Two questions arrive together, usually from a student choosing next year's math course: what is the difference between algebra and calculus, and is calculus harder? The short answers are that algebra finds unknown numbers and calculus measures change, and that calculus is harder in ideas but most students who fail it fail on the algebra. This page shows the difference with one problem solved both ways, then gives an honest answer on difficulty.
Download the algebra vs calculus chart and the ten-question readiness check (PDF) (the side-by-side table, the problem solved both ways, the ten questions with answers and how to read the score; two pages, free).
The difference in one line each
Algebra asks: what number makes this true? It uses letters for unknowns, keeps equations balanced, and describes things that are fixed or change at a steady rate.
Calculus asks: how fast is this changing right now, and how much has built up? It deals with rates that are themselves changing, and it uses algebra in every step to do so.
One problem, both ways
The algebra version. A car travels at a steady 60 miles per hour. How far does it go in 3 hours? Distance = speed × time = 60 × 3 = 180 miles. Speed is constant, the graph of distance against time is a straight line, and its slope is 60 everywhere. Algebra is enough.
The calculus version. A car starts from rest and its distance after t seconds is s = 2t² metres. How fast is it going at t = 5? There is no single speed: it is getting faster the whole time. The graph is a curve, and the speed at t = 5 is the slope of that curve at one point. The derivative of 2t² is 4t, so the speed is 4 × 5 = 20 metres per second. And it goes the other way: given the speed 4t, the distance covered in the first 5 seconds is the area under that line, ½ × 5 × 20 = 50 metres, which is 2 × 5².
The first problem has a rate. The second has a rate that changes. That is the whole difference.
Side by side
| Algebra | Calculus | |
|---|---|---|
| Main question | What is the unknown number? | How fast is it changing; how much has built up? |
| Main objects | Equations, expressions, functions, lines and parabolas | Limits, derivatives, integrals |
| Kind of change | None, or steady (constant slope) | Changing (slope that varies from point to point) |
| Typical task | Solve 3x + 5 = 20; factor x² + 7x + 12 | Find the slope of y = x² at x = 3; find the area under it |
| Graph | A line has one slope | A curve has a slope at every point |
| When it is taught (US) | Algebra 1 in Grade 8 or 9, Algebra 2 in Grade 10 or 11 | Grade 12, or Grade 11 if accelerated |
| Invented | Developed over centuries; named from al-Khwarizmi's ninth-century book | Newton and Leibniz, late 1600s |
| Depends on | Arithmetic | Algebra, geometry and trigonometry |
The courses in order, and when each is taken, are in what grade do you take Algebra 1.
Is calculus harder than algebra? The honest answer
It depends on which kind of hard.
Calculus is harder in ideas. A limit, an instantaneous rate and an area as an infinite sum are new kinds of thinking. Algebra 1 has nothing as strange.
Calculus is easier in structure. Algebra 2 is a year of separate topics: logarithms, rational expressions, complex numbers, sequences, conic sections. Calculus has two ideas and keeps returning to them. Students who like to know why often prefer it.
Calculus is harder in length. A calculus problem is one line of calculus and five lines of algebra. A student whose factoring is right four times in five will get a problem that uses factoring three times wrong about half the time. This is why the usual report from a first calculus course is "I understand the lessons and fail the tests".
So:
| Compared with | Is calculus harder? |
|---|---|
| Algebra 1 | Yes, clearly: it comes four courses later |
| Algebra 2 | For most students yes; for some strong students Algebra 2 felt harder because it is more scattered |
| Pre-Calculus | About the same workload; Pre-Calculus has more memorising (trigonometric identities), calculus has more reasoning |
| Linear algebra (university) | Different, not harder: linear algebra is more abstract, calculus has more computation |
The best predictor of how hard calculus will feel is not talent. It is how automatic your Algebra 2 is.
Is calculus easy?
For some students, yes, and the reason is instructive. The students who find calculus easy are almost never the ones with the most talent; they are the ones whose Algebra 2 and trigonometry are automatic, so the one line of calculus in each problem is the only thing they have to think about. For them the first course has two ideas (the derivative and the integral), each with a handful of rules, and the applications repeat. A student who says "calculus is easy" is usually describing their algebra.
The students who find it hard are the ones doing two courses at once: the calculus, and the algebra they never made automatic. That is why the readiness check above matters more than any amount of worrying about limits, and why a month of algebra before the course is the single thing that makes calculus easy.
A ten-question readiness check
No calculator, fifteen minutes. These are the algebra skills a first calculus course uses every day.
- Factor x² − 9
- Factor x² − 5x + 6
- Simplify (x² − 4) / (x − 2)
- Write 1/x³ and √x as powers of x
- Expand (x + h)²
- Simplify (1/x) − (1/(x + h)) into one fraction
- Solve x² − 3x − 10 = 0
- Find the slope of the line through (1, 2) and (4, 11)
- What is sin 30° and cos 60°?
- If f(x) = x² + 1, find f(3) and f(a + 1)
Answers: 1) (x − 3)(x + 3) 2) (x − 2)(x − 3) 3) x + 2 4) x⁻³ and x to the power ½ 5) x² + 2xh + h² 6) h / (x(x + h)) 7) x = 5 or x = −2 8) 3 9) ½ and ½ 10) 10 and a² + 2a + 2.
Nine or ten right, quickly: you are ready. Six to eight: a month of algebra first will save you a term. Five or fewer: the problem to fix is in Algebra 2, and why am I bad at algebra shows how to find it.
Algebra-based and calculus-based physics
The same split appears in physics courses, and students choosing between them ask the same question.
| Algebra-based physics | Calculus-based physics | |
|---|---|---|
| US courses | AP Physics 1 and 2; most high school physics | AP Physics C: Mechanics, and Electricity and Magnetism; university physics for engineers |
| Mathematics used | Algebra and trigonometry; formulas for constant acceleration | Derivatives and integrals; forces and speeds that vary continuously |
| Who needs it | Life sciences, pre-med, general requirements | Engineering, physics, most physical science degrees |
| Taken alongside | Algebra 2 or Pre-Calculus | Calculus, in the same year or after it |
The physics is the same. Velocity is the derivative of position in both; one course says so and the other gives you the formula that results.
What calculus is, briefly
If the three ideas are new to you, what is basic calculus explains limits, derivatives and integrals with a worked example of each.
Download the algebra vs calculus chart and the ten-question readiness check (PDF) (the side-by-side table, the problem solved both ways, the ten questions with answers and how to read the score; two pages, free).
Where to get help
A 1-on-1 online class with a tutor who holds a master's in mathematics and teaches Algebra 2, Pre-Calculus and Calculus every week: the readiness check in the first class, the algebra gaps closed first, then the course itself with every line of working read. The free 30-minute first class is a real lesson on the problem you bring.
Questions parents ask
1What is the difference between algebra and calculus?
2Is calculus harder than algebra?
3Is calculus harder than Algebra 2?
4Do you need algebra for calculus?
5What is the difference between algebra-based and calculus-based physics?
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