# What axiom(s) assure(s) that Kirk and Holly both obtain the same sum whether added up or down the column?Kirk added a column of figures up while Holly added the column down.

kjcdb8er | Teacher | (Level 1) Associate Educator

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Both Kirk and Holly added the same numbers; the only thing that changed was the order in which they added them. In mathematics, an operation is commutative if changing the order of the numbers (or variables) does not change the end result. So, the axiom that assures that they both obtained the same sum is the axiom of commutativity.

Two elements of a set are said to be commutative under a binary operation if they satisfy x * y = y * x  and x + y = y + x

If these two conditions are met, then you have commutativity.

krishna-agrawala | College Teacher | (Level 3) Valedictorian

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There are five fundamental laws in or axioms in algebra that govern addition, subtraction, multiplication, and division operations. These laws are

3. Commutative Law of Multiplication
4. Associative Law of Multiplication
5. Distributive Law of Multiplication over Addition

The fact that figures in a column added from either top to bottom, or from bottom to top will result in the same sum is explained by two laws - the commutative law of addition and associative law of addition.

The commutative law of addition means that we can add two numbers in either order, and the sum will be the same. We can express this as:

x + y = y + x.

For example, 3 + 4 = 4 + 3 and (-7) + (-37) = (-37) + (-7).

The associative law of addition means that while adding several numbers, we can add any combination first, and the final sum will be the same.  We can express this as:

x + (y + z) = (x + y) + z.

For example, 3 + (4 + 5) = (3 + 4) + 5, or 3 + 9 = 7 + 5.