Numbers: Integers
The notion of integer
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, \[ \begin{array}{rcl}2 &=&1+1\\ 3 &=&1+1+1=2+1\\ 7 &= & 6+1\\ 2016 &= & 2015+1\\ \end {array}\]
Integers
Hence, the natural numbers are #1#, #2#, #\ldots# indicated by #\mathbb{N}#. These numbers are greater than zero, or in other words positive. The notation #12\gt0# expresses that #12# is positive. The notation #12\gt 11# expresses that #12# is bigger than #11#.
If we count backwards from #0#, then the negative integers arise: #-1#, #-2#, #-3#, #\ldots# WThe 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\ge 5# we indicate that #7# is greater than or equal to #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 are arranged as follows \[\cdots\lt-5\lt-4\lt-3\lt-2\lt-1\lt0\lt 1\lt2\lt3\lt4\lt5\lt\cdots\]
For a negative number we can say that it has sign #-#. A positive number has sign #+#.
The set of all natural numbers is denoted by #\mathbb{N}#; the set of all integers by #\mathbb{Z}#.
The number #12.0000# is a decimal number that is equal to #12#. So it is another way to write the integer. Decimal numbers will be discussed later.
The number #\frac{4}{2}# is a rational number which is equal to #2#. Rational numbers are discussed later. Integers are special rational numbers.
Operations on integers
Addition, subtraction, multiplication, and exponentiation are called operations, sometimes arithmetic operations, because they create new integers from old ones.
Dividing is of course an operation. We did not mention it above because not every division of an integer by another one results in an integer.
We will encounter many other operations, starting with exponentiation below.
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 this way.
If we add #5# with #7#, we get #12#, the same number: #7+5=12#. After all, #7+5# and #5+7# are both created by adding #12# times #1# to #0#.
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 equavalent to adding #14#. Hence #5-(-14) = 5+14=19#. The subtraction of two integers can be explained this way.
Multiplication
The equality #3\times 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\times 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.
Instead of #\times# we often write #\cdot#, hence #3\times 2# and #3\cdot 2# are both #6#.
#3\times 4# equals #4\times 3# : \[4\times 3 = \underbrace{1+1+1}_{\text{one time}} + \underbrace{1+1+1}_{\text{two times}} + \underbrace{1+1+1}_{\text{three times}}+\underbrace{1+1+1}_{\text{vier keer}}=12\tiny.\]
Exponentiation
Finally, exponentiation does for multiplication what multiplication does for adding: #3^5# stands for #3\times 3\times 3\times 3\times 3#, which is the same as #243#.
In here #3# is the base and #5#, the exponent.
Exponentiation can be performed with any base, as long as the exponent is a natural number.
If the exponent is equal to #0# and the base distinct from #0#, then the result is #1#: This is the agreement that multiplying a number zero times with itself yields #1#.
The case where both the base and the exponent are #0#, is problematic. This will become clear when we look at powers with a negative exponent.
Or visit omptest.org if jou are taking an OMPT exam.
