Calculator Notation¶

When entering mathematical answers in Varsity Learning, use calculator-style notation. This guide shows how to enter common expressions correctly.

Basic Operations¶

Operation	Notation	Example
Addition	`+`	`2 + 3`
Subtraction	`-`	`5 - 2`
Multiplication	`*`	`3 * 4`
Division	`/`	`10 / 2`
Exponentiation	`^`	`2^3` (means 2³)

Important: Always use * for multiplication. 2x won't work; use 2*x.

Grouping & Order of Operations¶

Use parentheses to clarify your meaning:

Expression	Result	Why
`2 + 3 * 4`	14	Multiply first, then add
`(2 + 3) * 4`	20	Add first (parentheses), then multiply
`2 ^ 3 ^ 2`	512	Right-to-left: 2^(3^2) = 2^9
`(2 ^ 3) ^ 2`	64	Left-to-right: (2^3)^2 = 8^2

Variables & Algebraic Expressions¶

Expression	Means
`2*x + 3`	2x + 3
`x^2 - 5*x + 6`	x² – 5x + 6
`(x + 1)^2`	(x + 1)²
`sqrt(x)`	√x
`3ab - c^2`	3ab – c²

Functions¶

Trigonometric Functions¶

(Angles are in radians)

Function	Notation	Example
Sine	`sin(x)`	`sin(pi/2)`
Cosine	`cos(x)`	`cos(0)`
Tangent	`tan(x)`	`tan(pi/4)`
Arcsine	`asin(x)`	`asin(0.5)`
Arccosine	`acos(x)`	`acos(0)`
Arctangent	`atan(x)`	`atan(1)`

Other Common Functions¶

Function	Notation	Example
Square root	`sqrt(x)`	`sqrt(16)` → 4
Absolute value	`abs(x)`	`abs(-5)` → 5
Natural logarithm	`ln(x)`	`ln(e)` → 1
Base-10 logarithm	`log(x)`	`log(100)` → 2
Exponential	`exp(x)`	`exp(1)` ≈ 2.718
Power/Exponent	`^`	`e^x`

Special Constants¶

Constant	Notation	Approximate Value
Pi	`pi`	3.14159...
Euler's number	`e`	2.71828...
Imaginary unit	`i`	√(–1)
Infinity	`oo` or `inf`	∞

Examples: - 2*pi*r (circumference of circle) - e^(-t) (exponential decay) - 3 + 4*i (complex number)

Fractions¶

You can enter fractions as decimals or division:

Notation	Result
`1/2`	0.5
`3/4`	0.75
`(x + 1)/(x - 1)`	(x+1)/(x–1)
`1/(2pir)`	1/(2πr)

Decimals & Scientific Notation¶

Standard Decimals¶

0.5
0.001
3.14159

Scientific Notation¶

1.5e-3    (means 1.5 × 10⁻³ = 0.0015)
2.3e5     (means 2.3 × 10⁵ = 230000)

Intervals & Sets¶

Interval Notation¶

[0, 5]      Closed interval (both endpoints included)
(0, 5)      Open interval (endpoints excluded)
[0, 5)      Half-open (left included, right excluded)
[5, oo)     From 5 to infinity
(-oo, 3)    From negative infinity to 3
(-oo, oo)   All real numbers

Union of Intervals¶

[0, 2) U (5, 10]
(-oo, -1] U [1, oo)

Set Notation¶

{1, 2, 3}           Set with elements 1, 2, 3
{x : x > 0}         Set-builder notation
{x ∈ ℝ : x ≠ 0}     All real numbers except 0

Complex Numbers¶

Enter in standard form a + bi:

3 + 2*i          (3 + 2i)
-1 + 1*i         (–1 + i)
5 - 3*i          (5 – 3i)
i                (same as 0 + 1*i)
4                (same as 4 + 0*i)

The system recognizes both i and j for the imaginary unit.

Matrices¶

For matrix answers, enter row-by-row:

[[1, 2], [3, 4]]       2×2 matrix: [1 2]
                                   [3 4]

[[1, 2, 3], [4, 5, 6]] 2×3 matrix: [1 2 3]
                                   [4 5 6]

N-Tuples & Ordered Pairs¶

(3, 5)          Ordered pair
(1, 2, 3)       Triple (3-tuple)
(a, b, c, d)    Using variables

Tips¶

Preview Before Submitting¶

Most questions show a preview of your answer. Check it before submitting to catch: - Missing multiplication signs - Misplaced parentheses - Typos in function names

Use the Math Palette¶

If available, click the Math Palette to: - Insert symbols graphically - Avoid typing mistakes - See correct notation

Spacing Doesn't Matter¶

These are all equivalent: - 2*x + 3 - 2 * x + 3 - 2*x+3

Order of Operations¶

The system follows standard mathematical order:

Parentheses — (...)
Exponents — ^
Multiplication & Division — *, / (left to right)
Addition & Subtraction — +, – (left to right)

Example: 2 + 3 * 4 = 2 + 12 = 14 (not 5 * 4 = 20)

When in Doubt, Use Parentheses¶

Bad:  2x + 3^2         (ambiguous: is 3 squared or whole thing?)
Good: (2*x + 3)^2      (clear: square the entire expression)

Common Mistakes¶

Wrong	Right	Issue
`2x`	`2*x`	Missing multiplication sign
`sin x`	`sin(x)`	Functions need parentheses
`1/2x`	`(1/2)x` or `1/(2x)`	Ambiguous division
`2^3^2`	`(2^3)^2`	Ambiguous exponentiation
`sqrt 4`	`sqrt(4)`	Functions need parentheses
`log_10(x)`	`log(x)`	No subscript notation; use `log` for base-10
`sin⁻¹(x)`	`asin(x)`	Use function name, not superscript

Degrees vs. Radians¶

Important: Trigonometric functions use radians, not degrees.

Angle	Radians	Result
0°	0	sin(0) = 0
30°	π/6	sin(π/6) = 0.5
45°	π/4	sin(π/4) ≈ 0.707
90°	π/2	sin(π/2) = 1
180°	π	sin(π) = 0

Conversion: degrees × (π/180) = radians

Example Question¶

Find sin(60°).

sin(60°) = sin(pi/3)  ← Convert 60° to π/3 radians
         = sqrt(3)/2
         ≈ 0.866