Chapter 1 Elementary Arithmetic

Section 1.1 Numbers, Variables, Terms

1.1.3 Transformation of terms

There is always more than one way to write the same term, although some are more natural than others. For example, x+x is a different arrangement of symbols than 2x, but describes the same term, i.e. if x is substituted with a specific number, then x+x and 2x provide the same value.
Info 1.1.17
Terms are related by an equals sign if they are always evaluated to the same value.

In general, new terms are created by transformation of existing terms:
Info 1.1.18
A transformation of a term is created by applying one or more calculation rules to the term:
  • Collecting: a+a++a=n·a ( n is the number of summands).
  • Distributive property ("expansion"): (a+b)·c=ac+bc and c·(a+b)=ca+cb.
  • Commutative property: a+b=b+a.
  • Associative property ("group numbers differently in operations of the same kind"):
    a+(b+c)=(a+b)+c=a+b+c, also possible in multiplications.
  • Calculation rules for powers and special functions.
  • Calculation rules for specific types of terms (e.g. the binomial formulas).
  • Calculation rules for fractions: 1    a b    = b a .

The rules will be presented in detail in the following sections. Often, the aim of this transformation is to simplify the term, to isolate individual variables, or to transform a term into a certain form:
Example 1.1.19
Examples of transformations and their uses:
  • a(a+a+a)+ a2 + a2 + a2 =6 a2 : the term on the right is simpler, since it requires fewer symbols.
  • (x+3 )2 -9= x2 +6x (first binomial formula): both terms describe a parabola. On the left, the vertex (-3,-9) of the parabola can be seen easily, on the right, the two roots ( x1 =0 and x2 =-6) can be seen easily.
  • 1+3x+3 x2 + x3 =(1+x )3 : on the right, it can be seen, for example, that the function described by the term has only the root x1 =-1.
  • a+1 a =1+ 1 a : on the left, it can be seen that the term has the root a1 =-1, on the right it can be seen that, for very large a, the term converges to 1 (since 1 a is very small in this case).

Exercise 1.1.20
Transform into a sum: a·(b+c)+c·(a+b) =

Exercise 1.1.21
Transform into a sum: (x-y)(z-x)+(x-z)(y-z) =

Exercise 1.1.22
Transform into a sum: (a+b+2)(a+1) =