Foundations

Numbers, functions and the families of functions that the rest of calculus works with — and how a computer represents them.

16 topics

Topics

Real numbers

The number line $\R$ : the rationals plus the limits of all their convergent sequences. Its defining property is completeness — every non-empty set bounded above has a least upper bound — and calculus depends on it.

Fundamental

Complex numbers

Numbers $z = a + bi$ with $i^2 = -1$ . Geometrically they are points of the plane, and multiplying by $e^{i\theta}$ is a rotation — which is why they are the natural language of oscillations, waves and signals.

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Intervals

Connected pieces of the real line: $[a,b]$ , $(a,b)$ , $[a,\infty)$ … Closed and bounded intervals are where the big theorems of calculus (Bolzano, Weierstrass) hold.

Fundamental

Absolute value

$|x|$ is the distance from $x$ to $0$ , and $|x - y|$ the distance between two numbers. "Close to" in every definition of calculus means a small absolute value.

Fundamental

Inequalities

Calculus is mostly the art of bounding: proving that an error is smaller than $\varepsilon$ , that a term is negligible, that an algorithm takes at most so many steps. The tools are the triangle inequality, AM–GM, Cauchy–Schwarz and Jensen.

Fundamental

Functions

A rule $f$ that assigns to each input $x$ of a set $A$ exactly one output $f(x)$ in a set $B$ . Calculus studies how outputs change when inputs change; computing is, quite literally, evaluating functions.

Fundamental

Domain and range

Where a function is defined and which values it actually takes. Getting them wrong is the mathematical version of a runtime error: $\sqrt{-1}$ or $\log 0$ have no real value.

Fundamental

Composition

Applying one function after another: $(g \circ f)(x) = g(f(x))$ . Deep networks, compilers and data pipelines are long compositions, and the chain rule tells how to differentiate them.

Fundamental

Inverse functions

$f^{-1}$ undoes $f$ : $f^{-1}(f(x)) = x$ . It exists when $f$ is one-to-one; its graph is the mirror image of $f$ 's across the line $y = x$ .

Fundamental

Polynomial functions

$p(x) = a_n x^n + \dots + a_1 x + a_0$ . Computed with additions and multiplications only, they are what a processor evaluates best — and other functions are approximated by them (Taylor, interpolation).

Fundamental

Rational functions

Quotients of polynomials $p(x)/q(x)$ . They blow up at the zeros of $q$ (poles) and are the exact form of the transfer functions of linear filters and control systems.

Fundamental

Exponential functions

$a^x$ , and above all $e^x$ : the only function (up to a factor) that is its own derivative. It models anything whose rate of change is proportional to its size — populations, interest, radioactive decay, and exponential algorithms.

Fundamental

Logarithmic functions

The inverse of the exponential: $\log_a x$ is the power you raise $a$ to in order to get $x$ . Logarithms turn products into sums, which is why they appear in algorithm costs, information theory and every loss function based on likelihood.

Fundamental

Trigonometric functions

$\sin$ and $\cos$ are the coordinates of a point turning around the unit circle. They describe every rotation and every oscillation, and therefore graphics, robotics, sound and signals.

Fundamental

Hyperbolic functions

$\sinh x = \frac{e^x - e^{-x}}{2}$ , $\cosh x = \frac{e^x + e^{-x}}{2}$ , $\tanh = \sinh/\cosh$ . They play for the hyperbola $x^2 - y^2 = 1$ the role $\sin$ and $\cos$ play for the circle; $\tanh$ is a classic activation function.

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Transformations of functions

$a\,f(bx + c) + d$ : shifting, stretching and reflecting a graph. A delay in a signal, a zoom in an image or a change of units are all such transformations.

Fundamental

Where this area leads in computing

Topics

Real numbers

Complex numbers

Intervals

Absolute value

Inequalities

Functions

Domain and range

Composition

Inverse functions

Polynomial functions

Rational functions

Exponential functions

Logarithmic functions

Trigonometric functions

Hyperbolic functions

Transformations of functions

Where this area leads in computing

λ Scientific computing and algorithms ★★★★★

ℒ AI and machine learning ★★★★★

3D Computer graphics ★★★★★

⚛ Physics and simulation ★★★★★

∿ Signals, media and vision ★★★★★

ψ Quantum computing and physics ★★★★★

⚙ Robotics and control ★★★★★

⊕ Cryptography and security ★★★★★