Difficulty: Introductory | Prerequisites: Basic algebra and an intuition for what "approaching a value" means.
Calculus grew out of questions that ordinary arithmetic cannot answer: what happens when you divide by zero, how motion works at an instant, and whether adding infinitely many numbers can produce a finite result. Zeno's paradoxes frame these problems in plain language, and the tools of limits and infinite series resolve them. These foundational ideas underpin everything else in MATH 104.
Division by zero
The expression a/0, where a is the dividend. In ordinary arithmetic, this is undefined because no number multiplied by 0 produces a non-zero result. In simple terms, it is the question "how many times does nothing fit into something?" and the answer is that the question breaks the rules of arithmetic.
Motion
A change in the position of an object with respect to time, described in terms of displacement, distance, velocity, acceleration and speed. Think of it as the most basic physical quantity that calculus was invented to analyse: Newton needed to describe how things move at every instant, not just on average.
Zeno's paradoxes
A set of philosophical problems attributed to the ancient Greek philosopher Zeno (c. 490-430 BC), arguing that motion is an illusion. Several of these paradoxes involve dividing distance or time into infinitely many parts, producing apparent contradictions that calculus tools, especially limits and infinite series, can resolve.
In simple terms, Zeno asked questions like "how can you ever cross a room if you always have to cover half the remaining distance first?" Calculus answers: because the sum of those infinite halves is finite.
Infinite series
A sequence of numbers in which an infinite number of terms are added successively in a given pattern; formally, the sequence of partial sums of a given sequence. Think of it as a running total that never stops being added to, yet can still settle on a definite value (converge).
In ordinary arithmetic, a/0 is undefined for any a ≠ 0.
The operation asks "what number, multiplied by 0, gives a?" No such number exists.
This creates paradoxes when we try to analyse continuous change (motion, area) using only algebra.
Calculus resolves this by introducing the limit: instead of dividing by zero, we examine what happens as a quantity approaches zero.
Motion is observed by fixing a frame of reference and tracking how position changes over time.
Key quantities: displacement, distance (scalar), velocity, acceleration, speed.
The central difficulty: average velocity = distance / time, but instantaneous velocity at a single moment appears to require dividing zero distance by zero time.
Calculus handles this through the derivative, which defines instantaneous rate of change as a limit.
Zeno's paradoxes were designed to argue that motion is impossible, supporting Parmenides's view that change is an illusion.
The Dichotomy Paradox: to travel any distance, you must first travel half, then half of what remains, and so on, producing infinitely many steps.
The Achilles Paradox: a faster runner can never overtake a slower one, because the faster runner must always first reach the point where the slower one was.
Both paradoxes hinge on the assumption that summing infinitely many quantities must produce an infinite result. Calculus shows this assumption is wrong.
An infinite series is a sum of the form a₁ + a₂ + a₃ + ..., studied through its sequence of partial sums: S₁ = a₁, S₂ = a₁ + a₂, S₃ = a₁ + a₂ + a₃, and so on.
If the partial sums approach a finite value L, the series converges to L.
The geometric series 1/2 + 1/4 + 1/8 + ... converges to 1, directly resolving Zeno's Dichotomy Paradox: the infinitely many half-distances add up to the full distance.
Whether a series converges or diverges is one of the central questions of calculus.
Geometric series formula:
\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \quad |r| < 1For Zeno's Dichotomy (a = 1/2, r = 1/2):
\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = \frac{1/2}{1 - 1/2} = 1Average velocity:
v_{avg} = \frac{\Delta x}{\Delta t}Instantaneous velocity (the limit that resolves the 0/0 problem):
v(t) = \lim_{\Delta t \to 0} \frac{x(t + \Delta t) - x(t)}{\Delta t}Students often think dividing by zero gives infinity. It does not. The expression is undefined, which is a different thing from being infinitely large. Limits can approach infinity, but the bare operation a/0 has no value.
Students sometimes believe that adding infinitely many positive numbers must always give an infinite result. Convergent series disprove this: 1/2 + 1/4 + 1/8 + ... = 1.
Zeno's paradoxes are sometimes dismissed as "just philosophy." In a calculus course, they are the motivation for the formal definition of limits and series convergence. Expect them in conceptual exam questions.
Instantaneous velocity is not "velocity at zero time." It is the limit of average velocity as the time interval shrinks toward zero. The distinction matters.
⚠️ Zeno's paradoxes are a classic source of conceptual exam questions. Be ready to explain how limits and convergent series resolve them.
⚠️ Understanding why a/0 is undefined (not infinite) is foundational. It underpins the entire definition of the derivative.
⚠️ The geometric series formula is one of the most commonly tested formulas in Calculus 1. Know it cold, including the convergence condition |r| < 1.
⚠️ The transition from average velocity to instantaneous velocity is the conceptual bridge to derivatives. Exam questions often frame this as a word problem involving position and time.
True or false: a/0 = ∞ when a ≠ 0. _______ (False. The expression is undefined.)
The sum 1/2 + 1/4 + 1/8 + 1/16 + ... converges to _______. (1)
True or false: Zeno's Dichotomy Paradox proves that motion is impossible. _______ (False. It shows that pre-calculus reasoning cannot handle infinite processes; the limit resolves it.)
Instantaneous velocity is defined as the _______ of average velocity as Δt approaches zero. (limit)
A series converges if its sequence of _______ approaches a finite value. (partial sums)
Q: Explain why division by zero is undefined, not infinite.
A: Division asks "what number, multiplied by the divisor, gives the dividend?" For a/0 with a ≠ 0, no number multiplied by 0 gives a non-zero result. The operation has no answer, so it is undefined. Limits can describe behaviour approaching a point where the denominator reaches zero, but the bare expression a/0 has no value.
Q: How does the geometric series resolve Zeno's Dichotomy Paradox?
A: Zeno argues you must traverse 1/2, then 1/4, then 1/8, and so on, producing infinitely many steps. The geometric series 1/2 + 1/4 + 1/8 + ... = 1/(1 - 1/2) · (1/2) = 1 shows the infinite sum is finite and equals the full distance. The runner does complete the journey.
Q: What is the difference between average velocity and instantaneous velocity?
A: Average velocity is total displacement divided by total time (Δx/Δt). Instantaneous velocity is the limit of this ratio as Δt approaches zero. The limit avoids the undefined 0/0 by examining the trend rather than the value at zero.
Q: Give an example of an infinite series that diverges and one that converges.
A: The harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (grows without bound). The geometric series 1/2 + 1/4 + 1/8 + ... converges to 1.
This material connects directly to limits (the formal tool that resolves the 0/0 problem), derivatives (instantaneous velocity is the first derivative of position), and series convergence tests (which extend the geometric series idea to harder cases). If you are reviewing this, the next logical step is the formal epsilon-delta definition of a limit.
Division by zero, undefined expression, indeterminate form, Zeno's paradoxes, Dichotomy Paradox, Achilles and the Tortoise, infinite series, convergent series, divergent series, geometric series, partial sums, limits, instantaneous velocity, average velocity, rate of change, motion, displacement, calculus foundations, MATH 104, Calculus 1, Penn State