Understanding Integers Operations

Integers

Integers, often denoted by the symbol I or Z, refer to whole numbers including zero along with their opposites. They can be positive, negative, or zero themselves, but they cannot be fractions or decimals. This makes them distinct from other types of numbers like real numbers, rational numbers, and irrational numbers which do have decimal parts.

Adding Integers

To add two integers together, you simply sum up their positive values if both signs are identical (either both plus or minus), while subtracting one value's absolute magnitude from another when the signs differ (plus and minus). For example, -7 + (-9) = -16 since removing any negative sign results in a higher number. Similarly, -8 + +9 = 1 due to changing it into its corresponding positive part. If there is no change in sign, you just add the numbers together without worrying about the negativity. Hence, 5 + 3 equals 8.

Subtracting Integers

When it comes to subtraction, a negative number will result if taking away more units would lead to going below zero. Otherwise, you take away less units and stay above zero. So, when we calculate -9 - 5, we find out that taking the difference leaves us at -14 because it exceeds zero; however, if we were to perform the calculation between +9 and -5, we would end up at 4 since it doesn't go beyond zero.

Multiplying Integers

Multiplying integers involves combining quantities having similar properties: if either multiplier has a positive sign, so does their product; otherwise, when both multipliers are negative, their product becomes positive. Let's consider some examples here too: multiplying -7 * 3 produces -21 since keeping all three terms negative leads to a larger quantity being formed, whereas multiplying 3 * (+2) results in +6 where each term remains unchanged yet adds up nicely.

Dividing Integers

Dividing an integer by another integer means splitting one unit into several equal parts according to how many times the divisor fits inside the dividend. Therefore, 4 divided by 2 yields 2, meaning we successfully split 4 into halves twice and got rid of 2 units completely. However, this rule gets slightly complicated when dealing with mixed quantities such as dividing 8 by -3; here lies one major distinction among different numeric systems: some allow division by certain numbers regardless of whether they are negative or positive ones (like Rational Numbers), while others don't permit operations involving arithmetic patterns that generate 'illegal' outcomes (such as Integer Arithmetic System).

Properties of Integers

Some key properties associated with integers include closure under addition (meaning that the sum of any two integers is still an integer); commutativity of addition (adding them in reverse order gives the same answer); associativity (grouping them differently results in same value); identity property of zero (any integer added to zero remains itself); inverse property of zero (zero property applies only for addition). Additionally, multiplication of non-negative integers forms a commutative monoid, making multiplication distributive over addition. As for ordering, every pair of consecutive integers differs exactly by one unit, forming what's known as an ordered set.