Scientific calculator

How it works

It works like a pocket scientific calculator: type the whole line, then press = to work it out. Each key has up to three meanings. Its own is printed on it; the gold one above it on the left needs SHIFT first, and the green one on the right needs ALPHA first. SHIFT and ALPHA last for one key, so SHIFT sin types sin⁻¹ and the next key is a plain key again. SHIFT or ALPHA pressed by mistake is switched off by pressing it a second time. A menu on the screen (MODE, SETUP, and SHIFT with 1 to 8) is answered with a number key or a tap on the line.

The line under the screen says what the calculator is doing. While the cursor is inside a function, it shows what that function takes and gives, written as the screen draws it and with the box the cursor is in in bold: Pol(x, y) with the y bold means the next number is y. When the calculator waits for one of a few keys, after STO, RCL, hyp or a menu, those keys are lit, so the next press is found on the keys. On a computer, the pointer held over a key or over a gold or green label shows what it does.

Brackets still open at the end of the line are closed for you, so √(2 = works. A key goes in front of the cursor; SHIFT DEL (INS) switches to typing over the key at the cursor instead, to change one number in a long line without deleting it first. On a computer the keyboard types too: digits, + − * / ^ ( ) , ! %, Enter for =, Backspace for DEL, Esc for AC, and the arrow keys. A function is typed by its name, sqrt( or sin(, and becomes its key at the bracket; Shift pressed and let go alone is SHIFT, twice ALPHA. Keyboard, or ?, lists the keys.

The shop's own formulas, such as spindle speed, sine bar height or the weight of a bar, are worked out by name on the machinist's calculator.

Worked examples from the shop

Each link opens the calculator with the line typed and worked out, in degrees, so the keys can be followed on the screen; ◀ then opens the line to change a number for your own job.

• Sine bar height: a 100 mm sine bar set to 15°30′ needs a gauge block stack of 100×sin(15°30′) = 26.724 mm. The bar is the long side of a right triangle and the stack lifts one roll by the side opposite the angle, so the height is the length times the sine. The angle is typed as on the drawing, with °′″.
• Hole to hole on a bolt circle: 8 holes on a Ø120 bolt circle are 120×sin(180÷8) = 45.922 mm apart, centre to centre. The line between two neighbouring holes is a chord; half of it is the radius times the sine of half the angle between the holes, so the whole of it is the diameter times sin(180°/n). It is the distance a caliper checks between two holes.
• The angle of a taper: a taper from Ø40 to Ø30 over 50 mm is at tan⁻¹((40−30)÷(2×50)) = 5.7106° to the axis; °′″ then gives 5°42′38″. Each side drops half the difference in diameter over the length, which is why the difference is halved; this half-angle is the one the compound rest is set to.
• A hole from its radius and angle: a hole at R50 and 135° is at Rec(50, 135): X = −35.355, Y = 35.355. The angle is counted from +X, anticlockwise, as on a drawing, and the signs of X and Y come out right for every quarter.
• Ten measured shafts against their tolerance: ten shafts of Ø25 ±0.02, made one after another from one setup, go into STAT, with the limits 24.98 and 25.02. The mean is 25.0022 and sx 0.0038, so nearly all parts fall between 24.9909 and 25.0135, well inside the tolerance: Cp 1.77, Cpk 1.58. Cpk is lower because the mean sits 0.0022 above the middle; an offset of −0.002 would bring it close to Cp. This replaces the data in the STAT table.

Typing in textbook form

In MthIO, the setting it starts in, the line is typed as it is written on paper. The fraction key puts in a fraction with an empty box above and below the bar and the cursor in the top one; √, xⁿ, 10ˣ, logₐb, Abs, ∫, d/dx, Σ and Π put in their shape the same way, and SHIFT a⌟b a mixed number. A tap or a click in the line puts the cursor at that place, which is quicker than ◀ ▶ across a long line. ▶ moves on to the next box and out of the shape at the end, ◀ moves back, and ▲ ▼ go where they point: between the top and the bottom of a fraction, and in ∫, Σ and Π up to the upper limit and down to the lower one. Typing it this way means the line on the screen is the one on the drawing or in the handbook, so a misplaced bracket shows at a glance: 1+2 over 3 is drawn with the 1+2 above the bar, and cannot be mistaken for 1+2÷3, which is 1⅔.

DEL on the edge of a shape steps into it instead of breaking it. DEL on its start removes the shape and keeps what is in its first box; a fraction stays as top÷bottom, so its value does not change. In LineIO the line is typed on one line, with the fraction typed as 1⌟3⌟4.

Ans, PreAns and the history

Every = keeps its result as Ans, and the one before as PreAns. An operator pressed straight after = carries on from the result, so 25.4 = then ÷ 8 = gives Ans÷8 without typing 25.4 again. Pressing = again works the same line out again with the new Ans: Ans×1.1 = = = adds 10 % three times. ▲ and ▼ bring back earlier lines with their results, and ◀ or ▶ after a result opens the line to correct it, so a mistake deep in a long line costs one key, not the whole line.

Ans, PreAns, the variables, matrices, vectors, data and the history are kept in this browser, on this device, so they are still there tomorrow. CLR (SHIFT 9) clears them.

Memory: STO, RCL, the variables and M

STO then a key with a green letter stores the result in that variable: A to F on (−) to tan, X on ), Y on S⇔D and M on M+. RCL then the same key shows what is in it. ALPHA then the key types the letter into a line, so 2A+B works with stored values; X and Y also have their own key. Keep a measured value or a constant there instead of copying it by hand from one line to the next: a digit copied wrong is the commonest mistake on a calculator.

M+ adds the result to M and M− takes it away, to total a column of values: the weight of several parts, or the lengths of a cut list. M keeps its value, even until tomorrow, so an old total left in it is added into the new one. While M is not zero, the top row of the screen shows what is in it, as in M=12.5. After M+, M− or RCL M, the line under the screen gives the count of the values added (n), their mean (x̄), the smallest and the largest, so a value keyed twice or left out shows in n before the total is used. M− of a value that M+ put in takes that value out of the count again. Before starting a new total, press 0, STO, M+ to set M to zero; this leaves the other variables and the history alone, where CLR clears everything.

CALC and SOLVE

CALC works one formula out for many values. Type it with letters, π×D×N÷1000 with D and N from ALPHA, and press CALC: the calculator asks for each letter in turn, showing the value it holds, and = takes it. Press CALC again for the next set. A line such as Y=2X+1 (ALPHA CALC types =) stores what it works out in the letter on the left.

Several steps go in one line with ALPHA ∫ (:) between them, worked out in turn, each one's result the next one's Ans: A=25.4:B=A÷8:A×B. The letters a step sets are not asked for by CALC.

SOLVE (SHIFT CALC) finds the X that makes a line true: X²=2, or a line without = taken as equal to 0. It asks for X first, as the place to start: an equation with two answers gives the one nearer the start, so start near the answer you want. It shows X and L−R, how far the two sides still differ; L−R near 0 is the check that the answer is real. With more than one letter in the line, it asks which one to find, and asks for the others' values first.

VERIFY (SHIFT 6) checks a statement: pick =, ≠, >, ≥, < or ≤ from its menu, as in √2×√2=2, and = answers TRUE or FALSE. Both sides are compared to the ten digits the display shows, so 0.1+0.2=0.3 is TRUE although the computer's binary fractions differ in the 17th digit. Use it to check a dimension chain or a rearranged formula against the original.

∫, d/dx, Σ and Π

∫(f(X), a, b) is the area under f from a to b, d/dx(f(X), a) the slope of f at a, and Σ(f(X), a, b) and Π(f(X), a, b) the sum and product of f for the whole numbers from a to b. They work f out at many X, so f is typed with X. The integral halves its steps until two answers agree to about 11 digits; a function with a jump or a spike inside the range needs the range split at it.

How results are shown

• MthIO (SETUP 1) draws the line as a textbook does and gives an exact result where there is one: 1÷3 as a stacked fraction, √8 as 2√2, 3π÷4 as a fraction of π. A line with a decimal point, a constant or a conversion gives a decimal, because those values were measured, not exact: 25.4÷8 reads 3.175, not 127⌟40. LineIO (SETUP 2) keeps both on one line with decimals. Beside an exact result the screen also shows its decimal, 0.375= in front of 3⌟8, because a drawing and a micrometer read in decimals. In either, S⇔D switches between the exact and the decimal result, and SHIFT S⇔D between 7⌟4 and 1⌟3⌟4. A fraction whose decimal never ends shows it repeating first, the repeating digits under a bar: 1⌟6, then 0.16̅, then 0.1666666667. Long division makes a decimal repeat: each digit depends only on the remainder before it, so once a remainder comes back, every digit after it comes back too. It ends only when the denominator has no factor but 2 and 5, the factors of ten: 3⌟8 is 0.375, 1⌟6 never ends.
• SETUP's second page (▼ in the SETUP menu) sets fractions to show as 1⌟3⌟4 (ab/c) or 7⌟4 (d/c), complex results as a+bi or r∠θ, the STAT frequency column, whether TABLE also asks for g(X), and the decimal mark: a comma, with ; between values, as in much of Europe.
• Fix, Sci and Norm (SETUP 6–8) set the decimals: Fix keeps a number of places, as on a drawing, Sci a number of significant digits, and Norm 1 or 2 switches to ×10ˣ below 0.01 or below 10⁻⁹.
• ENG writes the result with an exponent that is a multiple of 3, the steps of k, M, m and µ: 0.0047 becomes 4.7ᴇ−3, 4.7 thousandths. Each press moves the point three places; SHIFT ENG moves it back.
• °′″ after = shows an angle in degrees, minutes and seconds, and back. In a line it marks degrees, minutes and seconds as a protractor or drawing gives them: 30°15°20° is 30°15′20″.
• FACT (SHIFT °′″) splits a whole-number result into prime factors: 360 is 2³×3²×5.

Angles and coordinates

DRG steps the angle unit through degrees (D), radians (R) and grads (G, 400 to a turn). A value with its unit marked is changed to the unit in use: DRG▶ (SHIFT Ans) then 1, 2 or 3 types °, ʳ or ᵍ, so (π/2)ʳ is 90 in degrees.

Pol(x, y) and Rec(r, θ) change between the X–Y position of a point and its distance and angle from the origin. A hole on a bolt circle at R40 and 30° is at Rec(40, 30): X = 34.641, Y = 20. Pol does the reverse, from coordinates measured on a drawing. The second result goes into Y and the first into X, so both can be used in the next line.

CONST and CONV

CONST (SHIFT 7) lists 40 physical constants, the CODATA 2022 values published by NIST. CONV (SHIFT 8) lists unit conversions, the shop's first: inches, feet per minute, pound-force and pound-force feet, psi and bar. Pick one from the list: a constant goes in as its symbol, and a conversion goes after a value, 10in▶mm. Every factor comes from the unit's definition, the inch as exactly 25.4 mm and the pound as exactly 0.45359237 kg, so the conversion adds no error.

The modes

• CMPLX: complex numbers, typed with 𝑖, or as length∠angle with SHIFT 𝑖. SHIFT 2 gives arg, Conjg and ▶r∠θ or ▶a+bi to show the result either way. In AC circuits a voltage or an impedance is a length and an angle, and this keeps both through every step.
• STAT: type the data in the table, X alone or X and Y, with a frequency column for repeated values. The table under it gives the count, mean, both standard deviations (σx divides by n, sx by n − 1 for a sample), quartiles and, with Y, the regression coefficients and r, with ŷ and x̂ for any value. SHIFT 1 types any of them into the line by name, and so does a tap on a result: x̄+3sx uses the data as it is now, so it changes when a value in the table does. 5ŷ is the fitted y at x = 5, and 20x̂ the x where the fit gives 20.
• BASE-N: whole numbers in decimal, hex, binary or octal, chosen with the DEC HEX BIN OCT keys (x² xⁿ log ln). Hex digits A–F are on the keys marked with them. SHIFT 3 gives and, or, xor, xnor, Not and Neg on the 32 bits, and d h b o to type one number in another base.
• EQN: two or three linear equations, or a quadratic or cubic. Type the coefficients and press = to solve; complex roots are shown as a+bi.
• MATRIX and VECTOR: set the size and values of MatA–MatC or VctA–VctC in the table, then use them in the line from SHIFT 4 or SHIFT 5: +, −, ×, det, Trn and x⁻¹ for matrices; + −, × (cross product), Dot and Abs (length) for vectors.
• TABLE: type f(X) and press =, then set the start, end and step: the values of f come out as a table, to check a curve or pick values to plot. With f(X),g(X) set in SETUP, = after f(X) asks for g(X), and the table has both side by side, to compare two formulas or see where two curves cross.

Measured parts against the tolerance

Under the STAT results, type the lower and upper limits of the size. The calculator then gives x̄ − 3sx and x̄ + 3sx, the band where nearly all parts of a steady process fall (99.73 % when the sizes spread in the usual bell shape), and two figures for how well that band fits the tolerance:

• Cp = (upper − lower) ÷ 6sx: how many times the spread fits in the tolerance, as if it were centred. Below 1, some parts will be out of tolerance wherever the offset is set; only a steadier process (a sharper tool, a firmer setup) brings it up.
• Cpk = the distance from x̄ to the nearer limit ÷ 3sx: how well it fits where it is. When Cpk is well below Cp, the spread is fine but off centre, and an offset moves it back.

Cp and Cpk, or Pp and Ppk: which spread goes in

The two pairs use the same formulas. What differs is the spread put in for σ, and that depends on which parts were measured. Every part's size is pulled two ways: by the small part-to-part scatter that is there even when nothing changes, and by slow changes between one batch and the next, such as tool wear, a new bar of material that is a little harder, the machine warming up through the morning, or an offset changed after a tool change. A short run of parts sees only the first; parts gathered over days see both.

• Cp and Cpk use the short-term spread: parts made one after another from one setup, with the same tool and the same material. They show what the machine and process can do at their best.
• Pp and Ppk use the long-term spread: parts taken across batches, shifts, tool changes and days. Their sx also holds the shifts between batches, so it is larger and Pp and Ppk come out lower. They show what the customer actually receives.

The calculator works out both pairs the same way, from sx of the parts in the table, so the name depends on how the parts were picked. Ten parts in a row off the machine give Cp and Cpk; ten parts picked from ten days' boxes give Pp and Ppk. The gap between the two pairs is itself the useful figure: when Cp is high but Pp is low, the machine is fine and the losses come from what changes between batches, so the fix is a steadier routine (warming up, setting offsets after a tool change), not a better machine.

A common customer requirement is 1.33, and PPAP asks for a Ppk above 1.67 in a new part's first study. With one limit only, as a maximum roughness or a minimum hardness, leave the other empty: there is then only Cpk. Treat figures from a short run as a first look: from ten parts, sx itself is only good to about a quarter either way, so a Cpk of 1.58 could as well be 1.2 or 2. The more parts, the firmer it gets.

Syntax ERROR and Math ERROR

A Syntax ERROR is a line that cannot be read, such as 2×÷3 or a missing value. A Math ERROR is a line that reads correctly but has no value: dividing by zero, the square root of a negative number, sin⁻¹ of 1.2. The two need different fixes, a typing slip or a wrong number, so they are told apart. The line under the display says what is wrong; ◀ or ▶ opens the line with the cursor where the mistake was found, and AC clears it.

Full screen and as an app

Full screen gives the calculator the whole screen, so the keys are large enough for a finger. Where the browser can install it as an app (Chrome or Edge, on a phone or a computer), the page offers it in a bar above the calculator; once that bar is closed it does not come back, and the browser's own menu installs the app: it then opens from the home screen straight into the calculator, without the browser's bars, and works with no signal, because the phone keeps a copy of the page. A connected phone loads the current version each time it opens the app; an app left open for days says when a new version is ready and offers to reload, which keeps the memory and history. On an iPhone, install it from Safari: Share, then Add to Home Screen.