Order is essential in the definition of multiplication because not all forms of multiplication are commutative, such as matrix multiplication. This is why it is taught as a separate property.
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Why Was 5 x 3 = 5 + 5 + 5 Marked Wrong?
It seems absurd at first glance: we all know that 5 x 3 is equal to 3 x 5, which is 15. But check out the formal definition of multiplication:
The multiplication of two whole numbers, when thinking of multiplication as repeated addition, is equivalent to adding as many copies of one of them (multiplicand, written second) as the value of the other one (multiplier, written first).
In other words, 5 x 3 = 3 + 3 + 3 + 3 + 3
Why should this matter? It matters because the term equal is not the same as equivalent. Although 5 x 3 is equal to 5 + 5 + 5 it is not equivalent to 3 + 3 + 3 + 3 + 3. Suppose you were buying chocolates for your sweethearts on Valentine’s Day. You would have 3 boxes of 5 chocolates each in one case, and 5 boxes of 3 chocolates in the other case. What you choose to buy depends on how many sweethearts you are trying to impress, right?
Perhaps more importantly, the difference is also a fundamental concept in computer science.
Notice that the second problem is marked incorrect as well. That’s because keeping rows and columns straight in matrix multiplication is important. As explained in the link below: “Order is essential in the definition of multiplication because not all forms of multiplication are commutative, such as matrix multiplication. This is why it is taught as a separate property.”
So, what do you think? Do you agree with the teacher or not?