| Level 1: The basic
operations and grouping symbols. |
- Addition: $a+b$
- Subtraction: $a-b$
- Opposite: $-b$ (will be surcharged)
- Multiplication: $a \times b$
- Division: $\dfrac{a}{b}$
- (Parentheses)
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| Level 2: Common operations
involving place value. |
- The decimal point: $.a$ (2.3 will not be surcharged, but .3 will)
- The bar for a repeating decimal: $.\overline{a}$ (will be surcharged)
- The percent sign: $a\%$ (will be surcharged, equivalent to $0.01a$)
- Juxtaposition: $ab$ (equivalent to $10a+b$)
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| Level 3: Exponents,
radicals, factorials, and per mille. |
- Exponents: $a^b$
- Radicals: $\sqrt{a}$, $\sqrt[b]{a}$ (if the index is assumed, it will be surcharged)
- Factorials: $a!$ (will be surcharged)
- Per mille: $a ‰$ (will be surcharged, equivalent to $0.001a$)
|
| Level 4: Basic algebraic
functions. (All will be surcharged except as noted.) |
- Logarithms: $\log_b a$ (will not be surcharged)
- Specific Logarithms: $\log a$, $\ln a$ (logs base 10 and $e$, respectively)
- Exponentials: $\operatorname{antilog} a$, $\exp a$ (equivalent to $10^a$ and $e^a$, but $10$ and $e$ are constants, and therefore prohibited)
- Trigonometrics: $\sin a$, $\cos a$, $\tan a$, $\cot a$, $\sec a$, $\csc a$ (where $a$ is assumed to be in radians, but if degrees are desired, the degree symbol will receive a second surcharge)
- Inverse trigonometrics: $\arcsin a$, $\arccos a$, $\arctan a$, $\operatorname{arccot} a$, $\operatorname{arcsec} a$, $\operatorname{arccsc} a$ (their values are assumed to be in radians, but if degrees are desired, then you need to divide by a quantity in degrees to remove the unit of degrees)
- Hyperbolics: $\sinh a$, $\cosh a$, $\tanh a$, $\coth a$, $\operatorname{sech} a$, $\operatorname{csch} a$
- Inverse hyperbolics: $\operatorname{arsinh} a$, $\operatorname{arcosh} a$, $\operatorname{artanh} a$, $\operatorname{arcoth} a$, $\operatorname{arsech} a$, $\operatorname{arcsch} a$
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| Level 5: Some operations related to factorials. |
- Binomial coefficients: ${}_a C_b$
- Permutations: ${}_a P_b$
- Double factorial: $a!! = a \times (a-2) \times (a-4) \times ... \times n$,
where $n$ is 1 if $a$ is odd, and $c$ is 2 if $a$ is even. (will be surcharged)
- Triple factorial: $a!!! = a \times (a-3) \times (a-6) \times ... \times n$,
where $n$ is 1, 2, or 3. (will be surcharged)
- Gamma: $\Gamma(a)$. When $a$ is a positive integer, then $\Gamma(a) = (a - 1)!$ (will be surcharged)
|
| Level 6: Some other basic
math functions. |
- Sequence of prime numbers: $p_a$ (the $a$th prime number, will be surcharged)
- Fibonacci sequence: $f_a$ (the $a$th term of the Fibonacci sequence, will be surcharged)
- Floor function: $\lfloor a \rfloor$. This is equivalent to integer division, and to the greatest integer function. (will be surcharged)
- Ceiling function: $\lceil a \rceil$ (will be surcharged)
- Division modulo $b$: $a \mod b$ (the remainder after dividing $a$ by $b$)
- Greatest common divisors and least common multiples: $\gcd(a,b)$, $\operatorname{lcm}(a,b)$
- Summations: $\sum_a^b c$ (but an index variable is a variable, and therefore prohibited)
- Determinants: $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$
|
| Level 7: Other advanced
functions. (All will be surcharged, except as noted. Others are possible.) |
- Bernoulli numbers: $B_a$. Since several different indexing approaches are used, we use that which provides $B_a = 0$ when $a>2$ and odd, and the signs of $B_a$ alternate when $a$ is even.
- Change of base: $a_b =$ the number obtained by using the numeral $a$ in the base $b$ numeration system (Will not be surcharged.)
- Derivative: $a'$, the derivative of $a$. When $a$ is a number, then $a'=0$.
- Euler numbers: $E_a$. Since several different indexing approaches are used, we use that which provides $E_a = 0$ when $a$ is odd, and the signs of $E_a$ alternate when $a$ is even.
- Euler's totient function: $\varphi(a)$
- Grad: $a \text{ grad}$ (1 grad is a hundredth of a right angle,
hence $1 \text{ grad} = \dfrac{\pi}{200}$).
- Haversine function: $\operatorname{hav}(a)$
- Hyperfactorial sequence: $Hyperfact_a$
- Kronecker delta: $\delta_a^b$ (equal to 1 if $a = b$, 0 if $a \ne b$) (Will not be surcharged.)
- Lucas sequence: $L_a = L_{a-1} + L_{a-2}$, $L_1 = 1$, $L_2 = 3$
- Mersenne Prime sequence: $MP_a$
- Multifactorials other than double or triple: $a!!!!$, $a!!!!!$, etc.
- Number of divisors of a number: $d(a)$
- Perfect number sequence: $Perf_a$
- Prime Counting Function: $\pi(a)$, the number of primes at most $a$
- Sum of divisors of a number: $\sigma(a)$
- Triangular numbers: $T_a = {}_{a+1} C_2$
- Zeta function: $\zeta(a) = \sum\limits_{k=1}^{\infty} \dfrac{1}{k^a}$
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