Astronomy Answers: Mathematical Notation

# Astronomy AnswersMathematical Notation

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$$\DeclareMathOperator{\arsinh}{arsinh} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\artanh}{artanh} \DeclareMathOperator{\sgn}{sgn}$$

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.

• Variables usually have names that are made up of a single letter, which may come from the Latin (ordinary) or Greek alphabets. For example, the variable with the name "x" is written 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).
• Characteristics or indices in mathematical formulas are usually written a bit smaller and half a line lower, directly after the name of the variable. The second element of array a or $$a$$ is displayed thus: a₂ or $$a_2$$. The element with index i or $$i$$ is ai or $$a_i$$. The next element is a(i+1) or $$a_{i+1}$$.
• Powers are usually written a bit smaller and half a line higher, directly after the name of the variable. The third power of x or $$x$$ is x³ or $$x^3$$. The n-th or $$n$$-th power of x or $$x$$ is xn or $$x^n$$, and the one greater power is x(n+1) or $$x^{n + 1}$$.
• If a variable is a vector, then this is sometimes shown by writing the variable in a bold font, and sometimes by writing a right-pointing arrow above the variable. In these pages we try to put an arrow above the variable (but many browsers show that arrow not right above but instead in front of or behind the variable). The vector with name x is shown as x⃑ of $$\vec{x}$$.
• Multiplication is often not explicitly marked. Two variables immediately adjacent to each other are multiplied. Sometimes, multiplications are explicitly indicated, with a * 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)nsummation for [i] from 1 through [n]
√2the 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
πratio 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:

$$y = x + 3 + \text{ smaller terms}$$

• 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 × 105 = 1.2 * 105 = 120,000 or $$1.2×10^{5}$$.

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Last updated: 2017-12-28