4   Calculations in ALG Mode

In ALG, you enter an expression in its written order and press = to evaluate it. Parentheses identify groups, and precedence determines which operations are carried out first. You can enter a function before its argument, apply it to an operand already entered, or reuse the answer to an earlier calculation.

ALG has its own working area. It does not use the RPN stack, and selecting ALG does not transfer the RPN display into an expression. The scientific keyboard, ordinary memories, and settings are shared; the available calculations are summarized in section 4.10.

4.1   Entering and evaluating an expression

Select ALG above the display and show the Scientific panel. The Expression row contains (, ), FUNC, CPT, and CLEAR. In portrait it is above the Shift and Memory row: f, g, ANS, STO, and RCL. In landscape Expression sits below the Scientific and Financial panels on the left, while Shift and Memory sit above entry on the right. The left entry column has normal-size EEX, \(1/x\), +/\(-\), and = keys from top to bottom. The = key is on the bottom row. Orange \(1/x\) selects INT, which truncates toward zero; blue \(1/x\) selects FRAC, which returns the signed fractional part. Equals, ANS, and CPT have no shifted legends. CPT selects an explicit financial computation, described in section 4.11. The bottom line shows the expression in both real and complex mode and is available to VoiceOver. Calibration appears above it when digits overflow; drag the digits to scroll. Prompts and errors appear below the flags, above the numbers. The large number, or the two complex component rows, shows the number being entered or the most recent operand or function preview. It is not a running total of an unfinished expression.

Numbers in the expression line use four places after the decimal point, independently of the large display’s digit count. The selected FIX, SCI, or ENG notation still applies. In FIX, a number that needs more than four integer digits switches to scientific notation, also with four decimal places: for example, 12345 appears as 1.2345E+4. Very small nonzero values also use scientific notation if FIX 4 would round them to zero. This compact presentation applies to both complex components and to complete numbers still being typed. The large display keeps the exact entry text, and unfinished entries such as 3 E- retain their editing cues in the expression line. Only the presentation is rounded; calculations use the full operands.

The ENTRY indicator remains on while an expression is unfinished, including between numbers and after a closing parenthesis. Pressing = evaluates the whole expression, displays its result, and turns off ENTRY. The completed expression remains visible until you start or continue a calculation.

Before each independent example, make sure no shift, command, or digit-count prompt is pending, then press the unshifted CLEAR key. This clears the ALG working expression without clearing its previous answer or any memory bank. Prefix cancellation is explained in section 1.8; clearing in ALG is covered in section 4.9.

For the examples in this chapter, use real mode (COMPLEX off), DEG, FIX 4, and 600 working digits. These settings make the displayed answers easy to compare. Check working digits with PREC, as described in chapter 5. In ALG, select FIX 4 by pressing f 7 4 =. The orange legend on 7 is FIX; this particular equals press accepts the display setting and does not evaluate an expression. Set degrees with f SIN, selecting its orange DEG legend.

Enter integers and decimals with the digit keys and decimal point. To make a number negative, enter its digits and press +/\(-\). For scientific notation, enter the mantissa, press EEX, then enter the exponent. While the exponent is active, +/\(-\) changes the exponent’s sign rather than the mantissa’s sign.

For example, from the chapter’s cleared ALG starting state, 1.25 EEX +/\(-\) 2 + .5 = gives 0.5125. The calculation is \(1.25\times 10^{-2}+0.5\). If EEX starts a new number, the mantissa is supplied as 1: EEX 3 = gives 1,000.0000. An exponent needs at least one digit before evaluation; 2 EEX = does not mean \(2\times 10^0\).

Physical-keyboard shortcuts also enter digits, operators, parentheses, and commas. Return evaluates, ^ selects the power key, e selects EEX, and s selects +/\(-\). These are key shortcuts, not a general text editor for function names or pasted formulas. Use the visible function keys and FUNC for the procedures below. See table 1.4 for the common shortcuts.

4.2   Precedence and parentheses

Multiplication and division take priority over addition and subtraction. Operations at the same priority are normally evaluated from left to right. Thus \(8/4/2=(8/4)/2=1\), and \(10-3-2=5\). Powers are the exception: a chain of powers is evaluated from the right, as explained in section 4.3.

Parentheses make a group into one operand. Use them whenever the intended grouping would otherwise be unclear. The expression \((2+3)\times 4\) gives 20, whereas \(2+3\times 4\) gives 14. Every opening parenthesis needs its own closing parenthesis; = does not supply missing ones.

The multiplication-level scientific keys include indexed roots, base logarithms, permutations, combinations, atan2, the two remainders, and the three percentage operations. For a complicated expression involving these keys, an explicit FUNC call makes the two arguments clearer than a long chain of binary operations. Implicit multiplication has the same priority as ordinary multiplication; it does not receive extra priority because the multiplication sign was omitted.

4.3   Powers and negative operands

The \(y^x\) key enters a power between its base and exponent. With no parentheses, \(2^{3^2}\) means \(2^9=512\), not \(8^2=64\). To square the result of \(2^3\), enclose that first power in parentheses.

There are also two distinct ways to enter a negative operand. Pressing the subtraction key where an operand is expected inserts mathematical minus. It has lower priority than power. Changing the sign of a number while typing its digits makes the sign part of that number instead. The expression display puts a committed negative number in parentheses to make this distinction visible.

The same distinction applies to +/\(-\) when no number is being typed. At an empty operand position it inserts a leading minus; after an existing operand it negates that operand. During exponent entry it changes only the exponent’s sign. For an unmistakable negative base, enter the signed number first, or enclose the negative expression in parentheses before pressing \(y^x\).

4.4   Implicit multiplication

ALG supplies multiplication in several useful positions. An opening parenthesis after an operand starts a new factor. A number immediately after a closing parenthesis is another factor, as is a second group. A constant key or an explicit FUNC call after an operand also inserts multiplication. Selecting Re/Im to edit an existing complex operand is different: it replaces the trailing group or function value while preserving the surrounding expression. See section 8.2.

For example, with the chapter’s cleared real ALG starting state, 2 ( 3 + 4 ) = gives 14.0000: it means \(2\times (3+4)\). The expression line includes the inserted multiplication sign.

The orange function of 0 is \(\pi \), so the last row calculates \(2\pi \). This is display rounding of the result, not a four-place approximation used internally.

Do not extend this rule to every pair of adjacent key presses. Digits typed consecutively form one number: 2 3 enters 23. A unary function after a number acts on that number, as described next. ANS and ordinary memory recall can replace a trailing operand. Use an explicit \(\times \) before either when you want a product.

4.5   Functions before and after an operand

A single-operand function can be entered in two orders. If an operand is expected, pressing its function key opens a function call and its opening parenthesis. Enter the argument and close the parenthesis. At the start of an expression, for example, \(\sqrt {x}\) 9 ) = calculates \(\sqrt {9}\).

If an operand is already present, pressing the same function key applies it to that operand. This is called postfix use: 9 \(\sqrt {x}\) = also gives 3. The expression line marks a postfix function with square brackets, such as [sqrt]. Those brackets describe the function just selected; they are not extra keys to enter.

The affected operand is the immediately preceding number, closed group, or function result. A postfix function does not automatically apply to everything to its left. In \(2+9\), square root after 9 produces \(2+\sqrt {9}=5\). To take the square root of the sum, close \((2+9)\) first and then press square root. Likewise, in \(2^3\), square root pressed after the 3 applies to the exponent; use \((2^3)\) if the entire power is the intended argument.

Choose the angle unit before inserting a trigonometric function. The function entered into an expression retains the unit selected at that key press; changing the displayed angle setting afterward does not rewrite functions already entered. The expression’s function label identifies its unit. Hyperbolic functions do not use angle units. Function domains and complex extensions are covered in chapter 7 and chapter 8.

Successive postfix unary functions act in the order pressed. Shifted functions follow the same placement rule: for example, the orange \(x^2\) function on the square-root key squares the preceding operand. HYP followed by SIN, COS, or TAN selects a hyperbolic function; the resulting function opens a call when an argument is needed or acts on the preceding operand. Use orange HYP for the corresponding inverse-hyperbolic selection.

4.6   FUNC and multiple arguments

FUNC explicitly selects a function call before its arguments. Press FUNC, then a unary or binary mathematical function key. The FUNC indicator appears while the calculator waits for that selection. The selected function and its opening parenthesis are then inserted together; do not add another opening parenthesis unless you need a nested group within the argument.

Shifts can be used between FUNC and the function key. For example, FUNC g \(y^x\) selects the blue base-logarithm function. Constants and random numbers are values, so they do not use FUNC. HYP is itself a prefix rather than a mathematical function key: use the ordinary HYP sequence described above, without a preceding FUNC.

A unary function takes one argument. A binary function takes two, separated by a comma. Enter the comma with g FUNC, the blue legend on that key; a physical comma key does the same thing in ALG. A comma is not a thousands separator or a decimal point. It belongs inside a two-argument function call, and each argument can itself be a complete expression. Close the call with ), then continue the surrounding expression or press =.

The left argument of a binary function corresponds to the left operand of its ordinary ALG form. For functions also described in RPN notation, it is the Y (S1) operand; the second argument is the X (S0) operand. These are argument positions here, not registers in an ALG stack.

For a second two-argument example, use the same cleared starting state and enter FUNC f \(y^x\) 81 g FUNC 4 ) =. This calculates the fourth root of 81 and displays 3.0000.

After an existing operand, FUNC starts a new factor. Thus 2 FUNC \(\sqrt {x}\) 9 ) =, from the same cleared state, gives 6.0000: it means \(2\sqrt {9}\). Without FUNC, square root after 2 would act on 2 instead.

Percentage keys in an expression

The blue 1, 2, and 3 functions are two-operand operations, not a percent suffix on a single number. For example, from the chapter’s cleared starting state, 200 g 1 10 = gives 20.0000, which is 10 percent of 200. The equivalent call is FUNC g 1 200 g FUNC 10 ) =.

To add that amount to the base, enter it explicitly, for example \(200+200\times 10/100\), or continue from the percentage result with + 200 = to get 220.0000. ALG has no retained Y stack operand to supply the base automatically. Use the percentage keys for these functions; the MOD and REM keys are separate remainder operations, explained in chapter 7.

4.7   Editing an unfinished expression

Before editing, dismiss any pending shift or command with the header’s CANCEL. This preserves the expression; an unfinished expression alone does not make CANCEL appear. With no command pending, the editing keys work as described below. For alternate cancellation methods and their exceptions, see section 14.4. If a worksheet is open, follow its separate editing and cancellation rules in section 4.12.

Editing works at the right-hand end of the expression. While a number is being typed, \(\leftarrow \) removes its last character. It removes exponent digits before leaving exponent entry, and removes fractional digits before the decimal point. Deleting the entire number removes that operand; it does not insert a zero into the expression. The large display may show zero while it waits for a replacement.

When no number is being typed, backspace removes the last expression element, such as an operator, a closing parenthesis, or a postfix function. Removing a function’s opening parenthesis also removes its function name. A committed number is removed as one element, not reopened for character-by-character editing. There is no movable insertion cursor for selecting an earlier digit: delete back to the required position or re-enter the expression.

CLx, the blue function of backspace, removes the current entry or the trailing operand. A closed group or function call, together with its postfix functions, counts as one operand. The earlier expression remains. If the expression already ends with an operator, CLx leaves that operator waiting for its operand. It does not replace a missing operand with a literal zero.

You can also correct an operator before entering its right operand. A new binary operator replaces a preceding binary operator, except that subtraction in an operand position introduces a negative operand. From the chapter’s cleared starting state, 2 + \(\times \) 3 = gives 6.0000. But 2 \(\times \) \(-\) 3 = gives -6.0000. To replace a mistaken operator with subtraction, remove it with backspace first, then press subtraction.

Once a trailing number or postfix result has been committed, new digits replace that trailing operand. They do not append digits to its displayed result. A closing parenthesis is different: a number typed immediately after it starts an implicit product. For a reliable replacement of a closed group, press CLx first, then enter the new operand.

4.8   ANS, continuation, and ordinary memory

ANS supplies the saved ALG answer. It starts at zero in a new calculator state. Successfully evaluating the main expression with = updates it; a failed evaluation or CLEAR does not. It is independent of the RPN result and RPN’s saved operands, Last X and Last Y. Financial computations and worksheet actions can also update ANS, as explained in section 4.11 and section 4.12.

After a completed result, a binary operator continues from that result. A unary function can also act on it. New digits, a decimal point, or EEX instead begin a new expression. Opening a parenthesis, selecting an explicit FUNC call, inserting a constant, or recalling a value also begins new work after a completed result. To include the previous result in such a new expression, insert ANS at the appropriate operand position.

Repeated = leaves a completed result unchanged and records that result as ANS again. It does not repeat the previous arithmetic operation. Also distinguish a function preview from a completed answer: after applying a unary function to a result, press = to make the new value the saved answer.

Within an unfinished expression, ANS inserts a value where an operand is needed. If an operand is already present at the end, it replaces that operand, including a trailing closed group or function call. For example, to multiply a typed 2 by the previous answer, press 2 \(\times \) ANS; pressing ANS directly after 2 would replace the 2. The value inserted is the answer available at that moment.

With CMPLX off, inserting ANS copies only its real component into the expression and leaves the saved answer unchanged. A later successful evaluation replaces ANS with its new result. Merely toggling CMPLX or reopening the application preserves the saved answer and expression operands.

Ordinary RCL followed by a digit uses the same insertion and replacement rule. STO followed by a digit stores the current operand or preview without evaluating the entire unfinished expression. Both commands finish a number that is still being typed when the register is selected. To store the result of a whole expression, press = before STO.

For example, use the chapter’s settings and no pending prefix. This example deliberately replaces memory R0 with 3; the register need not have been empty. Press CLEAR 2 + 3 STO 0 =. The expression evaluates to 5.0000. Confirm the stored operand with CLEAR RCL 0 =: the display is 3.0000. For arithmetic storage and the shared memory banks, see chapter 6.

4.9   Clearing and recovering from an error

With no pending shift or command, the unshifted CLEAR key discards the whole ALG expression and any unfinished number, returning the display to zero. It preserves ANS, memories, settings, and the RPN workspace. CLx removes only the current entry or trailing operand. After a completed result, either CLx or backspace clears the completed working expression; neither changes the saved answer.

An incomplete expression is retained when evaluation fails. Read the message, then supply the missing item or correct the operand. The last successful ANS remains available. A valid number may have been committed before the error was detected, so subsequent backspace can remove a complete number rather than one digit.

A misplaced closing parenthesis, a comma outside a binary function call, or the wrong number of arguments is rejected. The failed structural change is not added to the expression. For example, if you try to close a two-argument call after its first argument, enter the comma and second argument before closing it again. An invalid postfix function similarly leaves its operand available for correction.

Real mode cannot return a finite real answer for expressions such as \(\sqrt {-1}\) or division by zero. Complex mode permits some additional function domains, but does not make every undefined calculation valid. Consult chapter 8 before using complex entry to address a domain error. Expressions also have limits of 4096 stored elements and 128 nested groups; an individual numeric entry is limited to 10 000 characters. If a limit is reached, split the work into smaller calculations and save useful results in memory.

4.10   Switching workspaces and choosing available functions

Switching modes preserves each workspace, including the ALG expression, unfinished number, completed-result state, and answer. It does not evaluate, finish entry, or transfer operands. To transfer a value, store it in ordinary memory, then recall it in the other mode.

Mode selection cancels shifts and pending prefixes. With an ALG \(\Delta \)FIX prompt, the first mode-button press only cancels that prompt; press again to switch.

Precision, angle and display settings, complex mode, and memory banks are shared. Both workspaces survive relaunches; see chapter 14. The table below identifies available functions. See chapter 15 for individual commands’ keys and operands.

Flags such as STAT, FIN, and ASSOC report shared occupied banks. Changing an input in either mode changes that shared bank; neither mode uses the other mode’s working operands.

4.11   Financial assignments and explicit computation

On the Financial panel, the unshifted n, i, PV, PMT, and FV keys always store the displayed operand. They never infer a solve from the preceding action. Explicit STO and arithmetic STO can also target these registers. Recall with RCL followed by the financial key; it inserts or replaces an operand just like ordinary memory recall.

CPT is a pending prefix: the display identifies the required target (a financial register, NPV, or IRR). CANCEL dismisses it; relaunch restores the selection without solving anything. A direct solve requires a completed or cleared main expression. If a compound expression is unfinished, the message Finish the expression first leaves it intact. A lone entered scalar does not block computation. A successful TVM solve stores the target, displays a completed result, and updates ANS. Direct NPV or IRR also displays a completed result and updates ANS, without overwriting a financial register. Digits start new work; an operator continues from that result.

The inputs retain SpicyRPN’s existing definitions. In particular, TVM i is a percentage per payment period, not an annual I/Y coupled to P/Y and C/Y settings. Solved period counts retain the rounding described in section 11.10; ALG changes the entry procedure, not that mathematics.

4.12   Working with labeled worksheets

Use a worksheet when an operation needs several inputs or has several outputs. Orange FUNC, labeled STAT, opens statistical worksheets; orange STO and RCL open associative storage and recall. Financial operations such as AMORT, bond price, cash-flow entry, and date arithmetic open their named worksheets. Each worksheet explains its inputs and uses the current angle, date, precision, and complex settings where relevant.

The main expression and its exact unfinished entry are suspended while a worksheet is open. The Operation selector switches among worksheets in the same category. On first use, fields contain displayed starting values or defaults; changing a field marks it unaccepted. Financial fields refresh from the shared registers when opened or after a computation, unless you have an unaccepted edit in that field. The fields are separate text expressions, so you can enter arithmetic such as 12*30 within a field. They use a decimal point and parentheses; for example, (12+3)*2 is 30. The worksheet uses text-field input, not the main calculator keyboard. Select a field and edit its text, then use these controls:

=

Evaluate the selected field and update ANS. This previews a value; it does not accept it as an input or write its mapped register.

ENTER

Evaluate and accept the selected field, updating ANS. Return while editing a field performs this same acceptance. A field labeled with a financial register also stores that register. Other accepted inputs are retained in their worksheet.

COMPUTE or a named action

Unaccepted edits must be accepted or discarded before computation; the calculator does not silently calculate from stale inputs. Use the accepted inputs. STORE, APPEND, REPEAT, ADD SAMPLE, REMOVE SAMPLE, and RECALL describe the corresponding operation. All outputs and bank changes are validated before the operation writes any bank.

Use value

Insert the selected labeled output into the suspended expression using RCL’s insertion/replacement rule, then close the worksheet. This transfer does not itself change ANS or evaluate the main expression.

CANCEL

In the worksheet or header, close the worksheet, discard unaccepted field edits, and return to the unchanged main expression. Earlier accepted values, ANS updates, and completed operations remain; cancellation is not undo.

Outputs remain individually labeled. For example, amortization returns Interest, Principal, and Balance; regression returns Slope and Intercept, with Correlation when defined. The declared primary result updates ANS after successful computation. Selecting another output for inspection does not update ANS. Results describe the last successful computation; editing inputs does not recompute them. Recomputing uses the accepted inputs and the current relevant bank; simply reopening or browsing a worksheet does not execute it.

Field evaluation and acceptance can change ANS before you compute, as described above. A failed action or output browsing does not change it, and CANCEL does not restore an earlier answer. If a worksheet has changed ANS, pressing = again on a completed main expression records that expression’s result as ANS again.

Worksheet state is saved with each action: its identity, accepted inputs, exact unfinished text, selected field, outputs, and selected output. Saved worksheet values retain their numerical components and precision, without retaining the full calculation history that produced them. Relaunch resumes an active worksheet without evaluating or accepting its draft. CANCEL then abandons only its unaccepted edits. In particular, accepting X for a statistical sample does not add half a sample: none of the six accumulators changes until both coordinates and the complete addition or removal validate successfully.

See section 6.7, section 10.10, section 11.12, section 12.9 and section 13.12 for the individual fields, outputs, and bank effects.