Test addition, multiplication, and more with switched operand order. See property checks and quick exports. Use examples, formulas, tables, and graphs for better understanding.
The graph compares the two evaluation orders for the current calculation.
| Operation | a | b | a op b | b op a | Commutative for the operation |
|---|---|---|---|---|---|
| Addition | 7 | 12 | 19 | 19 | Yes |
| Multiplication | 4 | 9 | 36 | 36 | Yes |
| Maximum | 6 | 14 | 14 | 14 | Yes |
| GCD | 18 | 24 | 6 | 6 | Yes |
| Subtraction | 14 | 6 | 8 | -8 | No |
| Division | 12 | 3 | 4 | 0.25 | No |
The commutative property asks whether changing operand order keeps the same result.
General test: a ⊙ b = b ⊙ a
Addition: a + b = b + a
Multiplication: a × b = b × a
Maximum: max(a, b) = max(b, a)
Minimum: min(a, b) = min(b, a)
Greatest common divisor: gcd(a, b) = gcd(b, a)
Least common multiple: lcm(a, b) = lcm(b, a)
Not generally commutative: subtraction, division, and exponentiation.
This calculator evaluates both orders, compares the outputs, and also reports whether the chosen operation is commutative as a general rule.
For GCD and LCM, use integers. For division, both operands must be nonzero so both orders can be evaluated correctly.
This calculator helps you test whether operand order changes the outcome of a mathematical operation. The idea is simple: evaluate the operation once as a op b and again as b op a. If the values match, the entered numbers satisfy the commutative check for that case.
The page also goes a step further. Some operations are commutative by definition, such as addition, multiplication, maximum, minimum, greatest common divisor, and least common multiple. Other operations are not commutative in general, including subtraction, division, and exponentiation. Because of that, two swapped inputs may occasionally match by coincidence, even when the selected operation is not commutative as a rule.
This distinction matters in algebra, arithmetic, logic design, programming, and proof writing. Students use it to simplify expressions. Teachers use it to demonstrate which transformations are valid. Developers use the idea when checking whether input order affects an algorithm or a helper function.
The calculator presents both direct outputs, a general property label, a difference value, a graph for visual comparison, and export options for quick documentation. The example table gives ready-made reference cases, while the formula and usage sections show how the property is tested step by step. Together, these features make the page useful for homework, revision, classroom demonstrations, and fast numeric checking.
The commutative property means operand order does not change the result. If a calculation gives the same value as a swapped version, the operation is commutative for that form.
Yes. For real numbers, integers, and many common number systems, a + b always equals b + a. Swapping the addends keeps the total unchanged.
Yes for ordinary number multiplication. Changing the order of factors does not change the product. That is why a × b equals b × a in standard arithmetic.
Subtraction depends on order. For example, 10 − 4 equals 6, but 4 − 10 equals −6. Because the results differ, subtraction is not commutative.
Yes. Some input pairs can produce equal swapped results by coincidence. That does not make the operation commutative in general. The rule must hold for all valid inputs.
Showing both orders makes the comparison direct. You can immediately see whether the numeric outputs match and how much difference the operand swap created.
Use integers for GCD and LCM. Those operations are defined for whole numbers in this calculator, so decimal entries will trigger a validation message.
The graph gives a quick visual comparison of the two evaluation orders. Equal bars suggest matching outputs, while different heights reveal a change after swapping operands.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.