0000

Step 9)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 64.0000

Step 10)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 128.0000

Step 11)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 256.0000

Step 12)
T: 2.0000
Z: 2.0000
Y: 2.0000
X: 512.0000


Now for something a bit more complex: the quadratic equation. For our example
we will only get the first of the two possible answers. Note, we will be using
a new key for this example,
, short for change sign.
most newer calculators have a +/ key instead.)
Let a = 2, b= 7, and c = 15
Step 1)
T: 0.0000
Z: 0.0000
Y: 0.0000
X: 7.0000

Step 2)
T: 0.0000
Z: 0.0000
Y: 7.0000
X: 7.0000

Step 3)
T: 0.0000
Z: 0.0000
Y: 7.0000
X: 7.0000

Step 4)
T: 0.0000
Z: 7.0000
Y: 7.0000
X: 7.0000

Step 5)
T: 0.0000
Z: 0.0000
Y: 7.0000
X: 49.0000

Step 6)
T: 0.0000
Z: 7.0000
Y: 49.0000
X: 4.0000

Step 7)
T: 7.0000
Z: 49.0000
Y: 4.0000
X: 4.0000

Step 8)
T: 7.0000
Z: 49.0000
Y: 4.0000
X: 2.0000

Step 9)
T: 7.0000
Z: 7.0000
Y: 49.0000
X: 8.0000

Step 10)
T: 7.0000
Z: 49.0000
Y: 8.0000
X: 15.0000

Step 11)
T: 7.0000
Z: 7.0000
Y: 49.0000
X: 120.0000

Step 12)
T: 7.0000
Z: 7.0000
Y: 7.0000
X: 169.0000

Step 13)
T: 7.0000
Z: 7.0000
Y: 7.0000
X: 13.0000

Step 14)
T: 7.0000
Z: 7.0000
Y: 7.0000
X: 20.0000

Step 15)
T: 7.0000
Z: 7.0000
Y: 20.0000
X: 2.0000

Step 16)
T: 7.0000
Z: 20.0000
Y: 2.0000
X: 2.0000

Step 17)
T: 7.0000
Z: 20.0000
Y: 2.0000
X: 2.0000

Step 18)
T: 7.0000
Z: 7.0000
Y: 20.0000
X: 4.0000

Step 14)
T: 7.0000
Z: 7.0000
Y: 7.0000
X: 5.0000



In this example all 4 positions on the
stack were used. However if we were to move one step, the T position
on the stack would not have been needed. By moving step 1 to just
before step 14, the T position would not have been needed. This kind
of rearranging works fine for addition and multiplication, however if
subtraction or division are needed, it will leave us with the
operands in the wrong order. RPN calculators were designed with this
possibility in mind. That is what the
button is for, it
swaps the X and Y stack items. If we were to find the other solution
from the quadratic equation, step 14 would be a subtraction, In that
case, to avoid using the T position on the stack, insert step 1 and
an
operation before step 14.
The previous paragraph illustrates a simple rule to help make
the most of the 4 stack positions. Don't put a number on the stack
until you need it for a calculation.
