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Integers

Integers are numbers with no fractional part. They contain the positive integers, their additive inverses (the negative numbers) and zero.

Properties

Flying numbers

The integers form a set containing the numbers …, -3, -2, -1, 0, 1, 2, 3, … We usually write this set with the letter Z with a double diagonal stroke: ℤ, from the German “Zahlen” (numbers). The integers form an infinite and countable set.

Integers have, for example, these properties:

  1. They are closed under addition and multiplication, just like the positive integers. This means that if we add two integers, we get an integer again.
  2. Unlike the positive integers, the integers are also closed under subtraction. The positive integers were not, because the difference of two of them could be negative. That does not bother us for the integers, because they include the negative numbers.
  3. Like the positive integers, they are not closed under division. We can still get a non-integer after dividing.
  4. For every integer c there is an additive inverse −c. If we have the number 10, its additive inverse is −10. For 55 it is −55. The same holds for negative numbers: for −13 the inverse is 13. If we have an integer c and its additive inverse −c, their sum is zero: c + (−c) = 0. So the additive inverse of zero is zero again.

Even and odd numbers

Integers can be split into even and odd numbers. Even numbers are those divisible by 2, for example 2, −4, −8, 40, 124 and so on. Odd numbers leave a remainder of 1 when divided by 2, so they are the numbers −1, 1, 5, 19, −41 and so on. Notice that we really do distinguish even and odd numbers among the negative numbers too (so −5 is odd), and that zero is an even number.

Properties with respect to addition: if you add two even numbers, you get an even number again. The following table shows the other cases:

even+even=eveneven+odd=oddodd+odd=even

A similar table for multiplication:

even⋅even=eveneven⋅odd=evenodd⋅odd=odd

Division with remainders

On the set of integers we can define division with remainders in much the same way as for the positive integers, but we have to deal with negative numbers. So this time the basic definition looks like this:

a=q⋅b+r,a,q∈Z,b∈Z−{0},0≤r<|b|

In this expression we divide a÷b. The number q is the quotient and the number r is the remainder. The number b must be non-zero, because we cannot divide by zero. The remainder must be non-negative and smaller than the absolute value of b, which makes the quotient and the remainder unique.

What would such a division look like? Let’s try −21÷4. The numbers would look like this:

−21=−6⋅4+3

The quotient is −6 and the remainder is 3. It may surprise you that the quotient differs from the one we get when all the numbers are positive:

21=5⋅4+1

Here the quotient would be 5 and the remainder 1. The difference comes from the negative numbers. With positive numbers, in the first step we look for the largest integer that is not greater than 21 and is exactly divisible by 4. That is 20. So we work out 20÷4=5 and get our 5. We then get the remainder as the difference 21 − 20.

With negative numbers we proceed in the same way. We look for the largest number that is not greater than −21 and is exactly divisible by 4. But −20 is greater than −21. So the largest number that is not greater than −21 and is exactly divisible by 4 is −24. Again we get the remainder as the difference −21 − (−24) = −21 + 24 = 3.