Aregnometry

Aregnometry is a process used for solving mathematical equations using five basic operations: summation (sum), afertation (aft), iteration summation (itersum), iteration afertation (iteraft), and circation (circ), and one special process called anti-sign (anti).

= History = Aregnometry was first described and used by the Eurik mathematician Wilhelm Gotthard in the Second Era. In his textbook, A Guide to the Five Operators, Gotthard provides many examples of and practice problems using his newly created five operators (described above).

Anti-sign was formally described later in the Second Era by Pordician mathematician and alchemist Borin Halfeks. This is why the process is sometimes referred to as the "Halfeks Operator" despite it not being a true operator.

In the Third Era, Romasti mathematician Norme Atticus expanded on aregnometry through a process he called signometry where symbols other than numerals were used in equations.

= Five Operators =

Summation
Summation is the process of combining two numerical values (called summators). When writing equations, it is customary to use the symbol 's' or '+' to show summation is occurring.

For example, 1s2 is equal to 3. 2s4s5 is equal to 2s4 s 5, 6s5, which is 11.

Itersum
Itersum is the process of repeated summation. The value before the itersum is the base and the value after the itersum is the iterator. The process of itersum is repeatedly adding the base to itself an iterator number of times. The customary symbol for itersum is 'z', 'ss' (also incorrectly called "super-sum"), or most commonly '· '.

For example, 1z2 is equal to 1s1 which is 2. 5z6 is equal to 5s5s5s5s5s5 which is 30.

Iteritersum
Iteritersum, also referred to as Duplesum, is the process of repeated itersum using a single value. Duplesum shows that a base is itersummed with itself a 'duplex' number of times. Duplesum is usually denoted with the value 'i'.

For example, 4i3 is equal to 4z4z4 which is equal to (4s4s4s4)s(4s4s4s4)s(4s4s4s4)s(4s4s4s4) which is 64.

Iteritersum can also be performed using a duplex of a non-whole value. This relies on more complex mathematics and is traditionally reserved for table-aided computations.

When the process of iteritersum is used on the duplex of another iteritersum, the duplex of the duplex is called a triplex. The duplex of a triplex is called a quadraplex. This naming scheme goes on as one continues to perform iteritersum on duplex's of other iteritersums. The naming scheme follows the Plex Tree.

Afertation
Afertation is the anti-process of summation where rather than combining the summators, the second summator (also known as the afertator) is removed from the first. This can also be explained as summation of the first summator and the anti-sign of the second summator. The symbol for afertation is traditionally 'a' or '-'.

For example, 5a3 is equal to 2. 5a3 is also equal to 5s('3) or five summation anti-three. Similarly, 3a5 would yield '2 (anti-two). This is because when one removes 5 from 3, there are still 2 removals that need to occur. Thus, the removals end up becoming their own value, an anti-value (an anti-signed value).

Iteration Afertation
Perhaps the most complex of Gotthard's five operators is iteration afertation, also referred to as dimication for this reason (dimication deriving from Old-Romasti dimicus meaning "struggle"). Dimication is most correctly described as the process of how many times the dimicator (the value after the dimication symbol) goes into the base (the value before the dimication symbol). It is most common for mathematicians to use 'd', '=', or rarely 'zz' when writing the operation.

Truncated Dimication
Truncated Dimication is a process of performing dimication followed by a down circation (see Down Circation). This process yields numbers that are significantly cleaner (having no partials, see Partial Dimication); yet, leaving values that are inaccurate.

Clean Dimication
Clean dimication occurs when the dimicator goes into the base a whole number of times. This process yields no partial and is usually taught first to young children.

For example, 4d2 equals 2. 9d3 equals 3.

Partial Dimication
Partial dimication is the most frequently used form of dimication. The process occurs when the dimicator does not go into the base a whole number of times. The result is what is known as a partial. Partials are values that exist between the number zero and the number one and are tagged on after a whole number to show their existence. Partials are more thoroughly explained in the section labeled Partials.

For example, 5d2 equals 2 and a half, often written simply as 2:5. 7d3 equals 2:333... followed by an infinite number of threes.

Remainders
Instead of writing the answer with partials, one may choose to use remainders. A remainder shows what is left after the dimicator dimicates the base as many whole times as it can.

For example, 5d2 yields a remainder of 1 because 2 goes in twice to 5 and can no longer go into that 1 a whole number of times. 7d4 would yield a remainder of 3 because 4 can go into 7 one time with 3 left over.

Circation
Circation is the process of replacing a number with the closest whole value (a number lacking any form of partial) to it. Circation is shown by surrounding a value with vertical lines "|".

For example, |3:5| is equal to 3. |4.92524| is equal to 5. The circation of a whole value yields the same whole value, for example, |5| is still 5.

Down Circation
Down Circation is a process shown by surrounding a value with down arrows ('↓'). Down Circation circates the value to the closest whole value below itself.

For example, ↓3:2↓ is equal to 3.

Up Circation
Up Circation is the opposite of down circation and is denoted with up arrows ('↑'). The process replaces the original value with the closest whole value above itself.

For example, ↑9:12↑ would be 10.

= Other Processes and Symbols =

Anti-Sign
Anti-Sign, also known more formally as Anti-Signature is the process of showing a value is a lack of a number. This is used to denote values that are less than zero. Anti-sign is usually shown by drawing a straight line above the value on wishes to anti-sign or by placing an apostrophe (') before the value. Anti-Signed numbers can also be the result of performing afertation where the afertator is greater in magnitude than the summator left of the symbol. Summation of two anti-signed values or one anti-signed value and a normal-sign value can also occasionally yield an anti-sign sum.

The process of antisigning an anti-signed value yields a normal-sign value.

Partials
Partials are the method of showing the existence of a value between zero and one that is attached to a whole value. Partials are usually denoted by numbers following a colon (':'), also known as the partial symbol.

Perpetual Partials
Perpetual Partials are partials that never cease. These are usually the values that occur when one performs dimication with a dimicator that does not go into the base a whole number of times. Perpetual partials are usually shown by placing an ellipses after a partial. For example, 3:333... shows that there should be an infinite number of threes after the partial symbol.

Residio
Residio, also known as Residual Dimication, is the process of showing the remainder after performing dimication. Residio is shown with the symbol 'r'.

For example, 5r2 would be 1 because 2 dimicates into 5 twice with 1 left over.

Medius
Medius was a process commonly used by Gotthard and is prevalent in the field of statistics. Medius yields the average value given a list of values. The process occurs by summing up all the given values, then dimicating by the number of values. Medius is often shown by placing a thin 'M' before a list of numbers.

For example, M[1,4,5,11,15] yields a value of 7:2 because (1s4s5s11s15)d5 is 7:2.

Magnitude
Magnitude is a process used to show the distance between a number and zero. The process is often denoted with the symbols surrounding the value one wants to find the magnitude of.

For example, <5> is the same as <'5> because both have a distance of 5 from zero.

Equals
Equals, Equals to, or Yields, is a symbol used to show that a previously described process provides the following result. Equals is usually written with the right arrow symbol ('→').