Basic Mathematical Expressions

Basic mathematical expressions are the building blocks of mathematical typesetting in LaTeX. They include common elements like symbols, operators, fractions, superscripts, subscripts, roots, and more. Understanding how to create these expressions allows you to write clean and precise mathematical content, whether for simple equations or more complex formulas.

Operators and symbols

Operators and symbols are fundamental components of mathematical expressions. They allow you to represent arithmetic, relational, and logical operations, as well as special mathematical symbols, with clarity and precision.

Arithmetic operators

Arithmetic operators in LaTeX allow you to represent basic mathematical operations. LaTeX offers commands for commonly used symbols such as:

Operator	Latex Command	Code Example	Rendered Output
Addition	+	\[ a+b \]
Subtraction	-	\[ a-b \]
Multiplication	\times	\[ a \times b \]
Division	\div	\[ a \div b \]
Dot product	\cdot	\[ a \cdot b \]

Relational operators

Relational operators are used to create clear and accurate comparisons in equations or inequalities. Below are common relational operators:

Operator	Latex Command	Code Example	Rendered Output
Equality	=	\[ a = b \]
Inequality	\neq	\[ a \neq b \]
Less than	<	\[ a < b \]
Greater than	>	\[ a > b \]
Less than or equal to	\leq	\[ a \leq b \]
Greater than or equal to	\geq	\[ a \geq b \]

Logical operators

logical operators are used to express statements involving conjunctions, disjunctions, negations, implications, and equivalences. These operators are typically used in set theory, proofs, and logic-based problems. Below are common logical operators:

Operator	Latex Command	Code Example	Rendered Output
And	\land	\[ P \land Q \]
Or	\lor	\[ P \lor Q \]
Not	\neg	\[ \neg P \]
Implies	\implies	\[ P \implies Q \]
If and only if	\iff	\[ P \iff Q \]

Fractions

Fractions are written using the \frac{numerator}{denominator} command which produces a clear horizontal bar between the numerator and denominator.

Concept	Code Example	Rendered Output
Simple Fraction	\[ \frac{a+b}{c+d} \]
Nested Fraction	\[ \frac{\frac{a}{b}}{c} \]

Ratios

Ratios are typically represented using a colon : or a fraction. While the colon is straightforward, ratios expressed as fractions are typeset using \frac.

Concept	Code Example	Rendered Output
Ratio (Colon Format)	\[ a : b \]
Ratio as Fraction	\[ \frac{a}{b} \]

Subscripts and superscripts

Subscripts and superscripts are used in mathematical expressions to denote indices, powers, exponents or additional contextual information. In LaTeX, these are added using the followings:

• _ (underscore) → for subscripts.
• ^ (caret) → for superscripts.

Concept	Code Example	Rendered Output
Subscript	\[ x_i \]
Superscript	\[ y^j \]
Multi-Character Subscripts and Superscripts	\[ x_{ij} + y^{i+j} \]
Combined Subscript and Superscript	\[ x_{i+1}^{j+2} \]
Nested Subscript and Superscript	\[ x_{i_{j}}^{k^l} \]
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Roots

Roots in LaTeX are mathematical expressions typeset using \sqrt{} for square roots and \sqrt[n]{} for n-th roots, for example:

Concept	Code Example	Rendered Output
Square Root	\[ \sqrt{x} \]
Cube Root	\[ \sqrt[3]{x} \]
General n-th Root	\[ \sqrt[n]{x+y} \]
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Logarithms

Logarithms in LaTeX are represented using \ln for natural logs, \log for base-10 or unspecified bases, and \log_{base}{} to define logarithms with a specific base.

Concept	Code Example	Rendered Output
Natural Logarithm (ln)	\[ \ln{x} \]
Base-10 Logarithm (log)	\[ \log{x} \]
Logarithm with Base	\[ \log_{2}{x} \]
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Summation and products

Summation (∑) and product (∏) notations are essential for representing series and multiplicative sequences in mathematical expressions.

Concept	Code Example	Rendered Output
Summation without limits	\[ \sum x_i \]
Summation with upper limit	\[ \sum^{n} x_i \]
Summation with lower limit	\[ \sum_{i=1} x_i \]
Summation with both limits	\[ \sum_{i=1}^{n} x_i \]

Concept	Code Example	Rendered Output
Product without limits	\[ \prod x_i \]
Product with upper limit	\[ \prod^{n} x_i \]
Product with lower limit	\[ \prod_{i=1} x_i \]
Product with both limits	\[ \prod_{i=1}^{n} x_i \]

Trigonometric functions

Trigonometric functions playing a crucial role in analyzing angles, ratios, and periodic phenomena:

Concept	Function	Code Example	Rendered Output
Basic Trigonometric Functions	Sine	\[ \sin(x) \]
	Cosine	\[ \cos(x) \]
	Tangent	\[ \tan(x) \]
Reciprocal Trigonometric Functions	Cotangent	\[ \cot(x) \]
	Secant	\[ \sec(x) \]
	Cosecant	\[ \csc(x) \]
Inverse Trigonometric Functions	Arcsine	\[ \arcsin(x) \]
	Arccosine	\[ \arccos(x) \]
Hyperbolic Functions	Arctangent	\[ \arctan(x) \]
	Hyperbolic Sine	\[ \sinh(x) \]
	Hyperbolic Cosine	\[ \cosh(x) \]
	Hyperbolic Tangent	\[ \tanh(x) \]
	Hyperbolic Cotangent	\[ \coth(x) \]

Integral

Integrals are a core concept in calculus, representing the area under a curve, accumulation of quantities, or anti-derivatives of functions.

Concept	Code Example	Rendered Output
Indefinite integral	\[ \int f(x) \, dx \]
Definite integral	\[ \int_a^b f(x) \, dx \]
Double integral	\[ \iint_R f(x, y) \, dx \, dy \]
Triple integral	\[ \iiint_V f(x, y, z) \, dx \, dy \, dz \]
Contour integral	\[ \oint f(z) \, dz \]
Line integral	\[ \int_C f(x, y) \, ds \]

Limits

Limits describe the behavior of a function as the input approaches a specific value. The basic syntax for expressing a general limit is \lim_{<variable> \to <value>} <expression>, for example:

Concept	Code Example	Rendered Output
Limit as x approaches a	\[ \int f(x) \, dx \]
Limit to infinity	\[ \lim_{x \to \infty} f(x) \]

Derivatives

Derivatives measure the rate of change of a function with respect to its variable and are essential for understanding slopes, rates, and instantaneous changes.

Concept	Code Example	Rendered Output
First derivative (Prime notation)	\[ f'(x) \]
First derivative (Leibniz notation)	\[ \frac{d}{dx}f(x) \]
Second derivative(Prime notation)	\[ f''(x) \]
Second derivative (Leibniz notation)	\[ \frac{d^2}{dx^2}f(x) \]
Higher-order derivative	\[ f^{(n)}(x) \]
Partial derivative	\[ \frac{\partial f}{\partial x} \]
Higher-order partial derivative	\[ \frac{\partial^2 f}{\partial x^2} \]
Gradient	\[ \nabla f \]

Greek Letter

Below is a table of important Greek letters frequently used in mathematical and scientific contexts.

Greek Letter	Lowercase Command	Uppercase Command	Rendered Output
Alpha	\alpha	A
Beta	\beta	B
Gamma	\gamma	\Gamma
Delta	\delta	\Delta
Epsilon	\epsilon	E
Theta	\theta	\Theta
Lambda	\lambda	\Lambda
Pi	\pi	\Pi
Sigma	\sigma	\Sigma
Omega	\omega	\Omega

In conclusion, the "Basic Mathematical Expressions" section provides essential tools for writing a wide range of mathematical notations in LaTeX. By understanding how to use operators, symbols, fractions, ratios, subscripts, superscripts, and other key expressions, you can easily structure a variety of mathematical expressions.