We explain what integers are and what their properties are. Additionally, we give you some examples.
What are integers?
The whole numbers are the numerical set that covers all of the natural numbers its negative inverses and zero. That is, these are the numbers used to count, along with their negative opposites (1 and -1). Normally, negative integers are written with their sign (-), which is not necessary for positive integers, but can sometimes be done to highlight the difference (+1 and -1).
The set of integers corresponds to the letter Z coming from the German word zahl (“number” or “quantity”). It is usually represented as a number line, with zero located in the middle and, from it, the positive numbers (Z+) displayed to the right and the negative numbers (Z-) displayed to the left, in both cases extending to infinity.
In this way, positive integers grow to the right, while negative integers grow to the left. This means that the quantities are larger towards positive infinity (+∞) and smaller towards negative infinity (-∞).
The appearance of whole numbers allowed to increase the number of possible operations with natural numbers since negative figures allow more complex operations to be carried out, such as subtracting a larger number from a number (5 – 7 = -2).
This is extremely useful for calculating and recording profits and losses, debts, and even certain quantities such as temperature, in which values above zero (positive) and below zero (negative) are used.
Since the natural numbers (N) are contained in the integers (Z), the natural numbers are considered to be a subset of the integers (N⊂Z). In turn, integers are a subset of rational numbers (Q), since they do not take fractions (Z⊂Q) into account.
See also: Mathematics
Properties of integers
Integers, except zero, must be positive (+) or negative (-), but at the same time they have a absolute value. The absolute value (represented between bars: |z|) is the distance between the location of a number on the number line and zero, regardless of whether it is positive or negative. For example, the absolute value of 5 and -5 is the same: |5|.
On the other hand, with integers it is possible to perform the same operations as with natural numbers, that is, they can be added, subtracted, multiplied or divided. However, in your case, The rules that determine the sign of the result must always be followed.
These standards vary according to the operation and can be understood as follows:
1. Sum
When adding whole numbers, you must pay attention to the addends to calculate the result:
- If both numbers are positive or one of the two is zero, their absolute values should normally be added and the positive sign retained. For example: 1 + 3 = 4; 6 + 0 = 6.
- If both numbers are negative or one of the two is zero, their absolute values should normally be added and the negative sign will be retained. For example: -1 + -1 = -2 ; -6 + 0 = -6.
- If the numbers have different signs, however, the absolute value of the smaller number must be subtracted from the larger number, and the result will have the sign of the larger number. For example: -4 + 5 = 1 ; -8 + 4 = -4.
2. Subtraction
When subtracting integers, you must also pay attention to the signs of the minuend and subtrahend, and which of the two has the greatest absolute value, as follows:
If they have a positive sign:
- If the minuend (positive) is greater than the subtrahend (positive), the subtraction will be performed normally and the difference will have a positive sign. For example: 8 – 5 = 3; 7 – 1 = 6.
- If the minuend (positive) is less than the subtrahend (positive), the subtraction will be equivalent to the difference between both numbers, but will have a negative sign. For example: 5 – 8 = -3 ; 2 – 9 = -7.
- If both figures are positive and equal, the result will be zero. For example: 5 – 5 = 0 ; 2 – 2 = 0.
If they have a negative sign:
- If the minuend (negative) is greater than the subtrahend (negative), the subtraction will be performed normally and the result will have a negative sign. For example: (-5) – (-3) = -2 ; (-9) – (-1) = -8.
- If the minuend (negative) is less than the subtrahend (negative), the subtrahend will be considered a positive number and the operation will be solved as if it were a sum. For example: (-2) – (-3) = 1 ; (-5) – (-8) = 3.
- If both figures are negative and equal, their absolute values will be added and the result will have a negative sign. For example: (-1) – (-1) = -2 ; (-5) – (-5) = -10.
If they have different signs:
- If the minuend (positive) is greater than, equal to, or less than the subtrahend (negative), their absolute values will normally be added and the result will have a positive sign. For example: 9 – (-1) = 10 ; 5 – (-5) = 10 ; 1 – (-9) = 10.
- If the minuend (negative) is greater than, equal to, or less than the subtrahend (positive), their absolute values will normally be added and the result will have a negative sign. For example: -8 – 2 = -10 ; -2 – 2 = -4 ; -2 – 8 = -10.
3. Multiplication
When integers are multiplied, their absolute values are normally multiplied, and then the sign of the product is calculated according to the following:
- Positive times positive equals positive. For example: 2 x 2 = 4.
- Positive times negative equals negative. For example: 2 x -2 = -4.
- Negative times positive equals negative. For example: -2 x 2 = -4.
- Negative times negative equals positive. For example: -2 x -2 = 4.
4. Division
When dividing between whole numbers, we proceed in the same way as in the case of multiplication: we normally operate with absolute values and apply the principle that determines the sign of the result. For example:
- Positive between positive equals positive. For example: 10 / 2 = 5.
- Positive divided by negative equals negative. For example: 10 / -2 = -5.
- Negative between positive equals negative. For example: -10 / 2 = -5.
- Negative among negative equals positive. For example: -10 / -2 = 5.
Examples of integers
It is not difficult to find examples of integers, since any natural number is in turn an integer: 1, 2, 3, 4, 5, 10, 125, 590, 1,926, 76,409 or 9,483,920. And, at the same time, are their corresponding negative inverses: -1, -2, -3, -4, -5, -10, -125, -590, -1,926, -76,409 and -9,483,920. The only condition is that they are not fractional numbers (such as ½ or 4.4).
Another possible example of an integer is zero (0).
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References
- Huete de Guevara, M. (1996). The set of integers. Editorial State Distance University (UNED).
- Núñez Cabello, R. (2007). Integers and divisibility. Publicatuslibros.com.
