Mathematics. Odds and ends
ed⁄dx f (x) = f (x+1)
This page isn’t finished,
and probably never will be.
The grammar of mathematical operations has been partly covered in the tutorial; some links to the relevant places are given below. Notation is a different matter, as it is internationally standardised.
- General notation
- Polish notation (prefix notation: ÷ + 11 7 √ 9 ‘divide the sum of 11 and 7 by the square root of 9’) would be the logical thing to do from a Lemizh viewpoint, but not from an international one; so there is no good reason why it should be the standard. Reverse Polish notation (postfix notation: 11 7 + 9 √ ÷ ‘take 11 and 7 and add them up, take 9 and calculate its square root, then divide’) is the most practical method for pocket calculators. The negation sign _ would have to follow negative numbers instead of preceding them, though. In short, the evidence is inconclusive.
- Basic operations
- Addition is expressed with a partitive ‘and’. The tentative addition symbol + is either derived from the partitive case suffix n or else from Greek τ (τε ‘and’), or it is a compromise between them.
- Subtraction is expressed with a partitive ‘and’ and a negative number. The subtraction symbol is tentatively !.
- Values are multiplied using a consecutive case. There’s much to be said for multiplication without a symbol, just by stringing things together: 2Á x ‘2 × 65536 × 22 mm’. The explicit multiplication symbol is *.
The way Lemizh grammar represents multiplication with an asymmetric structure – one factor has a consecutive, the other an accusative – corresponds to the fact that multiplication is not necessarily commutative. When multiplying, say, matrices or quaternions, the result generally depends on which factor has the consecutive.
- Division and fractions are expressed like multiplication together with the verb làgz. ‘calculate the reciprocal’; see Fractions in the tutorial. They are written by putting the divisor below the dividend. Again, this is the same that we do – and similar to what the Egyptians did –, but it’s kind of natural. The horizontal dividing bar is optional. The explicit division symbol is / and symbolises moving the object in front of it on top of the object behind it.
The ‘backwards division’ symbol, needed for non-commutative objects, is its mirror image \.
- Exponentiation and its inverse operations
- The grammar of exponentiation is explained here: the exponent is in the nominative, the base in the dative, and the result in the accusative. Regarding notation, exponentiation is written with the exponent as superscript to the left of the base, as we have seen on the previous page. This aligns nicely with the Lemizh plot.
- Roots are expressed with the same verb as exponentiation, having the index (e.g. 2 for square roots) in the nominative, the radicand in the accusative and the result in the dative. They are probably written with the index as subscript to the left. In combination with the way division is written, this means that a superscript and a subscript to the left of some value can equivalently be read as an exponent and a root, or as a fractional exponent: ∛8̅² = 82⁄3 is 2
38 in this notation. On the downside, this is inconsistant with subtraction and division, where the ‘accusative’ value (minuend, dividend and radicand, respectively) is on the left.
- Accordingly, logarithms have the argument in the accusative, the base in the dative and the result in the nominative. Notation is an open question.
- Trigonometric and hyperbolic functions
- Verbs for trigonometric functions are compounds from dà. and spatial verbs; the argument is in the dative and the result in the accusative. Their symbols are derived from Greek letters: α (from ἄνω ‘up’) for the sine, π (from πρόσω ‘forward’) for the cosine, and μ (from μακρῶς ‘long, far’) for the tangent; each with a line extending under the argument.
- Their inverse functions are expressed of course with the same verbs but reversed cases. They are symbolised by the same letters, but with lines extending over the argument.
- Verbs for hyperbolic functions and their inverses are compounds with xprà.. Their symbols have a ῾ above them (a Greek /h/ for ὑπερβολή ‘hyperbola’).
- Constants and variables
- The convention is to write constants or known values in Greek letters, and variables or unknown values in Waldaiic letters.
- The circle constant in the Lemizh world is 2π, as it should be, and is written τ. It is named for τόρνος ‘lathe, compass (drawing tool)’ and Τερψιχόρη, the muse of dancing.
- e, the base of the natural logarithm, is called ε. It is named after Εὐτέρπη, the muse of music, because of the logarithmic nature of musical scales.
- Having been developed in northern Europe, differentiation and integration probably have symbols derived from Waldaiic letters.
- Additional mathematical terms in the dictionary: lÙr. ‘dimension’, dnà. ‘vector’, krÌj. ‘group’, pxlÌj. ‘plane’, xàxs. ‘curl’, qmÌ. ‘set’.