โœ–

Whole numbers

Integers are numbers without a decimal part; they include natural numbers, their negative counterparts, and zero.

Properties

Flying numbers

An integer is a set that includes the numbers ..., -3, -2, -1, 0, 1, 2, 3, .... This set is usually denoted by the letter Z, with a double middle line: โ„ค, from the German "Zahlen" (numbers). An integer is an infinite and countable set.

For example, integers possess the following properties:

  1. They are closed under the operations of addition and multiplication, just like natural numbers. This means that if we add two integers, we get an integer again.
  2. Unlike natural numbers, integers are also closed under the operation of subtraction. Natural numbers are not, because subtracting one natural number from another can result in a negative number, which is acceptable in the case of integers since they include negative numbers.
  3. Just like natural numbers, which are not closed under the operation of division, we can still obtain a non-integer result after dividing.
    4. For every integer c, there exists an inverse โˆ’c. For example, if we have the integer 10, the inverse is โˆ’10. For 55, it is โˆ’55, and similarly, for negative numbers: โˆ’13 has the inverse 13. If we have the integer c and its inverse โˆ’c, adding them results in zero: c+(โˆ’c) = 0. Therefore, zero is its own inverse.

Even and odd numbers

We can divide integers into even and odd numbers. Even numbers are divisible by two; examples include 2, โˆ’4, โˆ’8, 40, 124, etc. Odd numbers have a remainder of one when divided by two; examples include โˆ’1, 1, 5, 19, โˆ’41, etc. Note that we distinguish between even and odd numbers even for negative integers (for example, โˆ’5 is indeed odd), and that zero is considered an even number.

Properties with respect to the addition operation: If you add two even numbers, the result is also an even number. Additional properties are shown in the following table:

$$\begin{eqnarray} \mbox{ Even }+\mbox{ Even }&=&\mbox{ Even }\\ \mbox{ Even }+\mbox{ Odd }&=&\mbox{ Odd }\\ \mbox{ Odd }+\mbox{ Odd }&=&\mbox{ Even } \end{eqnarray}$$

Similar table for multiplication:

$$\begin{eqnarray} \mbox{ Even }\cdot\mbox{ Even }&=&\mbox{ Even }\\ \mbox{ Even }\cdot\mbox{ Odd }&=&\mbox{ Even }\\ \mbox{ Odd }\cdot\mbox{ Odd }&=&\mbox{ Odd } \end{eqnarray}$$

Division with remainder

Even on the set of integers, we can define division with a remainder, just as we do with natural numbers, but we must consider negative numbers. Thus, the basic definition this time will look as follows:

$$a=q\cdot b+r,\qquad a,q\in\mathbb{Z}, b\in\mathbb{Z}-\left\{0\right\}, 0\le r<|b|$$

In this expression where we divide a:b, the number q represents the result (quotient), and the number r represents the remainder after division. The divisor b must not be zero, as division by zero is undefined. The remainder must be positive and less than the absolute value of b, thereby excluding cases where division by a negative number occurs.

What would such a division look like? Let's try dividing โˆ’21:4. The numbers would then appear as follows:

$$-21=-6\cdot4+3$$

The result (the quotient) is โˆ’6, and the remainder is 3. It may surprise you that the result differs from what it would be if all the numbers were positive.

$$21=5\cdot4+1$$

Here, the quotient is 5, and the remainder is 1. The only difference lies with negative numbers. With positive numbers, the first step is to find the largest integer that is smaller than 21 and divisible by four without a remainder, which is 20. Thus, we divide 20:4 = 5 to obtain five. We then determine the remainder by calculating 21 โˆ’ 20.

In negative numbers, we proceed similarly. We look for the largest number that is smaller than โˆ’21 and divisible by 4 without a remainder. However, the number โˆ’20 is not smaller than โˆ’21; it is actually larger. Therefore, the largest number smaller than โˆ’21 and divisible by 4 without a remainder is โˆ’24. We determine the remainder by calculating โˆ’21โˆ’(โˆ’24) = โˆ’21 + 24 = 3.