Difficulty: Beginner | Prerequisites: Basic arithmetic (addition, subtraction, multiplication, division).
This material sits at the very start of a MATH 104 sequence, bridging secondary-school arithmetic and the more formal language of algebra and calculus. It matters because every later topic in the course, from limits to integration, rests on the vocabulary and conceptual scaffolding introduced here. If you are comfortable with basic arithmetic operations and the idea of "solving for x," you are ready for this.
Elementary algebra extends arithmetic by introducing variables in place of fixed numbers, giving you a language for expressing general relationships and solving equations. The document also surveys the broader landscape of mathematical analysis, formal logic (propositional calculus), and several specialised branches of calculus (lambda, process, variations) that you will encounter later in the course or in adjacent fields.
Elementary algebra
The branch of mathematics that generalises arithmetic by replacing specific numbers with variables, allowing you to write and solve equations. Think of it as arithmetic with letters: instead of "3 + 5 = 8," you write "a + b = c" and work out what the letters stand for.
Variable
A symbol (usually a letter) representing a quantity that does not have a fixed value. In simple terms, it is a placeholder for a number you have not pinned down yet.
Function
A relation between a set of inputs and a set of permissible outputs where each input maps to exactly one output. Think of it as a machine: you feed in one number, and it spits out exactly one result, every time.
Mathematical analysis
The branch of mathematics concerned with continuous change, encompassing differentiation, integration, limits, infinite series, measure theory, and analytic functions. In simple terms, it is the rigorous framework behind everything you will do in calculus.
Infinitesimals
Quantities so small they cannot be measured, yet are not zero. Calculus was historically called "the calculus of infinitesimals" or "infinitesimal calculus." Think of them as the conceptual building blocks that let you talk about instantaneous rates of change.
Propositional calculus
The branch of mathematical logic dealing with formal statements (propositions) that are either true or false, combined using logical connectives (and, or, not, if-then). In simple terms, it is the algebra of true/false reasoning.
Calculus of variations
A field of mathematical analysis that finds functions which maximise or minimise functionals (mappings from functions to real numbers, often expressed as definite integrals). Think of it as optimisation, but instead of finding the best number, you are finding the best curve or path.
Lambda calculus
A formal system in mathematical logic for expressing computation through function abstraction (defining functions) and application (using them), built on variable binding and substitution. In simple terms, it is the mathematical blueprint behind how programming languages define and call functions.
Process calculus
A family of formal approaches for modelling concurrent systems, describing how independent agents or processes interact, communicate, and synchronise. Think of it as a language for writing down how multiple programs talk to each other at the same time.
Functionals
Mappings from a set of functions to the real numbers, often expressed as definite integrals involving functions and their derivatives. In simple terms, a functional takes an entire function as its input and returns a single number.
Name binding
The association of entities (data or code) with identifiers. When an identifier is bound to an object, it is said to reference that object. Think of it as giving a name tag to a piece of data so you can refer to it later.
From arithmetic to algebra
Arithmetic works with specific, known numbers (e.g. 3 + 5 = 8).
Elementary algebra introduces variables, symbols that stand for unknown or general quantities.
This shift lets you express general rules: instead of checking that 2 + 3 = 3 + 2, you can state a + b = b + a for all numbers.
What variables buy you
You can write equations (statements that two expressions are equal) and solve for unknowns.
You can state properties of operations (commutativity, associativity, distributivity) once and apply them universally.
You inherit all the rules of arithmetic operators (addition, subtraction, multiplication, division) and extend them to expressions with variables.
Scope and boundaries
Elementary algebra deals with real and complex numbers. It does not venture into more abstract algebraic structures (groups, rings, fields) covered in abstract algebra courses.
It is the standard prerequisite for pre-calculus and calculus, which is why MATH 104 starts here.
Functions: the bridge to calculus
A function is a rule that assigns each input exactly one output.
Notation: f(x) means "the output of function f when the input is x."
Understanding functions is essential before studying limits, derivatives, and integrals, because those operations act on functions.
Mathematical analysis
The umbrella field covering differentiation, integration, limits, infinite series, measure theory, and analytic functions.
Studied in the context of real and complex numbers and functions.
This is the theoretical backbone of your entire calculus sequence: MATH 104 covers the introductory layer.
Propositional calculus (formal logic)
Deals with propositions: statements that are definitively true or false.
Propositions are combined using logical connectives: AND, OR, NOT, IF...THEN.
The truth value of a compound statement depends entirely on the truth values of its components.
You will encounter this reasoning style whenever you write proofs or work with conditional statements in maths.
Calculus of variations
Instead of optimising a number ("find the x that minimises f(x)"), you optimise a function ("find the curve y(x) that minimises a given integral").
Functionals are the objects being optimised: they take a function as input and return a real number.
Real-world application: this is how physicists derive equations of motion and how engineers find the shape of a cable hanging under its own weight (a catenary).
Lambda calculus
A formal system for describing computation using function definition (abstraction) and function application.
Core idea: you bind a variable to an expression, then substitute values in to evaluate.
Real-world application: this is the theoretical foundation of functional programming languages (Haskell, Lisp, ML). Every time you write a function in code, you are using ideas from lambda calculus.
Process calculus
A toolkit for modelling systems where multiple agents run at the same time (concurrency).
Describes how processes interact, communicate, and synchronise.
Real-world application: this is how computer scientists reason about distributed systems, network protocols, and multi-threaded programs.
Students often think a variable must stand for one specific unknown number. It can also represent a general quantity in a rule or identity (e.g. "for all x, x + 0 = x").
Students often confuse "function" with "equation." A function is a mapping (input to output); an equation is a statement that two expressions are equal. You can write an equation involving a function, but they are different things.
Students often assume "calculus" means only differentiation and integration. The word "calculus" simply means a system of calculation. Propositional calculus, lambda calculus, and process calculus are entirely separate formal systems that happen to share the name.
Students often think infinitesimals are just "very small numbers." In the classical sense, an infinitesimal is smaller than any positive real number yet is not zero. Modern analysis usually avoids infinitesimals in favour of epsilon-delta limits, but the concept is worth understanding historically.
⚠️ Know the precise definition of a function (each input maps to exactly one output). Exam questions often test whether a given relation qualifies as a function.
⚠️ Be able to distinguish between arithmetic (specific numbers) and algebra (variables and general rules). This distinction frequently appears in short-answer or multiple-choice form.
⚠️ Understand that mathematical analysis is the parent field of calculus. If asked to place differentiation, integration, or limits within the broader discipline, the answer is mathematical analysis.
⚠️ The phrase "calculus of infinitesimals" (or "infinitesimal calculus") is a classic fill-in-the-blank question. Memorise both forms.
⚠️ Do not confuse the various "calculi": propositional calculus is logic, lambda calculus is computation theory, process calculus is concurrency, and the calculus of variations is optimisation of functionals. Exam questions may ask you to match each to its domain.
True or false: A variable always represents a single unknown number.
False. A variable can represent a general quantity in a rule or identity.
Fill in the blank: A function maps each input to exactly ______ output.
one
True or false: Propositional calculus is a branch of calculus that deals with derivatives.
False. It is a branch of mathematical logic dealing with true/false statements.
Fill in the blank: Calculus has historically been called "the calculus of " or " calculus."
infinitesimals; infinitesimal
True or false: The calculus of variations optimises numbers rather than functions.
False. It optimises functionals, which take entire functions as inputs.
Q: What is the key difference between arithmetic and elementary algebra?
A: Arithmetic operates on specific, known numbers. Elementary algebra introduces variables (symbols without fixed values), allowing you to express general rules, write equations, and solve for unknowns.
Q: Define "function" in precise mathematical terms.
A: A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
Q: What does mathematical analysis study, and what are its major sub-theories?
A: Mathematical analysis studies continuous change. Its major sub-theories include differentiation, integration, measure theory, limits, infinite series, and analytic functions, studied in the context of real and complex numbers.
Q: Explain what a "functional" is in the context of the calculus of variations.
A: A functional is a mapping from a set of functions to the real numbers, often expressed as a definite integral involving functions and their derivatives. The calculus of variations seeks to find which function maximises or minimises a given functional.
Q: How does lambda calculus relate to computer science?
A: Lambda calculus provides a formal system for expressing computation through function abstraction and application. It is the theoretical foundation of functional programming languages and underpins how programming languages define and evaluate functions.
Q: What is the purpose of process calculus?
A: Process calculus provides formal approaches for modelling concurrent systems, describing how independent agents or processes interact, communicate, and synchronise. It is used to reason about distributed systems, network protocols, and multi-threaded programs.
Elementary algebra feeds directly into pre-calculus and calculus. Every technique for working with limits, derivatives, and integrals assumes fluency with algebraic manipulation: factoring, expanding, simplifying rational expressions, and solving equations.
Mathematical analysis is the parent field for the entire MATH 104 sequence. Understanding that calculus is a subset of analysis helps you see why rigour (epsilon-delta proofs, convergence tests) becomes important later in the course.
Lambda calculus and process calculus connect mathematics to computer science. If you take courses in programming language theory, compilers, or distributed systems, these formal systems will reappear.
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