Ask HN: What maths are critical to pursuing ML/AI?
Part I<p>(1) Calculus<p>Generally should have college freshman and
sophomore calculus.<p>(1.1) Functions<p>So, there can understand better what a
<i>function</i> is.  E.g., function<p><pre><code>     f(x) = 3x^2 + 1.
</code></pre>
(1.2) Derivatives<p>Then will learn how to find the slope of
the graph of a function.  That is the
<i>derivative</i> of the function.  E.g., for
function f with f(x) = 3x + 2, as in high
school algebra, the slope is 3. Then for
each x, the derivative of f at x is just
3.<p>The derivative of function f is denoted by
either of<p><pre><code>     f&#x27;(x) = d&#x2F;dx f(x)
</code></pre>
E.g., for function f(x) = 3x^2 + 1 it turns
out that<p><pre><code>     f&#x27;(x) = 6x.
</code></pre>
(1.3) Integration<p>For function<p><pre><code>     g(x) = 6 x
</code></pre>
maybe we want to know what function f(x)
will give us<p><pre><code>     f&#x27;(x) = g(x)
</code></pre>
Finding such a function f is
<i>anti-differentiation</i>, that is, undoes
differentiation.  So, sure,<p><pre><code>     f(x) = 3x^2 + C
</code></pre>
for any constant C.<p>Such anti-differentiation is also the way
to find the area under a curve.  So, can
use that to find the area of a circle,
volume of a cylinder, etc.  Doing that the
anti-differentiation is <i>integration</i>.<p>The fundamental theorem of calculus shows
how differentiation and integration are
related.<p>(1.4) Analytic Geometry<p>Commonly taught at the beginning of a
calculus course is <i>analytic</i> geometry.<p>So, take a cone an cut it.  Then the cut
surfaces will be one of a circle, an
ellipse, a parabola, a hyperbola, or just
two crossed straight lines.  So, those
curves are from a cone and are the <i>conic
sections</i>.<p>There is some simple associated algebra.<p>Conic sections are important off and on;
e.g., applied math is awash in circles;
the planets move in ellipses; a baseball
moves in a parabola or nearly so; an
electron moving toward a negative charge
will turn away from that charge in a
hyperbola.<p>It turns out that in linear algebra
(below) circles and ellipses are important.<p>(1.5) Role of Calculus<p>Calculus was invented by Newton as part of
working with force and acceleration for
understanding the motion of the planets.<p>E.g., if at time t function d(t) gives
distance traveled, then function v(t) =
d&#x27;(t) is the velocity at time t and
function a(t) = v&#x27;(t) is the acceleration
at time t.<p>Then Newton&#x27;s second law is<p><pre><code>     F(t) = m a(t)
</code></pre>
where F(t) is the force at time t applied
to mass m.<p>Calculus is the first approach to the
analysis of continuous change and is a
pillar of civilization.<p>Knowledge of calculus will commonly be
assumed in work in ML&#x2F;AL, data science,
statistics, optimization, applied math,
engineering, etc.<p>E.g., a lot in ML, AI, and data science is
getting best fits to data; best fitting is
to minimize errors in the fit; such
minimization is mostly a calculus problem;
one of the main steps in ML is steepest
descent, and that is from a derivative.<p>Probability theory (e.g., evaluating coin
tossing, poker hands, accuracy in ML) will
be important in ML&#x2F;AI, etc.; two of the
basic notions in probability are
cumulative distributions and density
distributions; the cumulative is from an
integration, and the density is from a
differentiation.