Point calculation before line calculation

The rule of point calculation before line calculation , also called point before line for short , is a convention in the operator precedence of mathematics . It says that in a mathematical expression, unless there are parentheses , multiplications and divisions must be carried out before additions and subtractions . This convention makes it possible in many cases to dispense with parentheses, which improves the legibility of the expressions.

The generic term "point calculation" for multiplication and division refers to the individual point as a multiplication symbol and the colon as a division symbol. But even if other symbols such as or or are used that are typographically not pure "point symbols ", they are considered to be point calculations in the sense of the rule. ${\ displaystyle \ cdot}$ ${\ displaystyle:}$ ${\ displaystyle \ times}$ ${\ displaystyle /}$ ${\ displaystyle \ div}$

Examples

The dot-before-line rule forbids simply calculating expressions in which multiplication / division and addition / subtraction occur in a mixed manner from left to right. In the following examples, the correct result (rule is observed) and the wrong result (without observing the rule, calculation is carried out step by step from the left) are given:

Expression	invoice	correct result	wrong result
${\ displaystyle 1 + 2 \ cdot 3}$	First calculate the multiplication , which gives the total expression . ${\ displaystyle 2 \ cdot 3}$ ${\ displaystyle 1 + 6}$	${\ displaystyle 7}$	${\ displaystyle 9}$
${\ displaystyle 2 \ cdot 12 + 6: 3-7}$	Here the multiplication and division are to be calculated first (in any order) , which results in the total expression , which is then calculated from left to right (only line calculation). ${\ displaystyle 2 \ cdot 12}$ ${\ displaystyle 6: 3}$ ${\ displaystyle 24 + 2-7}$	${\ displaystyle 19}$	${\ displaystyle 3}$
${\ displaystyle 36- (4 + 7) \ cdot 3}$	Parentheses are always calculated first, i.e. the partial expression here . Overall, this gives the expression in which the multiplication is to be calculated next and the resulting subtraction last . ${\ displaystyle (4 + 7)}$ ${\ displaystyle 36-11 \ cdot 3}$ ${\ displaystyle 11 \ cdot 3}$ ${\ displaystyle 36-33}$ Alternatively you can also out the clip multiply out what about (stay put the bracket by 12 + 21 must, because this amount is the result of the multiplication) is also the right result leads. ${\ displaystyle 36- (12 + 21)}$	${\ displaystyle 3}$	With brackets: Without brackets: ${\ displaystyle 75}$ ${\ displaystyle 117}$

History of the convention

The mathematicians of antiquity and the Middle Ages formulated their findings as a linguistic text, whereby usually no misunderstandings about the groupings occur. The first of the above examples could then read: “The product of 2 and 3 is added to 1”, while the reverse grouping would be formulated as “The sum of 1 and 2 is multiplied by 3”.

Only in the modern age did the shorter formulaic representation of mathematical facts with numbers, identifiers and operators develop. The rule “dot before line” seems to have been assumed from the start. With René Descartes there are notations such as , which, as is still common today, simply omit the multiplication operator ( juxtaposition ) and assume that multiplication has priority over addition. ${\ displaystyle zz = az + bb}$

Further priority rules

Powers have priority over point calculation:

{\ displaystyle 4 \ cdot 2 ^ {3} = 4 \ cdot (2 ^ {3}) = 4 \ cdot 8 = 32}

The sides of a fraction line and the "bar" of the root sign are viewed as brackets:

{\ displaystyle 7 \ cdot {\ frac {5 + 3} {6-2}} = 7 \ cdot (5 + 3) :( 6-2) = 7 \ cdot 8: 4 = 14}

{\ displaystyle {\ sqrt {4 + 5}} \ cdot 7 = {\ sqrt {9}} \ cdot 7 = 3 \ cdot 7 = 21}

Technical implementation

Spreadsheet programs , programming languages and computer algebra systems naturally observe the dot-before-line rule, for example the open source system Maxima :

(%i1) 1+2*3;
(%o1)                                  7
(%i2) 2*12+6/3-7;
(%o2)                                 19
(%i3) 36-(4+7)*3;
(%o3)                                  3

Better pocket calculators have also long been paying attention to them; The TI-30 school computer , published in 1976, stood out among other things from competing models.

On the other hand, there were and still are computers that simply calculate an intermediate result for every new operator typed in, without taking into account that a higher-ranking operation could follow. This leads to incorrect results, for example in the case of the second example:

${\ displaystyle 2 \ cdot 12 = 24; 24 + 6 = 30; 30: 3 = 10; 10-7 = 3}$

Some computers offer the option of switching between the calculation modes "algebraic" (operator precedence is observed) and "sequential" (operations are carried out in the order in which they are entered).

The calculator app integrated in Windows 10 observes the dot-before-line rule in the "Scientific" mode. In the "Standard" mode, calculations are carried out sequentially, as in all Windows versions before.

Individual evidence

↑ René Descartes: La Géométrie , p. 302

[1] René Descartes: La Géométrie , p. 302