Rekenen met variabelen *: Variabelen *
Variables
We want to formulate general rules of calculation for calculating with numbers. We do this by using letters for randomly selected numbers. The fact that adding one number to the other yields the same result if you do it the other way around can be expressed by \[a+b=b+a\], meaning that it works for all #a# and #b#. This means that whatever numbers you for #a# and #b# fill, the equality always remains valid. Entering of #a=3# and #b=5# for example gives #3+5=5+3# and entering #a=2# and #b=-9# gives #2+(-9)=-9+2#.
Variables
Letters like #a# and #b# as above are called variables.
The letter is called the name of the variable.
Sometimes variable names are longer than one letter. If we want to use ten variables of the same type, we sometimes use an index: #a_1, a_2,a_3,\ldots,a_{10}#.
The name of the variable #a_3# is pronounced as "#a# of #3# "or simply" #a# three".
Some letters are often used as variable for the same purpose. Typical examples are
- #m#, #n# for natural numbers
- coordinates #x# and #y# in the flat plane
- spatial coordinates #x#, #y# and #z#
- time #t#
- the temperature in degrees Kelvin #T#
The basic arithmetic of mathematical expressions containing variables will be dealt with in the chapter Algebra.
Constants and symbols
Variables are also used to abbreviate complex expressions. You can write: \[ \text{Let }r \text{ be the real number }\frac{\sqrt{\pi}+\sqrt{3}}{2+\sqrt{5}}\tiny, \]and, after having made this statement, write down #r# instead of the complex expression itself. In this case, #r# is called a constant. Numbers themselves are also constants.
The expression #\pi# for the circumference of a circle of diameter #1# indicates a real number. Here #\pi# is no variable but a symbol. Symbols are used to call fixed mathematical expressions by a name.
The distinction between a variable that represents all the possible numbers, and a variable that represents a constant, is not always very clear. The name indicates what the intention is: basically you can let the value of a variable vary, while for a letter that is called a constant, you can assume ---at least for now--- that it does not vary.
Other examples of symbols that are later discussed are entier, sin, cos, log, and #\e#.
For, substituting #a=7# in #a\times a -13\times a+42# gives \[7\times 7 -13\times 7+42=0\tiny.\]
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