Commutative property
By reversing the order of the operands, the result does not change. The commutative property is a fundamental property of addition and multiplication
An operation is called binary when it combines two elements called operands to produce a third element: the result of the operation.
The commutative property applies to some binary operations. It states that the result of these operations does not change if we reverse the order in which the two operands are combined.
It applies to both addition and multiplication.
Addition
Given two numbers 𝑥 and 𝑦, the commutative property of addition states that 𝑥 + y = y + x. Its application is pretty simple. For example, 5 + 8 = 13; symmetrically, 8 + 5 = 13.
From an intuitive point of view, the truth of this identity can be verified through a number line, as shown in the following image.
If we make two consecutive jumps of 5 and 8 units on the number line, we arrive at 13. If we reverse the sequence, first making a jump of 8 units and then of 5, we also arrive at number 13.
We can draw another empirical demonstration of the validity of the commutative property from everyday experience. Imagine that we buy a good at the cost of 15 dollars and pay for it with a ten-dollar bill and a five-dollar bill. It is utterly indifferent to the storekeeper who receives the payment whether we give him the ten-dollar bill first and then the five-dollar bill or vice versa. The relevant thing is that he gets both.
Multiplication
Given two numbers 𝑥 and 𝑦, the commutative property of multiplication states that x ⋅ y = y ⋅ x. For example, 4 ⋅ 5 = 5 ⋅ 4 = 20.
Verifying the truth of this property has to do with how groups of elements are ordered. Let’s consider the following image.
In both the left and right sides of the image, precisely the same 20 orange circles are arranged in the same way. However, although the content of the two sides is identical, we can think of the organization of the circles in a rectangular shape in two different ways — on the left, as a series of 4 rows each containing five elements; on the right, as a series of 5 columns, each containing four elements.
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The arrangement by rows and columns is functional to the concept of multiplication as an addition repeated several times. The series of 4 rows of 5 elements is equivalent to the addition 5+5+5+5, which is equivalent to the multiplication 4 ⋅ 5. For its part, the series of 5 columns of 4 elements is equivalent to the addition 4+4+4+4+4, which in turn is equivalent to the multiplication 5 ⋅ 4. In both cases, the product is 20, which intuitively demonstrates the truth of the commutative property of multiplication.
The number line provides another useful tool for verifying the validity of this property. As you can see from the following image, making five jumps of 4 units each on the number line leads to 20. But making four jumps of 5 units also leads to 20. In mathematical symbols, 4 ⋅ 5 = 5 ⋅ 4 = 20.
Subtraction
Unlike addition and multiplication, subtraction is not commutative. Given two numbers 𝑥 and 𝑦, 𝑥–𝑦 ≠𝑦–𝑥. For example, 10−7 = 3; 7 –10 = −3 → 3 ≠−3.
Again, a number line provides a good help in forming an intuitive idea of why subtraction is not commutative. As the following two images show, subtracting 7 from 10 means returning to 3 on the number line. Differently, subtracting 10 from 7 means going beyond 0 and stopping at −3, on the side of the line occupied by negative numbers. They are evidently two different positions, which demonstrate that we cannot invert the operands of the subtraction without changing the result.
Division
Division, like subtraction, is also not commutative. In general, given two numbers 𝑥 and 𝑦 other than 0 (division by 0 has no meaning in mathematics), 𝑥 : y ≠ y : x. For example: 16 : 4 = 4, but 4 : 16 = 0.25, with the quotient 4 ≠ 0.25. A special case is when we divide a number by its opposite: 5 :(−5) =−1. Symmetrically, −5 : 5= −1.
The following two pictures provide an intuitive demonstration of the impossibility of applying the commutative property to division. As you can see, dividing a set of 16 elements — in this case, circles — into 4 parts produces 4 groups of 4 elements each.
Contrarily, dividing 4 objects (the same circles as above) into 16 parts produces 16 objects, each consisting of 1/4, that is, 0.25 parts, of each of the 4 objects to be divided.
The two operations do not produce the same result.
