Scientific Calculator - Trig, Logarithms & Advanced Math

Order of operations

First come brackets and the argument of a function such as sin(…), then factorial (!) and percent (%). Next come powers (^), right to left, so 2^3^2 is 2^9. Multiplication and division follow, left to right, and then addition and subtraction. A bracket left open at = is closed for you. A number or bracket next to a bracket is multiplied at the same step as ×, so 6÷2(1+2) is 9. A minus sign in front of a number applies after a power on its right, so -2^2 is -4 and (-2)^2 is 4; spreadsheets such as Excel read -2^2 as 4. A negative result or recalled number goes in brackets first, so it is raised as a whole.

Source: Standard rule; no single publisher. The leading minus follows written algebra; no standard states it. Implicit multiplication: CalcTrail’s own; no publisher.

Percent key (%)

B% on its own is B ÷ 100. After + or −, B% is B percent of everything to its left, so 200 + 10% is 220 and 200 + 10% + 5% is 231. After × or ÷ it is B ÷ 100, so 200 × 10% is 20.

Source: CalcTrail’s own rule; no publisher.

Trigonometric functions (Deg, Rad, Grad)

X

the angle, or for an inverse key the value

U

radians in one angle unit: π/180 in Deg, 1 in Rad, π/200 in Grad

cos and tan work as sin does, and acos and atan as asin. tan has no answer at 90°, 270° and every 180° from them (100 and 300 grad, and every 200 grad from them). π cannot be stored exactly as a double, so sin(180) shows 1.225e-16 rather than 0.

Source: Angle units as NIST defines them [1] [2]: 1° = π/180 rad and 1 gon = 0.9°. The functions as the DLMF defines them [3], computed as ECMA-262 allows [4].

Logarithms and exponentials

X

a positive number

b

another base, for a logarithm the keys do not offer

ln is the logarithm to base e, and eˣ and 10ˣ undo ln and log. A logarithm of 0 or a negative number has no real answer.

Source: Uses the definitions in the NIST Digital Library of Mathematical Functions [5].

Hyperbolic functions

X

a plain number

The inverse keys undo each one. The angle setting does not change them. sinh(1) + cosh(1) gives e (2.72 in Auto).

Source: Uses the definitions in the NIST Digital Library of Mathematical Functions [6].

Powers, roots and factorial

X

the number the key acts on

n

a whole number from 0 to 170; 0! is 1

√ needs a number of 0 or more; ∛ takes negatives, so ∛(−8) is −2. n! is a running product of doubles: exact up to 22!, and from 23! the last digits can differ from the exact value.

Source: Standard formulas; no single publisher. Factorial uses the definition in NIST’s Dictionary of Algorithms and Data Structures [7].

Constants

π is 3.141592653589793 and e is 2.718281828459045, every digit a double holds. φ is (1 + √5) ÷ 2, 1.618033988749895. The speed of light in vacuum is 299,792,458 m/s, the Planck constant 6.62607015 × 10⁻³⁴ J·s and the Avogadro constant 6.02214076 × 10²³ per mole; NIST lists all three as exact.

Source: π and e: [8]. c, h and NA, with NIST’s names: [9]. φ: standard definition; no single publisher.

Rounding and display

Each result is rounded to 15 significant digits before it is shown. Auto then pads no zeros. It shows up to two decimals from 1 up, so 2 + 3 shows 5, and up to ten significant digits below 1, so 0.5 shows 0.5 and 1 ÷ 3 shows 0.3333333333. From 10^15 up and from 10^-9 down it uses exponential form with up to three decimals, so 10^16 shows 1e+16. Fix, Sci and Eng show the digits you set. A division by zero shows Error with the reason, as does 0 ÷ 0 or a divisor that works out to zero, such as (1 − 1).

Source: CalcTrail’s own display rules; no publisher.

Memory keys

M+ adds the result of the calculation on the display to memory and M− subtracts it; MR puts the stored number into your entry and MC sets it to 0. Memory is kept with the tab.

Source: CalcTrail’s own; no publisher.