#### 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: $a+a+\dots +a=n·a$ ($n$ is the number of summands).
• Distributive property ("expansion"): $\left(a+b\right)·c=ac+bc$ and $c·\left(a+b\right)=ca+cb$.
• Commutative property: $a+b=b+a$.
• Associative property ("group numbers differently in operations of the same kind"):
$a+\left(b+c\right)=\left(a+b\right)+c=a+b+c$, 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: .

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:
• $a\left(a+a+a\right)+{a}^{2}+{a}^{2}+{a}^{2}=6{a}^{2}$: the term on the right is simpler, since it requires fewer symbols.
• $\left(x+3{\right)}^{2}-9={x}^{2}+6x$ (first binomial formula): both terms describe a parabola. On the left, the vertex $\left(-3,-9\right)$ of the parabola can be seen easily, on the right, the two roots (${x}_{1}=0$ and ${x}_{2}=-6$) can be seen easily.
• $1+3x+3{x}^{2}+{x}^{3}=\left(1+x{\right)}^{3}$: on the right, it can be seen, for example, that the function described by the term has only the root ${x}_{1}=-1$.
• $\frac{a+1}{a}=1+\frac{1}{a}$: on the left, it can be seen that the term has the root ${a}_{1}=-1$, on the right it can be seen that, for very large $a$, the term converges to $1$ (since $\frac{1}{a}$ is very small in this case).

##### Exercise 1.1.20
Transform into a sum: $a·\left(b+c\right)+c·\left(a+b\right)$$=$
.

##### Exercise 1.1.21
Transform into a sum: $\left(x-y\right)\left(z-x\right)+\left(x-z\right)\left(y-z\right)$$=$
.

##### Exercise 1.1.22
Transform into a sum: $\left(a+b+2\right)\left(a+1\right)$$=$
.