Examples... of the Simplified

Notation of Mathematical
Formulae

3,1 ab аЪ\с or abc 1

[a line must neither c

begin nor end with
the solidus]

3,2 CL±1 + d {a + b)\c + d or (a + b) c"1 + d

3,3 —^— aj{b + e) ór a(b + c)_1

/) -I с

3,4 í±± (a + b)l(c-d) or (a + b){c-dyl

с — d

- c a + b (a + b)jc + x

3,5 ----------h x------------------

_C______ b - У

• b-y

36 «J^ [(e + b)/(z + x)]:(i + y)

3,7

3,8

Z + X

b + у

e2 г

eia+*b or eila+b)

Nt

Nt ,N2

Q + v

Q + v Q

N2

Q

128

4 BRACKETS

33

There are six types of brackets: () parentheses, [] brackets, { } braces, | | vertical rules,
] double brackets, and < > angular brackets. The last three types are applied for special
purposes; vertical rules enclose absolute values and angular brackets, even when not
used in pairs, mark closed intervals, e.g. < ), ( > (In England, reversed parentheses are
sometimes used for this function, the reversion taking place at the open end of the
interval — ((, )), ) ( ). They can also be applied as brackets of the "first degree" within
the sequence {[( )]} which has been established as a standard convention in this country.

Double brackets, owing to their conspicuousness, can be successfully used for bracketing
analogical expressions at both sides of one equation or in a system of equations. An¬
gular and double brackets can be used to advantage by the author when correcting the
proofs, if he discovers, for example, that it is necessary to bracket within the sequence
of brackets some other expression whose involution was erroneously omitted. He does
not therefore need to rewrite all the brackets of the higher order.

The mathematical printer should have all the above-mentioned types of brackets at his
disposal with type faces of 12, 26, 38 and 50 points. For the rules, it will, of course, be
sufficient to be cut to appropriate lengths.

Brackets of a deeper body must be fitted together by hand. If, however, the author uses
the typographically simplified notation (see Sect. 3.), it will hardly ever be necessary to
take this step.

The usual order of brackets {[( )]} is not absolutely obligatory for the author. The printer
must not cling to their stereotyped application and order. In various fields of mathematics
it is customary to use certain types of brackets; thus e.g. the "higher" brackets in the
sequence are turned into "lower" ones, or square brackets are applied to emphasize
analogical, homogeneous expressions, even if, from the standpoint of the sequence, this
is not necessary in the particular case in the context, since the expression does not
contain parentheses:

{G'ilbí и G'itb2 и ...} , {aiß'ti и a2G'u и...}.- G'lt,

wi,x = к2 —, wa,x - blk0a2 Ci (cosh-----1 ) + C2 sinh - ,

b20 LI«/ «J

Tl,x — K2

'2 Г

a т0іХ - b\k0a

'o L

Ct sinh —HC, cosh -

;]

Щх)ах, №bx...dx

129