Scientific Calculator

How to use the scientific calculator

Write the whole calculation, then press = or Enter. You can tap the keys, type on a keyboard, or mix the two. The answer appears in the result panel together with the expression it belongs to, and every answer you work out is kept in a history list for this visit, so you can tap an earlier result to reuse it.

• Functions such as sin, log and √ open a bracket for you: tap sin, type 30 and the closing bracket is optional — missing brackets at the end are closed automatically.
• Ans is the previous answer. Pressing an operator straight after = continues from it, so “= then × 2” doubles the last result.
• x², 1/x and n! act on the number or bracket just before them: 5² = 25, 4⁻¹ = 0.25, 5! = 120.
• % divides by 100, so 15% is 0.15 and 200 × 15% = 30. It does not add a percentage on top (200 + 10% is 200.1); for increases and decreases use the percentage calculator.
• Typing works too: * and / for × and ÷, ^ for powers, “sqrt”, “pi”, “asin” and so on. Use a point for decimals and a capital E for powers of ten (1.5E3 = 1,500); a lowercase e is Euler’s number.

Order of operations (PEMDAS / BODMAS)

A calculator has to decide which part of an expression to work out first. The convention taught as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) is the same rule under two names:

1. Brackets first, innermost first.
2. Functions (sin, log, √) and the operators written after a number: ², ⁻¹, !, %.
3. Powers, worked from right to left: 2^3^2 = 2^9 = 512, not 8² = 64.
4. A minus sign in front: −2^2 = −(2²) = −4. To square −2, write (−2)^2 = 4.
5. Multiplication and division, from left to right: 100 ÷ 10 ÷ 5 = 2.
6. Addition and subtraction, from left to right: 10 − 4 − 3 = 3.

Multiplication does not come “before” division, and addition does not come before subtraction: each pair has equal rank and is read left to right. That is why 8 ÷ 4 × 2 = 4, not 1.

Implicit multiplication. You can leave out the × sign before a bracket, a constant or a function: 2π, 3(4 + 5) and 2sin(30) all work. This calculator gives the hidden × the same rank as a written one and reads left to right, so 1 ÷ 2π means (1 ÷ 2) × π ≈ 1.5708, and the much-argued 6 ÷ 2(1 + 2) is 9. Some handheld calculators give implicit multiplication higher priority and would answer 1 ÷ (2π) and 1. When it matters, add brackets: 1 ÷ (2π) leaves no doubt.

Worked example

The calculator opens with 2 + 3 × 4^2 − 10 ÷ 5. It is worked out in this order:

1. Power: 4^2 = 16
2. Multiply: 3 × 16 = 48
3. Add, left to right: 2 + 48 = 50
4. Divide: 10 ÷ 5 = 2
5. Subtract: 50 − 2 = 48

A trigonometry example: in degree mode, sin(30) = 0.5 exactly and cos(90) = 0. Switch to radians and sin(30) becomes −0.9880316241, because 30 radians is almost five full turns (about 1,718.9°). In radian mode, sin(π ÷ 6) gives the 0.5 instead.

Degrees or radians?

A full turn is 360 degrees or 2π radians, so 180° = π rad and 1 rad ≈ 57.2958°. The radian is the SI unit for plane angles and is what the formulas of calculus use; degrees are what most geometry, surveying and everyday problems use.

• Use degrees when the angle in your problem is written with a ° sign or comes from a protractor, a map bearing or a roof pitch.
• Use radians when the angle contains π (π/4, 2π/3) or comes from a physics or calculus formula such as angular speed.

The mode also changes what the inverse functions return: sin⁻¹(0.5) is 30 in degree mode and 0.5235987756 (π/6) in radian mode. Changing the mode recalculates the answer on screen.

How answers are rounded

Like almost all software, the calculator stores numbers in binary floating point (IEEE 754 double precision), which holds about 15 to 17 significant decimal digits. Some decimals, such as 0.1, have no exact binary form, so raw arithmetic can give 0.30000000000000004 for 0.1 + 0.2. To keep that noise off the screen:

• Intermediate results keep full precision; the final answer is rounded to 15 significant digits.
• A sum or difference smaller than the rounding error of the numbers it came from counts as exactly zero, so 0.1 + 0.2 − 0.3 = 0 and dividing by it is reported as division by zero.
• Sine, cosine and tangent return exact values at multiples of 30° and 45° (sin 30° = 0.5, tan 45° = 1, cos 90° = 0) and exact zeros at multiples of π/2 in radian mode, where plain floating point would give about 6 × 10⁻¹⁷ for cos(π/2).
• The display shows up to 10 digits, and switches to scientific notation (such as 1.2 × 10¹⁸) for numbers of 10¹⁵ and above or below 10⁻⁹. The ledger always shows the scientific form.

Limits

• The largest number is about 1.8 × 10³⁰⁸, so 170! (about 7.26 × 10³⁰⁶) is the largest factorial; 171! is reported as too large. Factorials need whole numbers.
• A negative number raised to a fractional power, such as (−8)^(1/3), is not calculated: 1/3 can’t be stored exactly, so the calculator can’t tell that you meant a cube root. Write −(8^(1/3)) = −2 instead.
• Tangent is not defined at 90°, 270° and so on. Values very close to those angles give very large, correct results rather than an error.
• Answers are real numbers only: no complex results (√−1), matrices or statistics. For a list of values use the standard deviation calculator; for exact fractions, the fraction calculator.
• The history is kept only while the page is open and is never sent anywhere.