# Mathematical formula

A mathematical formula represents a relationship between mathematical or z. B. physical or economic variables. It uses the form of an equation (rarely an inequality ) and is shorter and often clearer than the text form. It stands for a law, rule, regulation or definition.

The sizes are represented by formula symbols. This applies both to variables and to specially defined constants , e.g. B. the circle number . ${\ displaystyle \ pi}$

If the formula contains at least two variable quantities, it describes the interdependence between these quantities. If the sizes are fixed or known except for one, one speaks of a calculation formula for this one size; it is also determined by the formula.

The relationship between the variables is through their symbols, numbers and mathematical symbols shown, for example, equal sign , plus sign , integral sign or brackets . The graphic arrangement can also contain a calculation rule, for example for exponentiation .

In mathematics, the term “formula” is sometimes seen as slang, because it is only used to describe the actually intended (teaching) sentence or, more generally, for an equation or inequality. With these more precise terms, a technical distinction is achieved from the other meanings of " formula ".

## Well-known examples of formulas

• Pythagorean theorem :${\ displaystyle a ^ {2} + b ^ {2} = c ^ {2} \,}$
• pq formula :${\ displaystyle x_ {1,2} \; = \; - {\ frac {p} {2}} \ pm {\ sqrt {\ left ({\ frac {p} {2}} \ right) ^ {2 } -q}}}$
• Energy-mass relationship by Albert Einstein :${\ displaystyle E = mc ^ {2} \,}$
• Euler's identity :${\ displaystyle \ mathrm {e ^ {i \, \ pi}} = -1}$
• Gaussian empirical formula :${\ displaystyle 1 + 2 + \ ldots + n = {\ frac {n (n + 1)} {2}}}$

## Derivation of formulas

There are different ways to get a formula. In this context, one speaks of “deriving a formula”. The possibilities include:

• Derivation from other already existing formulas,
• Derivation from basic, non-derived assumptions, so-called axioms ,
• Observation and recording of the laws, e.g. B. in physics by experiments or in the financial sector by comparing numbers ( empirical formula ).

## Notation of formulas

Common Unusual
${\ displaystyle x = {\ frac {1} {2}} {\ sqrt {2}}}$ ${\ displaystyle x = {\ frac {\ sqrt {2}} {2}}}$ or ${\ displaystyle {\ frac {1} {\ sqrt {2}}}}$
${\ displaystyle v = s \ cdot t}$ ${\ displaystyle s \ cdot t = v}$
${\ displaystyle x = {\ frac {1} {2}} \ cdot {\ frac {3} {4}}}$ ${\ displaystyle x = {\ frac {\ frac {1} {2}} {\ frac {4} {3}}}}$

Roots and other compound parts are preferably written on the baseline in formulas instead of being written in the numerator or even denominator of a fraction.

If a new quantity is defined from already known quantities in physical formulas, the new quantity usually appears alone on the left.

Fractions from fractions are multiplied out as much as possible. As a result, the formula doesn't take up as much space vertically. In addition, it usually makes the formula easier to understand, since dividing divisions is complicated, much like double negative . Exceptions apply if the parts of a formula are related in terms of content (for example for a speed), then an artificial tearing apart would hide the relationship between the formula variables. ${\ displaystyle {\ frac {s} {t}}}$

## Forming

There are different spellings for a formula, all of which have the same meaning. Depending on which parts of the formula are already known and which are still unknown, one or the other spelling is suitable. The different spellings can be converted into one another using equivalence transformations.