Natural numbers and integers

Calculating with numbers: Computing with integers

Natural numbers and integers

When counting objects you usually start with \(1\) and continue adding \(1\). So you get in succession the numbers \(1,\;\;2=1+1,\;\; 3=2+1=1+1+1\), etcetera. In order to indicate the absence of an object one needs the number \(0\). You may also count down from \(0\) and then you get successively the numbers \(-1=0-1,\;\; -2=-1-1,\;\;-3=-2-1=-1-1-1\), etcetera. In this way all integers are constructed.

Natural numbers The sequence \(0, 1, 2, 3, \ldots\) is the sequence of the natural numbers. These are the numbers to count with or to index. The set of natural numbers is symbolized by \(\mathbb{N}\). So \[\mathbb{N}=\{0,1,2,3,\ldots\}\]

Yes or no zero You could argue whether the number 0 belongs to the natural numbers or not: In counting objects, after all, you start counting from 1. But since the axiomatic introduction of natural numbers by the 19th-century mathematician Peano zero is commonly considered a natural number. too Moreover, when using the natural numbers as index, it is convenient to take the lowest index 0. In modern programming languages, this convention almost always is used, sometimes in direct reference to the manucript EWD831 of the Dutch mathematician and computer scientist Edsger Wybe Dijkstra on this topic. According to the ISO 80000-2:2021 standard about quantities and units 0 belongs to the set of natural numbers. The Bourbaki school for mathematics uses the same convention. Enough reasons to also adopt the rule that 0 is a natural number.

A much used notation for the above choice is: \(\mathbb{N}=\{0,1,2,3,\ldots\}\) and \(\mathbb{N}_0=\{1,2,3,\ldots\}\). In other words: \(\mathbb{N}_0\) symbolises the set of all natural numbers without \(0\).

Integers The arithmetic of natural numbers goes well if you add such numbers. But subtraction of natural numbers sometimes causes problems. What, for example, would the outcome of \(1-2\) or \(2-3\) be and do we mean the same number with that? In other words, what is the solution of the equation \(x+2=1\) or the equation \(x+3=2\)? The wish to be able to solve such equations gives rise to the introduction of the set of integers, symbolized by \(\mathbb{ℤ}\).

Integers can be placed on a number line: Begin with a first point on a straight line and give it the label 0 and choose a second point that get the label 1. We agree that the distance between these two points is equal to 1. Now continue with steps of length 1 and place the next integer on the number line with its label. Each number has a successor. Do the same now in the other direction and let each number have a predecessor, and you get the following number line:

So \[\mathbb{ℤ}=\{\ldots, -2, -1, 0, 1, 2, \ldots\}\] The numbers from this collection can be added, subtracted and multiplied with each other.

Opposite The opposite of \(2\) is \(-2\) and, the other way around, \(2\) is the opposite of \(-2\). In short, \(2\) and \(-2\) are each other's oppositie. The number \(0\) is is its own opposite.

So we have three kinds of integers: the number zero \(0\), the positive natural numbers (integer greater than zero) and the negative integers (integers less than zero). The negative integers have a minus sign in their notation. Positive natural numbers have a plus sign, this is usually omitted (for example, \({}+2=2\)).

The opposite of an arbitrary integers \(n\) is denoted by \(-n\). For a positive number this is obvious. But also when \(n\) is negative, we mean by \(-n\) the opposite of \(n\), and then \(-n\) is positive. For example, we have the following equality: \(-(-2)=2\).

The opposite of an integer is obtained when switching the sign of the number: \(-\) in front of a digit becomes \(+\) and is eventually omitted. The other way around, \(+\) (which may or may not be written in front of the first digit) becomes \(-\).

Nonpositive and nonnegative We call a number nonpositive if it is negative or equal to \(0\). We call a number nonnegative if it is positive or equal to \(0\).

Absolute value The absolute value \(|a|\) of an integer \(a\) is equal to \(a\) if the number is nonnegatieve, and equal to \(-a\) if the number is negative. In the langauge of formulas: for any integer \(a\) we have \[ |a|=\begin{cases} \phantom{-}a\quad\text{if }a\ge 0\\ -a\quad\text{if }a\lt 0\end{cases}\] For example, \(|-2|=2\) and \(|2|=2\).

Integers as arrows along the number line The labelled positions in the number line represent integers. But you could also consider an arrow that starts at \(0\) and ends at a labelled point on the number line as a representation of an integer. A arrow to the right corresponds with a natural number greater than \(0\) and an arrow to the left corresponds with a negative number. Below you see the arrows corresponding with the integers \(-2\) and \(3\).

This representation with arrows suggests a movement: so many steps to the righ or to the left.

Addition of two integers is in this interpretation of the number line then the combination of arrows according to the head-tail method. Below you see the additions \(2+1\) and \(1+2\) visualised and the results are the same, namely the arrow representation of the number \(3\) on the number line. After all, first two steps to the right followed by one step to the right leads to the same movement of three steps to the right as first one step to the right followed bt two steps to the right.

This is an example of a general property of addition of natural numbers:the outcome of addition of two natural number does not depend when you exchange the two number in the operation. This is called the exchange property or, more formally, the commutative property of addition of natural numbers.