#### Chapter 1 Elementary Arithmetic

**Section 1.1 Numbers, Variables, Terms**

# 1.1.2 Variables and Terms

The use of variables, terms and equations is required to formalise expressions whose values have not been fixed.

##### **Info 1.1.6 **

A

**variable**is a symbol (typically a letter) used as a placeholder for an indeterminate value. A

**term**is a mathematical expression that can contain variables, arithmetic operations and further symbols and, after substituting variables with numbers, can be evaluated to a specific value. Terms can be combined into equations and inequalities, respectively, or they can be inserted into function descriptions, as we shall see later.

##### **Example 1.1.7 **

The word problem

can be formalised, for example, by introducing the variable $a$ for the number of girls and the variable $b$ for the number of boys in the class and setting up the two equations $a=b+4$ and $a+b=20$. These equations can be solved by inserting the first equation into the second which gives $a=12$ and $b=8$. Out of this, the full written answer

can be constructed. Here, for example, $b+4$ is a term, $b$ itself is a variable, and $a+b=20$ is an equation with a term on the left and a number on the right.

*In a school class there are four more girls than boys and in total there are 20 children. How many girls and boys are in the class, respectively?*can be formalised, for example, by introducing the variable $a$ for the number of girls and the variable $b$ for the number of boys in the class and setting up the two equations $a=b+4$ and $a+b=20$. These equations can be solved by inserting the first equation into the second which gives $a=12$ and $b=8$. Out of this, the full written answer

*In the school class there are $12$ girls and $8$ boys*can be constructed. Here, for example, $b+4$ is a term, $b$ itself is a variable, and $a+b=20$ is an equation with a term on the left and a number on the right.

##### **Info 1.1.8 **

This overview shows the (lowercase and uppercase) letters of the Greek alphabet in Greek alphabetical order:

$\alpha $, $A$ | "alpha" | $\beta $, $B$ | "beta" | $\gamma $, $\Gamma $ | "gamma" | $\delta $, $\Delta $ | "delta" | $\epsilon $, $E$ | "epsilon" |

$\zeta $, $Z$ | "zeta" | $\eta $, $H$ | "eta" | $\vartheta $, $\Theta $ | "theta" | $\iota $, $I$ | "iota" | $\kappa $, $K$ | "kappa" |

$\lambda $, $\Lambda $ | "lambda" | $\mu $, $M$ | "mu" | $\nu $, $N$ | "nu" | $\xi $, $\Xi $ | "xi" | $o$, $O$ | "omicron" |

$\pi $, $\Pi $ | "pi" | $\varrho $, $P$ | "rho" | $\sigma $, $\Sigma $ | "sigma" | $\tau $, $T$ | "tau" | $\upsilon $, $\Upsilon $ | "upsilon" |

$\phi $, $\Phi $ | "phi" | $\chi $, $X$ | "chi" | $\psi $, $\Psi $ | "psi" | $\omega $, $\Omega $ | "omega" |

In the exercises Greek letters can be entered with their description, e.g.

`alpha`instead of $\alpha $.

It is important that a term can be evaluated to a specific value if the variables occurring in the term are substituted with numbers:

##### **Example 1.1.9 **

The following expressions are terms:

- $x\xb7(y+z)-1$: for $x=1$, $y=2$, and $z=0$ one obtains, for example, the value $1$.

- $\mathrm{sin}(\alpha )+\mathrm{cos}(\alpha )$: for $\alpha ={0}^{\circ}$ and $\beta ={0}^{\circ}$ one obtains, for example, the value $1$ (for the calculation of sine and cosine refer to 5).

- $1+2+3+4$: no variables occur, however this is a term (which always gives the value $10$).

- $\frac{\alpha +\beta}{1+\gamma}$: for example, $\alpha =1$, $\beta =2$, and $\gamma =3$ give the value $\frac{3}{4}$. But $\gamma =-1$ is not allowed.

- $\mathrm{sin}(\pi (x+1))$: this term, for example, always gives the value zero if $x$ is substituted with an integer.

- $z$: a single variable is also a term.

- $1+2+3+\dots +(n-1)+n$ is a term, in which the variable $n$ occurs in the term itself and defines its length as well.

##### **Example 1.1.10 **

These expressions are not terms in a mathematical sense:

- $a+b=20$ is an equation (inserting values for $a$ and $b$ gives no number, but the equation is simply true or false).

- $a\xb7(b+c$ is not correctly bracketed,

- "
*The ratio of girls in the school class*" is not a term, but can be formalised by the term $\frac{a}{a+b}$,

- $\mathrm{sin}$ is not a term but a function name, in contrast $\mathrm{sin}(\alpha )$ is a term (which can be evaluated by inserting an angle for $\alpha $).

##### **Exercise 1.1.11 **

In each question, given a term and number values for the variables that occur in it, what is the evaluation of the term?

- $\frac{\alpha +\beta}{\alpha -\beta}$ takes the value

for $\alpha =6$ and $\beta =4$.

- ${y}^{2}+{x}^{2}$ takes the value

for $y=2x+1$ and $x=-1$.

- $1+2+3+\dots +(n-1)+n$ takes the value

for $n=6$.

##### **Exercise 1.1.12 **

Formalise, using the variables given, the proportion of girls and the proportion of boys, the number of girls being denoted by the variable $a$ and the number of boys by the variable $b$:

The proportion of girls is

and the proportion of boys is .

Fractions can be entered using the slash (above the 7-key on most keyboards), where numerator and denominator, respectively, should be bracketed if arithmetic operations occur. The fraction $\frac{1+x}{2+y}$ can be entered, for example, by

The proportion of girls is

and the proportion of boys is .

Fractions can be entered using the slash (above the 7-key on most keyboards), where numerator and denominator, respectively, should be bracketed if arithmetic operations occur. The fraction $\frac{1+x}{2+y}$ can be entered, for example, by

`(1+x)/(2+y)`.Terms can be inserted into other terms as well:

##### **Example 1.1.14 **

Substituting, for example, the right-hand side of $x=1+2+3$ into the term ${x}^{2}+{y}^{2}$ results in the new term ${x}^{2}+{y}^{2}=(1+2+3{)}^{2}+{y}^{2}=36+{y}^{2}$ and certainly not $1+2+{3}^{2}+{y}^{2}=12+{y}^{2}$.

##### **Exercise 1.1.15 **

Which term is formed if the following object is inserted into the term ${x}^{2}+{y}^{2}$?

- The angle $\alpha $ both for $x$ and $y$: Then ${x}^{2}+{y}^{2}$$\hspace{0.5em}=\hspace{0.5em}$ .

- The number $2$ for $y$ and the term $t+1$ for $x$: Then ${x}^{2}+{y}^{2}$$\hspace{0.5em}=\hspace{0.5em}$ .

- The term $z+1$ for $x$ and the term $z-1$ for $y$: Then ${x}^{2}+{y}^{2}$$\hspace{0.5em}=\hspace{0.5em}$ .

`alpha`.##### **Exercise 1.1.16 **

In the following figure, a square on the paper has side length $x$. What is the area of this figure (as a term in the variable $x$)?

Answer:

Answer:

- The large circle has a total area of ,

- each smaller circle has an area of ,

- the total area of the figure is .

`pi`.