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The counting of objects usually starts with \(1\). From here on you can reach all other natural numbers by counting up, or, in other words, by continuing to add \(1\) sufficiently often. For example,

\(\begin{array}{rcl}2 &=&1+1\\ 3 &=&1+1+1=2+1\\ 7 &= & 6+1\\ 2016 &= & 2015+1\\ \end {array}\)

The **natural numbers** are \( 1, 2, \ldots \) These numbers are greater than zero, that is, **positive.** The notation \(12\gt0\) expresses that \(12\) is bigger than \(11\).

If we count backwards from \(0\), then the **negative integers** arise: \( -1, -2, -3, \ldots\) The notation \(-10\lt0\) expresses that \(-10\) is negative. With \(-9\lt -8\) we indicate that \(-9\) is smaller than \(-8\).

To describe the set of all natural numbers and \(0\), we use the term **non-negative integers. **The notation \(10\ge0\) expresses that \(10\) is non-negative. With \(7\ge5\) we indicate that \(7\) is greater than \(5\).

The **non-positive integers **are \(0, -1, -2, \ldots \) The notation \(-3\le0\) expresses that \(-3\) is non-positive. By \(-9\le -8\) we indicate that \(-9\) is less than or equal to \(-8\).

The negative integers and the non-negative integers taken together are all **integers.** They arranged as follows

\(\cdots\lt-5\lt-4\lt-3\lt-2\lt-1\lt0\lt 1\lt2\lt3\lt4\lt5\lt\cdots\)

For negative number we can say that it has **sign **\(–\). A positive number has **sign** \(+\).

Addition, subtraction, multiplication, and exponentiation are called **operations**, sometimes **arithmetic operations**, because they create new integers from old ones.

**Addition**

If we add seven times \(1\) to \(5\), then the result is the number \(12\). Because seven times \(1\) is indicated by \(7\), we can write this activity briefly as \( 7 + 5 \). The equality \(7+5=12\) indicates the result of this operation. The **addition** of two numbers can be explained that way.

**Subtraction**

For counting backwards, we can use the minus sign. If we count backwards \(6\) starting from \(8\), we arrive at \(2\). This is denoted as \(8-6=2\). We can also use the negative number \(-6\) to describe this process: with \(8+ (-6)\) we mean the number you obtain from \(8\) by counting backwards \(6\). Hence \(8+(-6)=8-6=2\). This is convenient if we do not have enough natural numbers to make the counting backwards succeed: \(6 – 8 = -2\). After all, at \( 6 – 6\) we encounter \(0\), and from there we have to subtract \(2\) times \(1\) again, so that we end on the negative integer \(-2\). We can use the same convention when subtracting a negative integer \(-14\): which is equivalent to adding \(14\). Hence \(5-(-14) = 5+14=19\). The subtraction of two integers can be explained this way.

**Multiplication**

The equality \(3 \cdot 1 =3\) expresses that adding \(3\) times \(1\) to \(0\) yields the result \(3\). If instead of \(1\) we take for example \(4\), then \(3 \cdot 4\) expresses that we have to add \(4\) three times to \(0\). This result is \(12\), because:

\(3\times 4 = \underbrace{1+1+1+1}_{\text{one time}} + \underbrace{1+1+1+1}_{\text{two times}} + \underbrace{1+1+1+1}_{\text{three times}}=12\tiny.\)

The **multiplication **of integers can be explained this way.

**Exponentiation**

Finally, exponentiation does for multiplication what multiplication does for adding: \(3^5\) stands for \(3\cdot 3\cdot 3\cdot 3\cdot 3\), which is the same as \(243\).

In \(3^5\) , \(3\) is the base and \(5\) is the exponent.