The Math Behind Why Multiplying Two Negative Numbers is Positive.

And a case where (-n) does not turn positive if multiplied

The short answer is quite straightforward. We follow the order of operations and observe that numbers within groups will produce additional negatives which cancel each other out, but those that are not grouped between parentheses will not be associated with any other operator.

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If a group (nested mathematics between parenthesis) contains negative operators, and is multiplied, then it will multiply the number of negative operations as well equal to the multiplying integer. If an operation is performed directly on a number, no other operators are evaluated. If the operation is performed on a collection of integers enclosed in parenthesis, everything changes and is abruptly affected.

Contrary to common belief:

-5² does not equal 25. It is -25, yet (-5)² equals 25. This is because all operations work directly onto the number, and not actually the number and the operator behind it.

Upon investigation, I found that some educators claim that parentheses do not change any sort of outcome and can be factored out without any consequence. But this is wrong. The deepest nested groups are worked on first, so parentheses within parentheses, within parentheses would go through all order of operations before it would get to the next outside layer of parentheses.

-5 * 2 = -10
0-5 * 2 = 0-10 = -10

-5² = -25
0-5² = 0-25 = -25

(-5) * 2 = 10
(-5)(-2) = (- -5)(2) = (5)(2) = 10

(-5)² = 25
(-5)(-5) = (- - 5)(5) = (5)(5) = 25

Precedence Order: Follow the standard mathematical order of operations, also known as PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders (i.e., powers and roots), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Operator Isolation: Operators (+, -, *, /, ^) must not directly affect each other and should only be applied to operands, including numbers and parentheses-enclosed expressions.

Why does: 2 ⋅ -111-55 ⋅ 2 equal -332 instead of -664?

Because nothing is in parentheses. The multipliers do not affect the negative operators, and thus the equation is solved:

2 ⋅ -111 = -222
-55 ⋅ 2 = -110
-222–110 = -332

Summation:

1. The only time an operator can affect another operator is if the target operator are parentheses. You cannot use these operations with each other +-*/^, may only use them with +-*/^, ( n )
2. All nested equations must be completed before moving up a level: Must go in the correct order every time: Parentheses/Brackets, Exponents/Orders (i.e., powers and roots), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
3. Negative Signs in Multiplication: If a negative number in parentheses is multiplied, its negative sign is considered part of the multiplication operation. When simplifying expressions, negative signs are paired and canceled out, adhering to the rule of signs.
4. Multiplication and Division of Negative Numbers: When a negative number is involved in multiplication or division, its sign is integral to the operation. The rule of signs applies: a negative times a positive equals a negative, a negative divided by a positive equals a negative, and similarly for negative/negative combinations.
5. Multiplication, division, and exponentials do not affect the positive or negative value of a number unless it affects the entire area within parentheses.
6. Sequential Resolution: Nested expressions within parentheses must be completely resolved according to the order of operations before integrating their results into the broader expression.
7. Parenthetical Influence: Parentheses can modify the application of operators, particularly in altering the precedence of operations within them.
8. Exponentiation of Negative Numbers: The sign of a negative base raised to an exponent depends on the exponent’s parity (even or odd). This must be considered if the base is within parentheses.
9. Misinterpretation Avoidance: Be mindful of how calculators interpret operations, particularly with negative numbers and parentheses. Adhering strictly to the clarified rules helps avoid discrepancies between manual calculations and calculator results.