Go to the previous, next section.
This chapter explains how to use Calc and its many features, in a step-by-step, tutorial way. You are encouraged to run Calc and work along with the examples as you read (see section Starting Calc). If you are already familiar with advanced calculators, you may wish to skip on to the rest of this manual.
This tutorial describes the standard user interface of Calc only. The "Quick Mode" and "Keypad Mode" interfaces are fairly self-explanatory. See section Embedded Mode, for a description of the "Embedded Mode" interface.
The easiest way to read this tutorial on-line is to have two windows on your Emacs screen, one with Calc and one with the Info system. (If you have a printed copy of the manual you can use that instead.) Press M-# c to turn Calc on or to switch into the Calc window, and press M-# i to start the Info system or to switch into its window.
This tutorial is designed to be done in sequence. But the rest of this manual does not assume you have gone through the tutorial. The tutorial does not cover everything in the Calculator, but it touches on most general areas.
The Calc Summary at the end of the reference manual includes some blank space for your own use. You may wish to keep notes there as you learn Calc.
In this section, we learn how RPN and algebraic-style calculations work, how to undo and redo an operation done by mistake, and how to control various modes of the Calculator.
The central component of an RPN calculator is the stack. A calculator stack is like a stack of dishes. New dishes (numbers) are added at the top of the stack, and numbers are normally only removed from the top of the stack.
In an operation like 2+3, the 2 and 3 are called the operands and the + is the operator. In an RPN calculator you always enter the operands first, then the operator. Each time you type a number, Calc adds or pushes it onto the top of the Stack. When you press an operator key like +, Calc pops the appropriate number of operands from the stack and pushes back the result.
Thus we could add the numbers 2 and 3 in an RPN calculator by typing: 2 RET 3 RET +. (The RET key, Return, corresponds to the ENTER key on traditional RPN calculators.) Try this now if you wish; type M-# c to switch into the Calc window (you can type M-# c again or M-# o to switch back to the Tutorial window). The first four keystrokes "push" the numbers 2 and 3 onto the stack. The + key "pops" the top two numbers from the stack, adds them, and pushes the result (5) back onto the stack. Here's how the stack will look at various points throughout the calculation:
. 1: 2 2: 2 1: 5 .
. 1: 3 .
.
M-# c 2 RET 3 RET + DEL
The `.' symbol is a marker that represents the top of the stack. Note that the "top" of the stack is really shown at the bottom of the Stack window. This may seem backwards, but it turns out to be less distracting in regular use.
The numbers `1:' and `2:' on the left are stack level numbers. Old RPN calculators always had four stack levels called x, y, z, and t. Calc's stack can grow as large as you like, so it uses numbers instead of letters. Some stack-manipulation commands accept a numeric argument that says which stack level to work on. Normal commands like + always work on the top few levels of the stack.
The Stack buffer is just an Emacs buffer, and you can move around in it using the regular Emacs motion commands. But no matter where the cursor is, even if you have scrolled the `.' marker out of view, most Calc commands always move the cursor back down to level 1 before doing anything. It is possible to move the `.' marker upwards through the stack, temporarily "hiding" some numbers from commands like +. This is called stack truncation and we will not cover it in this tutorial; see section Truncating the Stack, if you are interested.
You don't really need the second RET in 2 RET 3 RET +. That's because if you type any operator name or other non-numeric key when you are entering a number, the Calculator automatically enters that number and then does the requested command. Thus 2 RET 3 + will work just as well.
Examples in this tutorial will often omit RET even when the stack displays shown would only happen if you did press RET:
1: 2 2: 2 1: 5
. 1: 3 .
.
2 RET 3 +
Here, after pressing 3 the stack would really show `1: 2' with `Calc: 3' in the minibuffer. In these situations, you can press the optional RET to see the stack as the figure shows.
(*) Exercise 1. (This tutorial will include exercises at various points. Try them if you wish. Answers to all the exercises are located at the end of the Tutorial chapter. Each exercise will include a cross-reference to its particular answer. If you are reading with the Emacs Info system, press f and the exercise number to go to the answer, then the letter l to return to where you were.)
Here's the first exercise: What will the keystrokes 1 RET 2 RET 3 RET 4 + * - compute? (`*' is the symbol for multiplication.) Figure it out by hand, then try it with Calc to see if you're right. See section RPN Tutorial Exercise 1. (*)
(*) Exercise 2. Compute @c{$(2\times4) + (7\times9.4) + {5\over4}$} 2*4 + 7*9.5 + 5/4 using the stack. See section RPN Tutorial Exercise 2. (*)
The DEL key is called Backspace on some keyboards. It is whatever key you would use to correct a simple typing error when regularly using Emacs. The DEL key pops and throws away the top value on the stack. (You can still get that value back from the Trail if you should need it later on.) There are many places in this tutorial where we assume you have used DEL to erase the results of the previous example at the beginning of a new example. In the few places where it is really important to use DEL to clear away old results, the text will remind you to do so.
(It won't hurt to let things accumulate on the stack, except that whenever you give a display-mode-changing command Calc will have to spend a long time reformatting such a large stack.)
Since the - key is also an operator (it subtracts the top two stack elements), how does one enter a negative number? Calc uses the _ (underscore) key to act like the minus sign in a number. So, typing -5 RET won't work because the - key will try to do a subtraction, but _5 RET works just fine.
You can also press n, which means "change sign." It changes the number at the top of the stack (or the number being entered) from positive to negative or vice-versa: 5 n RET.
If you press RET when you're not entering a number, the effect is to duplicate the top number on the stack. Consider this calculation:
1: 3 2: 3 1: 9 2: 9 1: 81
. 1: 3 . 1: 9 .
. .
3 RET RET * RET *
(Of course, an easier way to do this would be 3 RET 4 ^, to raise 3 to the fourth power.)
The space-bar key (denoted SPC here) performs the same function as RET; you could replace all three occurrences of RET in the above example with SPC and the effect would be the same.
Another stack manipulation key is TAB. This exchanges the top two stack entries. Suppose you have computed 2 RET 3 + to get 5, and then you realize what you really wanted to compute was 20 / (2+3).
1: 5 2: 5 2: 20 1: 4
. 1: 20 1: 5 .
. .
2 RET 3 + 20 TAB /
Planning ahead, the calculation would have gone like this:
1: 20 2: 20 3: 20 2: 20 1: 4
. 1: 2 2: 2 1: 5 .
. 1: 3 .
.
20 RET 2 RET 3 + /
A related stack command is M-TAB (hold META and type TAB). It rotates the top three elements of the stack upward, bringing the object in level 3 to the top.
1: 10 2: 10 3: 10 3: 20 3: 30
. 1: 20 2: 20 2: 30 2: 10
. 1: 30 1: 10 1: 20
. . .
10 RET 20 RET 30 RET M-TAB M-TAB
(*) Exercise 3. Suppose the numbers 10, 20, and 30 are on the stack. Figure out how to add one to the number in level 2 without affecting the rest of the stack. Also figure out how to add one to the number in level 3. See section RPN Tutorial Exercise 3. (*)
Operations like +, -, *, /, and ^ pop two arguments from the stack and push a result. Operations like n and Q (square root) pop a single number and push the result. You can think of them as simply operating on the top element of the stack.
1: 3 1: 9 2: 9 1: 25 1: 5
. . 1: 16 . .
.
3 RET RET * 4 RET RET * + Q
(Note that capital Q means to hold down the Shift key while typing q. Remember, plain unshifted q is the Quit command.)
Here we've used the Pythagorean Theorem to determine the hypotenuse of a right triangle. Calc actually has a built-in command for that called f h, but let's suppose we can't remember the necessary keystrokes. We can still enter it by its full name using M-x notation:
1: 3 2: 3 1: 5
. 1: 4 .
.
3 RET 4 RET M-x calc-hypot
All Calculator commands begin with the word `calc-'. Since it gets tiring to type this, Calc provides an x key which is just like the regular Emacs M-x key except that it types the `calc-' prefix for you:
1: 3 2: 3 1: 5
. 1: 4 .
.
3 RET 4 RET x hypot
What happens if you take the square root of a negative number?
1: 4 1: -4 1: (0, 2)
. . .
4 RET n Q
The notation (a, b) represents a complex number. Complex numbers are more traditionally written @c{$a + b i$} a + b i; Calc can display in this format, too, but for now we'll stick to the (a, b) notation.
If you don't know how complex numbers work, you can safely ignore this feature. Complex numbers only arise from operations that would be errors in a calculator that didn't have complex numbers. (For example, taking the square root or logarithm of a negative number produces a complex result.)
Complex numbers are entered in the notation shown. The ( and , and ) keys manipulate "incomplete complex numbers."
1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3)
. 1: 2 . 3 .
. .
( 2 , 3 )
You can perform calculations while entering parts of incomplete objects. However, an incomplete object cannot actually participate in a calculation:
1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
. 1: 2 2: 2 5 5
. 1: 3 . .
.
(error)
( 2 RET 3 + +
Adding 5 to an incomplete object makes no sense, so the last command produces an error message and leaves the stack the same.
Incomplete objects can't participate in arithmetic, but they can be moved around by the regular stack commands.
2: 2 3: 2 3: 3 1: ( ... 1: (2, 3)
1: 3 2: 3 2: ( ... 2 .
. 1: ( ... 1: 2 3
. . .
2 RET 3 RET ( M-TAB M-TAB )
Note that the , (comma) key did not have to be used here. When you press ) all the stack entries between the incomplete entry and the top are collected, so there's never really a reason to use the comma. It's up to you.
(*) Exercise 4. To enter the complex number (2, 3), your friend Joe typed ( 2 , SPC 3 ). What happened? (Joe thought of a clever way to correct his mistake in only two keystrokes, but it didn't quite work. Try it to find out why.) See section RPN Tutorial Exercise 4. (*)
Vectors are entered the same way as complex numbers, but with square brackets in place of parentheses. We'll meet vectors again later in the tutorial.
Any Emacs command can be given a numeric prefix argument by typing a series of META-digits beforehand. If META is awkward for you, you can instead type C-u followed by the necessary digits. Numeric prefix arguments can be negative, as in M-- M-3 M-5 or C-u - 3 5. Calc commands use numeric prefix arguments in a variety of ways. For example, a numeric prefix on the + operator adds any number of stack entries at once:
1: 10 2: 10 3: 10 3: 10 1: 60
. 1: 20 2: 20 2: 20 .
. 1: 30 1: 30
. .
10 RET 20 RET 30 RET C-u 3 +
For stack manipulation commands like RET, a positive numeric prefix argument operates on the top n stack entries at once. A negative argument operates on the entry in level n only. An argument of zero operates on the entire stack. In this example, we copy the second-to-top element of the stack:
1: 10 2: 10 3: 10 3: 10 4: 10
. 1: 20 2: 20 2: 20 3: 20
. 1: 30 1: 30 2: 30
. . 1: 20
.
10 RET 20 RET 30 RET C-u -2 RET
Another common idiom is M-0 DEL, which clears the stack. (The M-0 numeric prefix tells DEL to operate on the entire stack.)
If you are not used to RPN notation, you may prefer to operate the Calculator in "algebraic mode," which is closer to the way non-RPN calculators work. In algebraic mode, you enter formulas in traditional 2+3 notation.
You don't really need any special "mode" to enter algebraic formulas. You can enter a formula at any time by pressing the apostrophe (') key. Answer the prompt with the desired formula, then press RET. The formula is evaluated and the result is pushed onto the RPN stack. If you don't want to think in RPN at all, you can enter your whole computation as a formula, read the result from the stack, then press DEL to delete it from the stack.
Try pressing the apostrophe key, then 2+3+4, then RET. The result should be the number 9.
Algebraic formulas use the operators `+', `-', `*', `/', and `^'. You can use parentheses to make the order of evaluation clear. In the absence of parentheses, `^' is evaluated first, then `*', then `/', then finally `+' and `-'. For example, the expression
2 + 3*4*5 / 6*7^8 - 9
is equivalent to
2 + ((3*4*5) / (6*(7^8)) - 9
or, in large mathematical notation,
The result of this expression will be the number -6.99999826533.
Calc's order of evaluation is the same as for most computer languages, except that `*' binds more strongly than `/', as the above example shows. As in normal mathematical notation, the `*' symbol can often be omitted: `2 a' is the same as `2*a'.
Operators at the same level are evaluated from left to right, except that `^' is evaluated from right to left. Thus, `2-3-4' is equivalent to `(2-3)-4' or -5, whereas `2^3^4' is equivalent to `2^(3^4)' (a very large integer; try it!).
If you tire of typing the apostrophe all the time, there is an "algebraic mode" you can select in which Calc automatically senses when you are about to type an algebraic expression. To enter this mode, press the two letters m a. (An `Alg' indicator should appear in the Calc window's mode line.)
Press m a, then 2+3+4 with no apostrophe, then RET.
In algebraic mode, when you press any key that would normally begin entering a number (such as a digit, a decimal point, or the _ key), or if you press ( or [, Calc automatically begins an algebraic entry.
Functions which do not have operator symbols like `+' and `*'
must be entered in formulas using function-call notation. For example,
the function name corresponding to the square-root key Q is
sqrt. To compute a square root in a formula, you would use
the notation `sqrt(x)'.
Press the apostrophe, then type sqrt(5*2) - 3. The result should be 0.16227766017.
Note that if the formula begins with a function name, you need to use the apostrophe even if you are in algebraic mode. If you type arcsin out of the blue, the a r will be taken as an Algebraic Rewrite command, and the csin will be taken as the name of the rewrite rule to use!
Some people prefer to enter complex numbers and vectors in algebraic form because they find RPN entry with incomplete objects to be too distracting, even though they otherwise use Calc as an RPN calculator.
Still in algebraic mode, type:
1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1)
. 1: (1, -2) . 1: 1 .
. .
(2,3) RET (1,-2) RET * 1 RET +
Algebraic mode allows us to enter complex numbers without pressing an apostrophe first, but it also means we need to press RET after every entry, even for a simple number like 1.
(You can type C-u m a to enable a special "incomplete algebraic mode" in which the ( and [ keys use algebraic entry even though regular numeric keys still use RPN numeric entry. There is also a "total algebraic mode," started by typing m t, in which all normal keys begin algebraic entry. You must then use the META key to type Calc commands: M-m t to get back out of total algebraic mode, M-q to quit, etc.)
If you're still in algebraic mode, press m a again to turn it off.
Actual non-RPN calculators use a mixture of algebraic and RPN styles. In general, operators of two numbers (like + and *) use algebraic form, but operators of one number (like n and Q) use RPN form. Also, a non-RPN calculator allows you to see the intermediate results of a calculation as you go along. You can accomplish this in Calc by performing your calculation as a series of algebraic entries, using the $ sign to tie them together. In an algebraic formula, $ represents the number on the top of the stack. Here, we perform the calculation @c{$\sqrt{2\times4+1}$} sqrt(2*4+1), which on a traditional calculator would be done by pressing 2 * 4 + 1 = and then the square-root key.
1: 8 1: 9 1: 3
. . .
' 2*4 RET $+1 RET Q
Notice that we didn't need to press an apostrophe for the $+1, because the dollar sign always begins an algebraic entry.
(*) Exercise 1. How could you get the same effect as pressing Q but using an algebraic entry instead? How about if the Q key on your keyboard were broken? See section Algebraic Entry Tutorial Exercise 1. (*)
The notations $$, $$$, and so on stand for higher stack entries. For example, ' $$+$ RET is just like typing +.
Algebraic formulas can include variables. To store in a variable, press s s, then type the variable name, then press RET. (There are actually two flavors of store command: s s stores a number in a variable but also leaves the number on the stack, while s t removes a number from the stack and stores it in the variable.) A variable name should consist of one or more letters or digits, beginning with a letter.
1: 17 . 1: a + a^2 1: 306
. . .
17 s t a RET ' a+a^2 RET =
The = key evaluates a formula by replacing all its variables by the values that were stored in them.
For RPN calculations, you can recall a variable's value on the stack either by entering its name as a formula and pressing =, or by using the s r command.
1: 17 2: 17 3: 17 2: 17 1: 306
. 1: 17 2: 17 1: 289 .
. 1: 2 .
.
s r a RET ' a RET = 2 ^ +
If you press a single digit for a variable name (as in s t 3, you
get one of ten quick variables q0 through q9.
They are "quick" simply because you don't have to type the letter
q or the RET after their names. In fact, you can type
simply s 3 as a shorthand for s s 3, and likewise for
t 3 and r 3.
Any variables in an algebraic formula for which you have not stored values are left alone, even when you evaluate the formula.
1: 2 a + 2 b 1: 34 + 2 b
. .
' 2a+2b RET =
Calls to function names which are undefined in Calc are also left alone, as are calls for which the value is undefined.
1: 2 + log10(0) + log10(x) + log10(5, 6) + foo(3)
.
' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) RET
In this example, the first call to log10 works, but the other
calls are not evaluated. In the second call, the logarithm is
undefined for that value of the argument; in the third, the argument
is symbolic, and in the fourth, there are too many arguments. In the
fifth case, there is no function called foo. You will see a
"Wrong number of arguments" message referring to `log10(5,6)'.
Press the w ("why") key to see any other messages that may
have arisen from the last calculation. In this case you will get
"logarithm of zero," then "number expected: x". Calc
automatically displays the first message only if the message is
sufficiently important; for example, Calc considers "wrong number
of arguments" and "logarithm of zero" to be important enough to
report automatically, while a message like "number expected: x"
will only show up if you explicitly press the w key.
(*) Exercise 2. Joe entered the formula `2 x y',
stored 5 in x, pressed =, and got the expected result,
`10 y'. He then tried the same for the formula `2 x (1+y)',
expecting `10 (1+y)', but it didn't work. Why not?
See section Algebraic Entry Tutorial Exercise 2. (*)
(*) Exercise 3. What result would you expect 1 RET 0 / to give? What if you then type 0 *? See section Algebraic Entry Tutorial Exercise 3. (*)
One interesting way to work with variables is to use the evaluates-to (`=>') operator. It works like this: Enter a formula algebraically in the usual way, but follow the formula with an `=>' symbol. (There is also an s = command which builds an `=>' formula using the stack.) On the stack, you will see two copies of the formula with an `=>' between them. The lefthand formula is exactly like you typed it; the righthand formula has been evaluated as if by typing =.
2: 2 + 3 => 5 2: 2 + 3 => 5
1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b
. .
' 2+3 => RET ' 2a+2b RET s = 10 s t a RET
Notice that the instant we stored a new value in a, all
`=>' operators already on the stack that referred to a
were updated to use the new value. With `=>', you can push a
set of formulas on the stack, then change the variables experimentally
to see the effects on the formulas' values.
You can also "unstore" a variable when you are through with it:
2: 2 + 5 => 5
1: 2 a + 2 b => 2 a + 2 b
.
s u a RET
We will encounter formulas involving variables and functions again when we discuss the algebra and calculus features of the Calculator.
If you make a mistake, you can usually correct it by pressing shift-U, the "undo" command. First, clear the stack (M-0 DEL) and exit and restart Calc (M-# M-# M-# M-#) to make sure things start off with a clean slate. Now:
1: 2 2: 2 1: 8 2: 2 1: 6
. 1: 3 . 1: 3 .
. .
2 RET 3 ^ U *
You can undo any number of times. Calc keeps a complete record of all you have done since you last opened the Calc window. After the above example, you could type:
1: 6 2: 2 1: 2 . .
. 1: 3 .
.
(error)
U U U U
You can also type D to "redo" a command that you have undone mistakenly.
. 1: 2 2: 2 1: 6 1: 6
. 1: 3 . .
.
(error)
D D D D
It was not possible to redo past the 6, since that was placed there by something other than an undo command.
You can think of undo and redo as a sort of "time machine." Press U to go backward in time, D to go forward. If you go backward and do something (like *) then, as any science fiction reader knows, you have changed your future and you cannot go forward again. Thus, the inability to redo past the 6 even though there was an earlier undo command.
You can always recall an earlier result using the Trail. We've ignored the trail so far, but it has been faithfully recording everything we did since we loaded the Calculator. If the Trail is not displayed, press t d now to turn it on.
Let's try grabbing an earlier result. The 8 we computed was undone by a U command, and was lost even to Redo when we pressed *, but it's still there in the trail. There should be a little `>' arrow (the trail pointer) resting on the last trail entry. If there isn't, press t ] to reset the trail pointer. Now, press t p to move the arrow onto the line containing 8, and press t y to "yank" that number back onto the stack.
If you press t ] again, you will see that even our Yank command went into the trail.
Let's go further back in time. Earlier in the tutorial we computed a huge integer using the formula `2^3^4'. We don't remember what it was, but the first digits were "241". Press t r (which stands for trail-search-reverse), then type 241. The trail cursor will jump back to the next previous occurrence of the string "241" in the trail. This is just a regular Emacs incremental search; you can now press C-s or C-r to continue the search forwards or backwards as you like.
To finish the search, press RET. This halts the incremental search and leaves the trail pointer at the thing we found. Now we can type t y to yank that number onto the stack. If we hadn't remembered the "241", we could simply have searched for 2^3^4, then pressed RET t n to halt and then move to the next item.
You may have noticed that all the trail-related commands begin with the letter t. (The store-and-recall commands, on the other hand, all began with s.) Calc has so many commands that there aren't enough keys for all of them, so various commands are grouped into two-letter sequences where the first letter is called the prefix key. If you type a prefix key by accident, you can press C-g to cancel it. (In fact, you can press C-g to cancel almost anything in Emacs.) To get help on a prefix key, press that key followed by ?. Some prefixes have several lines of help, so you need to press ? repeatedly to see them all.
Try pressing t ? now. You will see a line of the form,
trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-
The word "trail" indicates that the t prefix key contains trail-related commands. Each entry on the line shows one command, with a single capital letter showing which letter you press to get that command. We have used t n, t p, t ], and t y so far. The `[MORE]' means you can press ? again to see more t-prefix comands. Notice that the commands are roughly divided (by semicolons) into related groups.
When you are in the help display for a prefix key, the prefix is still active. If you press another key, like y for example, it will be interpreted as a t y command. If all you wanted was to look at the help messages, press C-g afterwards to cancel the prefix.
One more way to correct an error is by editing the stack entries. The actual Stack buffer is marked read-only and must not be edited directly, but you can press ` (the backquote or accent grave) to edit a stack entry.
Try entering `3.141439' now. If this is supposed to represent pi, it's got several errors. Press ` to edit this number. Now use the normal Emacs cursor motion and editing keys to change the second 4 to a 5, and to transpose the 3 and the 9. When you press RET, the number on the stack will be replaced by your new number. This works for formulas, vectors, and all other types of values you can put on the stack. The ` key also works during entry of a number or algebraic formula.
Calc has many types of modes that affect the way it interprets your commands or the way it displays data. We have already seen one mode, namely algebraic mode. There are many others, too; we'll try some of the most common ones here.
Perhaps the most fundamental mode in Calc is the current precision. Notice the `12' on the Calc window's mode line:
--%%-Calc: 12 Deg (Calculator)----All------
Most of the symbols there are Emacs things you don't need to worry about, but the `12' and the `Deg' are mode indicators. The `12' means that calculations should always be carried to 12 significant figures. That is why, when we type 1 RET 7 /, we get 0.142857142857 with exactly 12 digits, not counting leading and trailing zeros.
You can set the precision to anything you like by pressing p, then entering a suitable number. Try pressing p 30 RET, then doing 1 RET 7 / again:
1: 0.142857142857
2: 0.142857142857142857142857142857
.
Although the precision can be set arbitrarily high, Calc always has to have some value for the current precision. After all, the true value 1/7 is an infinitely repeating decimal; Calc has to stop somewhere.
Of course, calculations are slower the more digits you request. Press p 12 now to set the precision back down to the default.
Calculations always use the current precision. For example, even though we have a 30-digit value for 1/7 on the stack, if we use it in a calculation in 12-digit mode it will be rounded down to 12 digits before it is used. Try it; press RET to duplicate the number, then 1 +. Notice that the RET key didn't round the number, because it doesn't do any calculation. But the instant we pressed +, the number was rounded down.
1: 0.142857142857
2: 0.142857142857142857142857142857
3: 1.14285714286
.
In fact, since we added a digit on the left, we had to lose one digit on the right from even the 12-digit value of 1/7.
How did we get more than 12 digits when we computed `2^3^4'? The answer is that Calc makes a distinction between integers and floating-point numbers, or floats. An integer is a number that does not contain a decimal point. There is no such thing as an "infinitely repeating fraction integer," so Calc doesn't have to limit itself. If you asked for `2^10000' (don't try this!), you would have to wait a long time but you would eventually get an exact answer. If you ask for `2.^10000', you will quickly get an answer which is correct only to 12 places. The decimal point tells Calc that it should use floating-point arithmetic to get the answer, not exact integer arithmetic.
You can use the F (calc-floor) command to convert a
floating-point value to an integer, and c f (calc-float)
to convert an integer to floating-point form.
Let's try entering that last calculation:
1: 2. 2: 2. 1: 1.99506311689e3010
. 1: 10000 .
.
2.0 RET 10000 RET ^
Notice the letter `e' in there. It represents "times ten to the power of," and is used by Calc automatically whenever writing the number out fully would introduce more extra zeros than you probably want to see. You can enter numbers in this notation, too.
1: 2. 2: 2. 1: 1.99506311678e3010
. 1: 10000. .
.
2.0 RET 1e4 RET ^
Hey, the answer is different! Look closely at the middle columns of the two examples. In the first, the stack contained the exact integer 10000, but in the second it contained a floating-point value with a decimal point. When you raise a number to an integer power, Calc uses repeated squaring and multiplication to get the answer. When you use a floating-point power, Calc uses logarithms and exponentials. As you can see, a slight error crept in during one of these methods. Which one should we trust? Let's raise the precision a bit and find out:
. 1: 2. 2: 2. 1: 1.995063116880828e3010
. 1: 10000. .
.
p 16 RET 2. RET 1e4 ^ p 12 RET
Presumably, it doesn't matter whether we do this higher-precision calculation using an integer or floating-point power, since we have added enough "guard digits" to trust the first 12 digits no matter what. And the verdict is... Integer powers were more accurate; in fact, the result was only off by one unit in the last place.
Calc does many of its internal calculations to a slightly higher precision, but it doesn't always bump the precision up enough. In each case, Calc added about two digits of precision during its calculation and then rounded back down to 12 digits afterward. In one case, it was enough; in the the other, it wasn't. If you really need x digits of precision, it never hurts to do the calculation with a few extra guard digits.
What if we want guard digits but don't want to look at them? We can set the float format. Calc supports four major formats for floating-point numbers, called normal, fixed-point, scientific notation, and engineering notation. You get them by pressing d n, d f, d s, and d e, respectively. In each case, you can supply a numeric prefix argument which says how many digits should be displayed. As an example, let's put a few numbers onto the stack and try some different display modes. First, use M-0 DEL to clear the stack, then enter the four numbers shown here:
4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000
2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450
1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345
. . . . .
d n M-3 d n d s M-3 d s M-3 d f
Notice that when we typed M-3 d n, the numbers were rounded down to three significant digits, but then when we typed d s all five significant figures reappeared. The float format does not affect how numbers are stored, it only affects how they are displayed. Only the current precision governs the actual rounding of numbers in the Calculator's memory.
Engineering notation, not shown here, is like scientific notation except the exponent (the power-of-ten part) is always adjusted to be a multiple of three (as in "kilo," "micro," etc.). As a result there will be one, two, or three digits before the decimal point.
Whenever you change a display-related mode, Calc redraws everything in the stack. This may be slow if there are many things on the stack, so Calc allows you to type shift-H before any mode command to prevent it from updating the stack. Anything Calc displays after the mode-changing command will appear in the new format.
4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345.
2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45
1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345
. . . . .
H d s DEL U TAB d SPC d n
Here the H d s command changes to scientific notation but without updating the screen. Deleting the top stack entry and undoing it back causes it to show up in the new format; swapping the top two stack entries reformats both entries. The d SPC command refreshes the whole stack. The d n command changes back to the normal float format; since it doesn't have an H prefix, it also updates all the stack entries to be in d n format.
Notice that the integer 12345 was not affected by any of the float formats. Integers are integers, and are always displayed exactly.
Large integers have their own problems. Let's look back at the result of 2^3^4.
2417851639229258349412352
Quick--how many digits does this have? Try typing d g:
2,417,851,639,229,258,349,412,352
Now how many digits does this have? It's much easier to tell! We can actually group digits into clumps of any size. Some people prefer M-5 d g:
24178,51639,22925,83494,12352
Let's see what happens to floating-point numbers when they are grouped. First, type p 25 RET to make sure we have enough precision to get ourselves into trouble. Now, type 1e13 /:
24,17851,63922.9258349412352
The integer part is grouped but the fractional part isn't. Now try M-- M-5 d g (that's meta-minus-sign, meta-five):
24,17851,63922.92583,49412,352
If you find it hard to tell the decimal point from the commas, try changing the grouping character to a space with d , SPC:
24 17851 63922.92583 49412 352
Type d , , to restore the normal grouping character, then d g again to turn grouping off. Also, press p 12 to restore the default precision.
Press U enough times to get the original big integer back. (Notice that U does not undo each mode-setting command; if you want to undo a mode-setting command, you have to do it yourself.) Now, type d r 16 RET:
16#200000000000000000000
The number is now displayed in hexadecimal, or "base-16" form. Suddenly it looks pretty simple; this should be no surprise, since we got this number by computing a power of two, and 16 is a power of 2. In fact, we can use d r 2 RET to see it in actual binary form:
2#1000000000000000000000000000000000000000000000000000000 ...
We don't have enough space here to show all the zeros! They won't fit on a typical screen, either, so you will have to use horizontal scrolling to see them all. Press < and > to scroll the stack window left and right by half its width. Another way to view something large is to press ` (back-quote) to edit the top of stack in a separate window. (Press M-# M-# when you are done.)
You can enter non-decimal numbers using the # symbol, too. Let's see what the hexadecimal number `5FE' looks like in binary. Type 16#5FE (the letters can be typed in upper or lower case; they will always appear in upper case). It will also help to turn grouping on with d g:
2#101,1111,1110
Notice that d g groups by fours by default if the display radix is binary or hexadecimal, but by threes if it is decimal, octal, or any other radix.
Now let's see that number in decimal; type d r 10:
1,534
Numbers are not stored with any particular radix attached. They're just numbers; they can be entered in any radix, and are always displayed in whatever radix you've chosen with d r. The current radix applies to integers, fractions, and floats.
(*) Exercise 1. Your friend Joe tried to enter one-third as `3#0.1' in d r 3 mode with a precision of 12. He got `3#0.0222222...' (with 25 2's) in the display. When he multiplied that by three, he got `3#0.222222...' instead of the expected `3#1'. Next, Joe entered `3#0.2' and, to his great relief, saw `3#0.2' on the screen. But when he typed 2 /, he got `3#0.10000001' (some zeros omitted). What's going on here? See section Modes Tutorial Exercise 1. (*)
(*) Exercise 2. Scientific notation works in non-decimal modes in the natural way (the exponent is a power of the radix instead of a power of ten, although the exponent itself is always written in decimal). Thus `8#1.23e3 = 8#1230.0'. Suppose we have the hexadecimal number `f.e8f' times 16 to the 15th power: We write `16#f.e8fe15'. What is wrong with this picture? What could we write instead that would work better? See section Modes Tutorial Exercise 2. (*)
The m prefix key has another set of modes, relating to the way Calc interprets your inputs and does computations. Whereas d-prefix modes generally affect the way things look, m-prefix modes affect the way they are actually computed.
The most popular m-prefix mode is the angular mode. Notice the `Deg' indicator in the mode line. This means that if you use a command that interprets a number as an angle, it will assume the angle is measured in degrees. For example,
1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5
. . . .
45 S 2 ^ c 1
The shift-S command computes the sine of an angle. The sine of 45 degrees is @c{$\sqrt{2}/2$} sqrt(2)/2; squaring this yields 2/4 = 0.5. However, there has been a slight roundoff error because the representation of @c{$\sqrt{2}/2$} sqrt(2)/2 wasn't exact. The c 1 command is a handy way to clean up numbers in this case; it temporarily reduces the precision by one digit while it re-rounds the number on the top of the stack.
(*) Exercise 3. Your friend Joe computed the sine of 45 degrees as shown above, then, hoping to avoid an inexact result, he increased the precision to 16 digits before squaring. What happened? See section Modes Tutorial Exercise 3. (*)
To do this calculation in radians, we would type m r first. (The indicator changes to `Rad'.) 45 degrees corresponds to pi/4 radians. To get @c{$\pi$} pi, press the P key. (Once again, this is a shifted capital P. Remember, unshifted p sets the precision.)
1: 3.14159265359 1: 0.785398163398 1: 0.707106781187
. . .
P 4 / m r S
Likewise, inverse trigonometric functions generate results in either radians or degrees, depending on the current angular mode.
1: 0.707106781187 1: 0.785398163398 1: 45.
. . .
.5 Q m r I S m d U I S
Here we compute the Inverse Sine of @c{$\sqrt{0.5}$} sqrt(0.5), first in radians, then in degrees.
Use c d and c r to convert a number from radians to degrees and vice-versa.
1: 45 1: 0.785398163397 1: 45.
. . .
45 c r c d
Another interesting mode is fraction mode. Normally, dividing two integers produces a floating-point result if the quotient can't be expressed as an exact integer. Fraction mode causes integer division to produce a fraction, i.e., a rational number, instead.
2: 12 1: 1.33333333333 1: 4:3
1: 9 . .
.
12 RET 9 / m f U / m f
In the first case, we get an approximate floating-point result. In the second case, we get an exact fractional result (four-thirds).
You can enter a fraction at any time using : notation. (Calc uses : instead of / as the fraction separator because / is already used to divide the top two stack elements.) Calculations involving fractions will always produce exact fractional results; fraction mode only says what to do when dividing two integers.
(*) Exercise 4. If fractional arithmetic is exact, why would you ever use floating-point numbers instead? See section Modes Tutorial Exercise 4. (*)
Typing m f doesn't change any existing values in the stack. In the above example, we had to Undo the division and do it over again when we changed to fraction mode. But if you use the evaluates-to operator you can get commands like m f to recompute for you.
1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3
. . .
' 12/9 => RET p 4 RET m f
In this example, the righthand side of the `=>' operator on the stack is recomputed when we change the precision, then again when we change to fraction mode. All `=>' expressions on the stack are recomputed every time you change any mode that might affect their values.
In this section, we explore the arithmetic and scientific functions available in the Calculator.
The standard arithmetic commands are +, -, *, /, and ^. Each normally takes two numbers from the top of the stack and pushes back a result. The n and & keys perform change-sign and reciprocal operations, respectively.
1: 5 1: 0.2 1: 5. 1: -5. 1: 5.
. . . . .
5 & & n n
You can apply a "binary operator" like + across any number of stack entries by giving it a numeric prefix. You can also apply it pairwise to several stack elements along with the top one if you use a negative prefix.
3: 2 1: 9 3: 2 4: 2 3: 12
2: 3 . 2: 3 3: 3 2: 13
1: 4 1: 4 2: 4 1: 14
. . 1: 10 .
.
2 RET 3 RET 4 M-3 + U 10 M-- M-3 +
You can apply a "unary operator" like & to the top n stack entries with a numeric prefix, too.
3: 2 3: 0.5 3: 0.5
2: 3 2: 0.333333333333 2: 3.
1: 4 1: 0.25 1: 4.
. . .
2 RET 3 RET 4 M-3 & M-2 &
Notice that the results here are left in floating-point form. We can convert them back to integers by pressing F, the "floor" function. This function rounds down to the next lower integer. There is also R, which rounds to the nearest integer.
7: 2. 7: 2 7: 2
6: 2.4 6: 2 6: 2
5: 2.5 5: 2 5: 3
4: 2.6 4: 2 4: 3
3: -2. 3: -2 3: -2
2: -2.4 2: -3 2: -2
1: -2.6 1: -3 1: -3
. . .
M-7 F U M-7 R
Since dividing-and-flooring (i.e., "integer quotient") is such a common operation, Calc provides a special command for that purpose, the backslash \. Another common arithmetic operator is %, which computes the remainder that would arise from a \ operation, i.e., the "modulo" of two numbers. For example,
2: 1234 1: 12 2: 1234 1: 34
1: 100 . 1: 100 .
. .
1234 RET 100 \ U %
These commands actually work for any real numbers, not just integers.
2: 3.1415 1: 3 2: 3.1415 1: 0.1415
1: 1 . 1: 1 .
. .
3.1415 RET 1 \ U %
(*) Exercise 1. The \ command would appear to be a frill, since you could always do the same thing with / F. Think of a situation where this is not true---/ F would be inadequate. Now think of a way you could get around the problem if Calc didn't provide a \ command. See section Arithmetic Tutorial Exercise 1. (*)
We've already seen the Q (square root) and S (sine) commands. Other commands along those lines are C (cosine), T (tangent), E (e^x) and L (natural logarithm). These can be modified by the I (inverse) and H (hyperbolic) prefix keys.
Let's compute the sine and cosine of an angle, and verify the identity @c{$\sin^2x + \cos^2x = 1$} sin(x)^2 + cos(x)^2 = 1. We'll arbitrarily pick -64 degrees as a good value for x. With the angular mode set to degrees (type m d), do:
2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1.
1: -64 1: -0.89879 1: -64 1: 0.43837 .
. . . .
64 n RET RET S TAB C f h
(For brevity, we're showing only five digits of the results here. You can of course do these calculations to any precision you like.)
Remember, f h is the calc-hypot, or square-root of sum
of squares, command.
Another identity is @c{$\displaystyle\tan x = {\sin x \over \cos x}$} tan(x) = sin(x) / cos(x).
2: -0.89879 1: -2.0503 1: -64.
1: 0.43837 . .
.
U / I T
A physical interpretation of this calculation is that if you move 0.89879 units downward and 0.43837 units to the right, your direction of motion is -64 degrees from horizontal. Suppose we move in the opposite direction, up and to the left:
2: -0.89879 2: 0.89879 1: -2.0503 1: -64.
1: 0.43837 1: -0.43837 . .
. .
U U M-2 n / I T
How can the angle be the same? The answer is that the / operation
loses information about the signs of its inputs. Because the quotient
is negative, we know exactly one of the inputs was negative, but we
can't tell which one. There is an f T [arctan2] function which
computes the inverse tangent of the quotient of a pair of numbers.
Since you feed it the two original numbers, it has enough information
to give you a full 360-degree answer.
2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180.
1: -0.43837 . 2: -0.89879 1: -64. .
. 1: 0.43837 .
.
U U f T M-RET M-2 n f T -
The resulting angles differ by 180 degrees; in other words, they point in opposite directions, just as we would expect.
The META-RET we used in the third step is the "last-arguments" command. It is sort of like Undo, except that it restores the arguments of the last command to the stack without removing the command's result. It is useful in situations like this one, where we need to do several operations on the same inputs. We could have accomplished the same thing by using M-2 RET to duplicate the top two stack elements right after the U U, then a pair of M-TAB commands to cycle the 116 up around the duplicates.
A similar identity is supposed to hold for hyperbolic sines and cosines, except that it is the difference cosh(x)^2 - sinh(x)^2 that always equals one. Let's try to verify this identity.
2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54
1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54
. . . . .
64 n RET RET H C 2 ^ TAB H S 2 ^
Something's obviously wrong, because when we subtract these numbers the answer will clearly be zero! But if you think about it, if these numbers did differ by one, it would be in the 55th decimal place. The difference we seek has been lost entirely to roundoff error.
We could verify this hypothesis by doing the actual calculation with, say, 60 decimal places of precision. This will be slow, but not enormously so. Try it if you wish; sure enough, the answer is 0.99999, reasonably close to 1.
Of course, a more reasonable way to verify the identity is to use a more reasonable value for x!
Some Calculator commands use the Hyperbolic prefix for other purposes. The logarithm and exponential functions, for example, work to the base e normally but use base-10 instead if you use the Hyperbolic prefix.
1: 1000 1: 6.9077 1: 1000 1: 3
. . . .
1000 L U H L
First, we mistakenly compute a natural logarithm. Then we undo and compute a common logarithm instead.
The B key computes a general base-b logarithm for any value of b.
2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077
1: 10 . . 1: 2.71828 .
. .
1000 RET 10 B H E H P B
Here we first use B to compute the base-10 logarithm, then use the "hyperbolic" exponential as a cheap hack to recover the number 1000, then use B again to compute the natural logarithm. Note that P with the hyperbolic prefix pushes the constant e onto the stack.
You may have noticed that both times we took the base-10 logarithm of 1000, we got an exact integer result. Calc always tries to give an exact rational result for calculations involving rational numbers where possible. But when we used H E, the result was a floating-point number for no apparent reason. In fact, if we had computed 10 RET 3 ^ we would have gotten an exact integer 1000. But the H E command is rigged to generate a floating-point result all of the time so that 1000 H E will not waste time computing a thousand-digit integer when all you probably wanted was `1e1000'.
(*) Exercise 2. Find a pair of integer inputs to the B command for which Calc could find an exact rational result but doesn't. See section Arithmetic Tutorial Exercise 2. (*)
The Calculator also has a set of functions relating to combinatorics and statistics. You may be familiar with the factorial function, which computes the product of all the integers up to a given number.
1: 100 1: 93326215443... 1: 100. 1: 9.3326e157
. . . .
100 ! U c f !
Recall, the c f command converts the integer or fraction at the top of the stack to floating-point format. If you take the factorial of a floating-point number, you get a floating-point result accurate to the current precision. But if you give ! an exact integer, you get an exact integer result (158 digits long in this case).
If you take the factorial of a non-integer, Calc uses a generalized factorial function defined in terms of Euler's Gamma function gamma(n) (which is itself available as the f g command).
3: 4. 3: 24. 1: 5.5 1: 52.342777847
2: 4.5 2: 52.3427777847 . .
1: 5. 1: 120.
. .
M-3 ! M-0 DEL 5.5 f g
Here we verify the identity @c{$n! = \Gamma(n+1)$} n! = gamma(n+1).
The binomial coefficient n-choose-m@c{ or $\displaystyle {n \choose m}$} is defined by n! / m! (n-m)! for all reals n and m. The intermediate results in this formula can become quite large even if the final result is small; the k c command computes a binomial coefficient in a way that avoids large intermediate values.
The k prefix key defines several common functions out of combinatorics and number theory. Here we compute the binomial coefficient 30-choose-20, then determine its prime factorization.
2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29]
1: 20 . .
.
30 RET 20 k c k f
You can verify these prime factors by using v u to "unpack" this vector into 8 separate stack entries, then M-8 * to multiply them back together. The result is the original number, 30045015.
Suppose a program you are writing needs a hash table with at least 10000 entries. It's best to use a prime number as the actual size of a hash table. Calc can compute the next prime number after 10000:
1: 10000 1: 10007 1: 9973
. . .
10000 k n I k n
Just for kicks we've also computed the next prime less than 10000.
See section Financial Functions, for a description of the Calculator
commands that deal with business and financial calculations (functions
like pv, rate, and sln).
See section Binary Number Functions, to read about the commands for operating
on binary numbers (like and, xor, and lsh).
A vector is a list of numbers or other Calc data objects. Calc provides a large set of commands that operate on vectors. Some are familiar operations from vector analysis. Others simply treat a vector as a list of objects.
If you add two vectors, the result is a vector of the sums of the elements, taken pairwise.
1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3]
. 1: [7, 6, 0] .
.
[1,2,3] s 1 [7 6 0] s 2 +
Note that we can separate the vector elements with either commas or spaces. This is true whether we are using incomplete vectors or algebraic entry. The s 1 and s 2 commands save these vectors so we can easily reuse them later.
If you multiply two vectors, the result is the sum of the products of the elements taken pairwise. This is called the dot product of the vectors.
2: [1, 2, 3] 1: 19
1: [7, 6, 0] .
.
r 1 r 2 *
The dot product of two vectors is equal to the product of their lengths times the cosine of the angle between them. (Here the vector is interpreted as a line from the origin (0,0,0) to the specified point in three-dimensional space.) The A (absolute value) command can be used to compute the length of a vector.
3: 19 3: 19 1: 0.550782 1: 56.579
2: [1, 2, 3] 2: 3.741657 . .
1: [7, 6, 0] 1: 9.219544
. .
M-RET M-2 A * / I C
First we recall the arguments to the dot product command, then we compute the absolute values of the top two stack entries to obtain the lengths of the vectors, then we divide the dot product by the product of the lengths to get the cosine of the angle. The inverse cosine finds that the angle between the vectors is about 56 degrees.
The cross product of two vectors is a vector whose length is the product of the lengths of the inputs times the sine of the angle between them, and whose direction is perpendicular to both input vectors. Unlike the dot product, the cross product is defined only for three-dimensional vectors. Let's double-check our computation of the angle using the cross product.
2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579
1: [7, 6, 0] 2: [1, 2, 3] . .
. 1: [7, 6, 0]
.
r 1 r 2 V C s 3 M-RET M-2 A * / A I S
First we recall the original vectors and compute their cross product, which we also store for later reference. Now we divide the vector by the product of the lengths of the original vectors. The length of this vector should be the sine of the angle; sure enough, it is!
Vector-related commands generally begin with the v prefix key. Some are uppercase letters and some are lowercase. To make it easier to type these commands, the shift-V prefix key acts the same as the v key. (See section General Mode Commands, for a way to make all prefix keys have this property.)
If we take the dot product of two perpendicular vectors we expect to get zero, since the cosine of 90 degrees is zero. Let's check that the cross product is indeed perpendicular to both inputs:
2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0
1: [-18, 21, -8] . 1: [-18, 21, -8] .
. .
r 1 r 3 * DEL r 2 r 3 *
(*) Exercise 1. Given a vector on the top of the stack, what keystrokes would you use to normalize the vector, i.e., to reduce its length to one without changing its direction? See section Vector Tutorial Exercise 1. (*)
(*) Exercise 2. Suppose a certain particle can be at any of several positions along a ruler. You have a list of those positions in the form of a vector, and another list of the probabilities for the particle to be at the corresponding positions. Find the average position of the particle. See section Vector Tutorial Exercise 2. (*)
A matrix is just a vector of vectors, all the same length. This means you can enter a matrix using nested brackets. You can also use the semicolon character to enter a matrix. We'll show both methods here:
1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
[ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
. .
[[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] RET
We'll be using this matrix again, so type s 4 to save it now.
Note that semicolons work with incomplete vectors, but they work better in algebraic entry. That's why we use the apostrophe in the second example.
When two matrices are multiplied, the lefthand matrix must have the same number of columns as the righthand matrix has rows. Row i, column j of the result is effectively the dot product of row i of the left matrix by column j of the right matrix.
If we try to duplicate this matrix and multiply it by itself, the dimensions are wrong and the multiplication cannot take place:
1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ]
[ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
.
RET *
Though rather hard to read, this is a formula which shows the product of two matrices. The `*' function, having invalid arguments, has been left in symbolic form.
We can multiply the matrices if we transpose one of them first.
2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ]
[ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ]
1: [ [ 1, 4 ] . [ 27, 36, 45 ] ]
[ 2, 5 ] .
[ 3, 6 ] ]
.
U v t * U TAB *
Matrix multiplication is not commutative; indeed, switching the order of the operands can even change the dimensions of the result matrix, as happened here!
If you multiply a plain vector by a matrix, it is treated as a single row or column depending on which side of the matrix it is on. The result is a plain vector which should also be interpreted as a row or column as appropriate.
2: [ [ 1, 2, 3 ] 1: [14, 32]
[ 4, 5, 6 ] ] .
1: [1, 2, 3]
.
r 4 r 1 *
Multiplying in the other order wouldn't work because the number of rows in the matrix is different from the number of elements in the vector.
(*) Exercise 1. Use `*' to sum along the rows of the above @c{$2\times3$} 2x3 matrix to get [6, 15]. Now use `*' to sum along the preted as a row or column as appropriate.
2: [ [ 1, 2, 3 ] 1: [14, 32]
[ 4, 5, 6 ] ] .
1: [1, 2, 3]
.
r 4 r 1 *
Multiplying in the other order wouldn't work because the number of rows in the matrix is different from the number of elements in the vector.
(*) Exercise 1. Use `*' to sum along the rows of the above @c{$2\times3$} 2x3 matrix to get [6, 15]. Now use `*' to sum along the preted as a row or column as appropriate.
2: [ [ 1, 2, 3 ] 1: [14, 32]
[ 4, 5, 6 ] ] .
1: [1, 2, 3]
.
r 4 r 1 *
Multiplying in the other order wouldn't work because the number of rows in the matrix is different from the number of elements in the vector.
(*) Exercise 1. Use `*' to sum along the rows of the above @c{$2\times3$} 2x3 matrix to get [6, 15]. Now use `*' to sum along the