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The Complete Guide to Mathematical Expressions

Mathematical expressions are the language of numbers. From simple arithmetic like 2 + 3 to complex formulas like sin(45) * sqrt(x^2 + y^2), understanding how to write and evaluate expressions is fundamental to science, engineering, finance, and everyday problem-solving.

Order of Operations: BODMAS/PEMDAS

The order of operations determines which calculations are performed first. Without these rules, 2 + 3 * 4 could equal 20 (left-to-right) or 14 (multiplication first).

PriorityOperationExample
1stBrackets / Parentheses(2 + 3) = 5
2ndOrders / Exponents2^3 = 8
3rdDivision & Multiplication12 / 4 * 2 = 6
4thAddition & Subtraction5 + 3 - 2 = 6

Scientific Functions Reference

Trigonometry

  • sin(45) = 0.707
  • cos(60) = 0.5
  • tan(45) = 1

Logarithms

  • log(100) = 2
  • ln(e) = 1

Roots & Powers

  • sqrt(144) = 12
  • 2^10 = 1024

Common Use Cases

Circle Calculations

Area: pi * r^2

Circumference: 2 * pi * r

Right Triangle

Hypotenuse: sqrt(x^2 + y^2)

Example: x=3, y=4 → 5

Compound Interest

Formula: P * (1 + r/100)^t

P=Principal, r=Rate%, t=Years

Distance Formula

2D: sqrt((x2-x1)^2 + (y2-y1)^2)

Frequently Asked Questions

What is an expression solver?

An expression solver is a calculator that evaluates mathematical expressions by parsing the entire string and applying the correct order of operations. Unlike basic calculators, it understands that 2 + 3 * 4 equals 14 (not 20), because multiplication has higher precedence than addition.

What is BODMAS/PEMDAS?

BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) and PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) are mnemonics for the mathematical order of operations. They ensure consistent evaluation of expressions.

What scientific functions are supported?

Supported functions include: Trigonometric (sin, cos, tan - input in degrees), Logarithmic (log for base-10, ln for natural), Roots (sqrt, cbrt), and Other (abs, floor, ceil, round, exp). All functions use standard notation like sqrt(16) or sin(45).

How do I use variables?

You can use x, y, and z as variables. Set their values in the Variables section, then use them in expressions like '2*x + y' or 'sqrt(x^2 + y^2)'. This is useful for testing formulas with different inputs.

What is the difference between log and ln?

log() is the common logarithm (base 10). For example, log(100) = 2. ln() is the natural logarithm (base e ≈ 2.718). For example, ln(e) = 1. Natural log is more common in science; base-10 is used in everyday calculations.

Are trigonometric functions in degrees or radians?

All trigonometric functions use degrees, not radians. This is more intuitive for most users. For example, sin(90) = 1.

What is batch mode?

Batch mode lets you solve multiple expressions at once. Enter each expression on a new line, and the solver evaluates all of them simultaneously. You can also upload a .txt file containing expressions.

Why do I get a syntax error?

Common causes include: unbalanced parentheses, missing operators (use '5*3' not '5(3)'), unsupported characters, division by zero, or negative square root. Check the expression syntax and try again.

How accurate are the calculations?

Calculations use JavaScript's 64-bit floating-point arithmetic (IEEE 754), providing approximately 15-17 significant decimal digits of precision. Results are displayed to 8 decimal places.

Can I use this offline?

Yes! Once the page is loaded, all calculations happen in your browser. No internet connection is needed for computation. Your calculation history is stored in browser memory.

What are the mathematical constants?

Two constants are available: pi = 3.14159... (ratio of circumference to diameter) and e = 2.71828... (Euler's number, base of natural logarithm). Use them directly in expressions like 'pi * 5^2'.

How do I calculate powers?

Use the ^ operator for powers. For example, 2^10 = 1024, 5^2 = 25. For roots, use sqrt() for square root, cbrt() for cube root, or x^(1/n) for nth root.

Can I see the calculation steps?

Yes! Click 'Show Steps' to see how the expression is processed: original input, constant/variable substitutions, function evaluations, and final result.

How do I export my calculations?

Click the 'Export' button to download your calculation history or batch results as a text file. Each line will contain the expression and its result.

What operators are supported?

Supported operators: + (addition), - (subtraction), * (multiplication), / (division), ^ (power/exponent), and parentheses () for grouping. Operators follow standard BODMAS/PEMDAS precedence.