### Calculating with numbers: Computing with integers

### Arithmetic calculations with integers

Arithmetic calculations Addition, subtraction, multiplication and exponentiation are called **arithmetic calculations** because they make new integers from given integers.

Adding natural numbers When you add to a bin containing four marbles three times one marble, then you have hereafter a bin that contains seven marbles. You could also have placed first three marbles in your hand or in an extra bin, and then you could have merged the contents in order to get a bin with seven marbles. The following figure visualises the process of emptying of the container with three marbles into the container with four marbles.

You have in fact added the numbers \(4\) and \(3\), leading to the outcome \(7\). This is expressed in the equality \(4+3=7\). We pronounce it as *"four plus three equals seven"* when we want to focus on the calculation of the final outcome. In the statement *"two plus three equals seven"* we emphasize that these are equivalent representations of the number seven: this may be the number \(7\), and the outcome of the sum \(3+4\), but also the outcome of say \(2+5\). The terms on the left- and right-hand side are permitted nametags for one and the same mathematical object.

Via tabs below you get access to more details about addition of natural numbers and to a systematic approach toward addition of natural numbers.

Subtracting natural numbers When you remove three times one marble from a bin with seven marbles and put it in a second bin, then you have afterwards a second bin with three marbles and only four marbles are left in the original bin. The figure below visualises the process of removing three marbles from a bin with seven marbles and the placement of three marbles in an initially empty second bin (the white circles in the middle bin only indicate how many marbles will be transferred into it).

You have basically subtracted the number \(3\) from \(7\). This you indicate by the equality \(7-3=4\) and you pronounce it as *"seven minus three gives four" or "seven minus three equals four,"* depending on whether you focus more on the operation or on the number four.

The subtraction \(3-7\) is not possible within the set of natural numbers because \(3\) is less than \(7\). This subtraction only makes sense within the set of integers and then the result is the negative number \(-4\). What you actually first do is swapping the two numbers in the subtraction: you get a subtraction that can be resolved within the set of natural numbers. Hereafter you take the opposite of the intermediate result.

Multiplication When you put three columns of four marbles together, you have built a structure with a total of \(3\times 4=12\) marbles.

This illustrates that multiplication of natural numbers not more or less than repeated addition: in our example: \(3\times 4=4+4+4=12\).

We pronounce \(3\times 4=12\) as *"three times four is twelve"* or *"three times for makes twelve"* when we want to focus more on the calculation of the final result. Through he statement *"three times four equals seven"* we emphasize that these are equivalent representations of the number 12: This may be the number \(12\) or the result of the calculation \(3\times 4\), but also the outcome of for example \(2\times 6\) or \(1\times 12\). The terms on the left and right are valid nametags for the same mathematical object.

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Addition, subtraction, and multiplication of natural numbers is most systematically done by placing the numbers underneath each other and aligned to the right aligned at each place and by calculating columnwise. If needed, look at a few examples and study the additional information below the tabs in the above frames.

The correct answer is \(1774\).

You learned in elementary school to add such numbers, but perhaps not in the most convenient and systematic way. The best recipe is as follows:

- Place the numbers right-aligned below each other
- Add the units (digit in the right column): \(2+4+5+0+3=14\), and write of this number only \(4\) on the bottom line and move \(1\) to the top of the next column (the tens).
- Add the tens, i.e. add the middle digits of a three-digit number, together with the additional \(1\): \(1+5+6+5+9+1=27\). Again you write \(7\) on the bottom line and move \(2\) to the top of the next column on the left.
- Add the hundreds together with the extra \(2\): \(2+8+2+5=17\). The outcome at the bottom line and you're done with the calculation of the outcome \(1774\).

The following diagram illustrates the steps in the calculation; we remove at the end the extra digits added on the top.

\[\begin{aligned}

&\textit{units:}\qquad \\ \begin{array}[t]{rl} & \\ 85\blue{2} & \\ 6\blue{4} & \\ 25\blue{5} & \\ 9\blue{0} & \\ 51\blue{3} & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ \blue1\blue{4} & \end{array} \quad &\longrightarrow\qquad \begin{array}[t]{rl} \phantom{8}1\phantom{2} & \\ 852 & \\ 64 & \\ 255 & \\ 90 & \\ 513 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ 4 & \end{array}

\\ \\

&\textit{tens:}\qquad \\

\begin{array}[t]{rl} \phantom{8}\blue{1}\phantom{2} & \\ 8\blue{5}2 & \\ \blue{6}4 & \\ 2\blue{5}5 & \\ \blue{9}0 & \\ 5\blue{1}3 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ \blue2\blue{7}4 & \end{array} \quad &\longrightarrow\qquad \begin{array}[t]{rl} 21\phantom{2} & \\ 852 & \\ 64 & \\ 255 & \\ 90 & \\ 513 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ 74 & \end{array}

\\ \\

&\textit{hundreds:}\qquad \\

\begin{array}[t]{rl} \blue{2}1\phantom{2} & \\ \blue{8}52 & \\ 64 & \\ \blue{2}55 & \\ 90 & \\ \blue{5}13 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ \blue{17}74 & \end{array} \quad &\longrightarrow\qquad \begin{array}[t]{rl} \phantom{2} & \\ 852 & \\ 64 & \\ 255 & \\ 90 & \\ 513 & \\ \overline{\phantom{xxxx}}& {}^{\displaystyle +} \\ 1774 & \end{array}

\end{aligned}\]

Experienced persons do not write the additional numbers at the top of the columns anymore, but they memorise then and add them immediately in their brains. They write:

\[\begin{aligned} 2+4+5+0+3=14,\quad &\blue{4\text{ written, }1\text{ memorised.}} \\ \blue{1}+5+6+5+9+1=27, \quad &\blue{7\text{ written, }2\text{ memorised.}} \\ \blue{2}+8+2+5=17.\quad &\blue{\text{ready; outcome}=1774.}\end{aligned}\]

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Calculations with negative integers So far we have focused on addition, subtraction, and multiplication of natural numbers. This forms the basis for calculating with integers (which may be negative).

Subtraction within the set of integers is rather easy when you apply the following rule:

*Subtraction of a number is exactly the same as adding the opposite number.*

For example: \(1-3=1+(-3)=-2\).

The result is easily understood by representing the numbers as arrows along the number line and by combining them via the head-tail method.

Multiplication within the set of integers is rather easy when you apply the following rule:

*Multiplying by \(-1\) is the same as the toggling of the sign, from plus to minus or from minus to plus.*

Examples: \(3\times (-3)=3\times 3\times (-1)=9\times (-1)=-9\) and \((-3)\times (-4)=3\times (-1)\times 4\times (-1)=3\times 4\times (-1)\times (-1)=12\times (1)=12\).

Summarized in words and easy to recall: \[\begin{array}{ll} \textit{plus times plus is plus} & \textit{min times plus is min} \\ \textit{plus times min is min} & \textit{min times min is plus}\end{array}\]