\( \DeclareMathOperator{\arsinh}{arsinh} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\artanh}{artanh} \DeclareMathOperator{\sgn}{sgn} \)
World Wide Web browsers often cannot show many of the symbols that one often finds in mathematical formulas. A mathematical formula from these web pages can therefore look different in one browser than in another browser. Some examples are shown below, so you can see what certain mathematical constructions from these pages look like in your browser. If your browser cannot display many special cases, then it can be difficult to understand the formulas. You can of course try to use a different browser, or you can study the source code of the page.
I am switching to using MathJax (//www.mathjax.org), but not all formulas are in MathJax format yet.
x
or
\( x \). Some browsers can show Greek letters, but others cannot, and
those may then show Latin letters instead that hopefully look a bit
like the intended Greek letters. Here is a list of Greek letters as
your browser shows them, with for each letter first the capital form,
then the small form, and then the English name between parentheses: Α
α (alpha), Β β (beta), Γ γ (gamma), Δ δ (delta), Ε ε (epsilon), Ζ ζ
(zeta), Η η (eta), Θ θ (theta), Ι ι (iota), Κ κ (kappa), Λ λ (lambda),
Μ μ (mu), Ν ν (nu), Ο ο (omicron), Π π (pi), Ρ ρ (rho), Σ σ (sigma), Τ
τ (tau), Υ υ (upsilon), Ξ ξ (xi), Χ χ (chi), Ω ω (omega).a
or \( a
\) is displayed thus: a₂
or \( a_2 \). The element
with index i
or \( i \) is a_{i}
or \( a_i
\). The next element is a_{(i+1)}
or \( a_{i+1} \).x
or \( x \) is x³
or \( x^3 \). The
n
-th or \( n \)-th power of x
or \( x \)
is x^{n}
or \( x^n \), and the one greater power is
x^{(n+1)}
or \( x^{n + 1} \).x⃑
of \( \vec{x} \).*
or \(
× \). For example, v x
or \( vx \) and v
∗ x
or \( v×x \) both indicate the product of
x
or \( x \) and v
or \( v \).A number of special characters and notations:
Table 1: Mathematical Notation
notation | meaning |
---|---|
≡ | is equivalent to |
| | | absolute value |
∑_{(i=1)}^{n} | summation for [i] from 1 through [n] |
√2 | the square root of 2 |
≈ | approximately equal to |
< | less than |
> | greater than |
⌈ ⌉ | rounded up (to a whole number) |
⌊ ⌋ | rounded down (to a whole number) |
[ ] | rounded to the nearest whole number |
π | ratio of the circumference to the radius of a circle |
∞ | infinity; an infinitely large number |
Special functions are indicated by their full name or by their usual abbreviation (like those used on electronic calculators). I've tried to use a different style for these than for the names of variables. Some examples:
Table 2: Mathematical Functions
\({\min(x,y)}\) | min(x,y) | the least (closest to \(-\infty\)) of \(x\) and \(y\) |
\({\max(x,y)}\) | max(x,y) | the greatest (closest to \(+\infty\)) of \(x\) and \(y\) |
\({\sin(x)}\) | sin(x) | the sine of angle \(x\) |
\({\cos(x)}\) | cos(x) | the cosine of angle \(x\) |
\({\tan(x)}\) | tan(x) | the tangent of angle \(x\) |
\({\arcsin(x)}\) | arcsin(x) | the arc sine of \(x\) |
\({\arccos(x)}\) | arccos(x) | the arc cosine of \(x\) |
\({\arctan(x)}\) | arctan(x) | the arc tangent of \(x\) |
\({\arctan(y,x)}\) | arctan(y,x) | the angle between the \(x\)-axis and the line from \((0,0)\) to \((x,y)\) |
\({\exp(x)}\) | exp(x) | \({e\) to the power \(x\); the natural antilogarithm of \(x}\) |
\({\log(x)}\) | log(x) | the decimal (base 10) logarithm of \(x\) |
\({\ln(x)}\) | ln(x) | the natural (base \(e\)) logarithm of \(x\) |
\({\sinh(x)}\) | sinh(x) | the hyperbolic sine of \(x\) |
\({\cosh(x)}\) | cosh(x) | the hyperbolic cosine of \(x\) |
\({\tanh(x)}\) | tanh(x) | the hyperbolic tangent of \(x\) |
\({\arsinh(x)}\) | arsinh(x) | the hyperbolic arc sine of \(x\) |
\({\arcosh(x)}\) | arcosh(x) | the hyperbolic arc cosine of \(x\) |
\({\artanh(x)}\) | artanh(x) | the hyperbolic arc tangent of \(x\) |
\({\sgn(x)}\) | sgn(x) | the sign of \(x\): −1 for \(x \lt 0\), 0 for \(x = 0\), +1 for \(x \gt 0\) |
For the arctan function with one argument and the one with two arguments we have \( \tan\left( \arctan \left( \frac{y}{x} \right) \right) = \tan(\arctan(y,x)) \), but \( \arctan\left( \frac{y}{x} \right) \) may differ by a multiple of 180° from \( \arctan(y,x) \).
Words that are not names of variables or of special functions are written just like the names of special functions, but only if the formula is displayed as a separate equation (with an equation number). For example, "smaller terms" shows up like this:
(Eq. 1)
y = x + 3 + smaller terms
or
\begin{equation} y = x + 3 + \text{ smaller terms} \end{equation}
In science, very large and very small numbers are often written in
exponential notation. In these pages we use the calculator or
computer notation with an "e" to introduce the power of 10:
1.2 × 10^{5} = 1.2 * 10^{5} = 120,000
or \( 1.2×10^{5} \).
//aa.quae.nl/en/reken/notatie.html;
Last updated: 2017-04-24