Why negative of a negative is a positive ( " – ( – Number ) = + Number" ), and why the use of parenthesis matters in mathematical expressions?

A None-Mathematical approach for those of you who are still confused.

There are more than few explanations and theories for the mathematical operation “negative –-> negative is positive” or: " – ( – Number ) = + Number". However, I believe that a none-mathematical explanation is less confusing.

Some light hearted lines related to “negative –-> negative is positive”:

"Minus times minus results in a plus,

The reason for this, we needn't discuss."

- Ogden Nash (An American Poet).

The famous British detective Sherlock Holmes observed:

"When you have excluded the impossible, whatever remains, however improbable, must be the truth."

Note the following:

The actual real life use and applications of the negative –-> negative principle should not give you pause (" Why do I need to understand this? ") about learning and understanding the principle. There is a need and applications (particularly in science, applied mathematics and engineering) for these principles.

For example, in science or engineering, you will find negative numbers essential. Even when quantities we measure, like mass or pressure, are always positive, we need negative numbers to describe changes or differences or rates of change (like temperature), because quantities can both increase and decrease.

For now, think about Algebra and algebraic signs as a necessary exercise in logic and common sense.

Use of Parenthesis

Use of parenthesis is recommended to make mathematical expressions clear and unambiguous, particularly when positive and negative signs are included in the expression.

Example:

12 – – 5 = ? – – 5 is NOT a meaningful mathematical operation.

The expression should be written with parenthesis around – 5 to make the mathematical operation unambiguous.

12 – (– 5) = 12 + 5 = 17

The negative ->negative principle is simple and we will explain it by a few examples:

Start with Positive –-> Positive is Positive ( + (+ Number) = + Number = Number )

Example 1: Positive –-> Positive is Positive.

Suppose I state that “I am going to watch TV”. (A positive statement).

Then I state again “I am going to watch TV”. (A positive confirmation of the first statement).

I can combine the first and second statements by stating that “I am indeed going to watch TV” (“Indeed” is the positive confirmation of the “I am” statement).

Or:

I can combine the first and second statements by stating that “Of course I am going to watch TV” (“of course” is the positive confirmation of the “I am” statement).

Numerical example:

2 + 4 = (+2) + (+4) = 6

The plus sign to the left of the parenthesis is the positive confirmation to (+4). You can think about this plus sign as stating that “It is TRUE that the number inside the parenthesis is positive.

Example 2: Negative ->Negative is Positive.

Suppose I state that “I am NOT (NOT going to watch TV)”.

This is like stating that “It is NOT true (that I am NOT going to watch TV).

The first NOT is included within the parenthesis and it states that “I am NOT going to watch TV”. The second NOT (left of the parenthesis) negates or cancels out the first NOT. The result is a positive statement: “I am going to watch TV”.

Example 3: Negative –-> Negative is Positive.

Suppose I state that “I am NOT (NOT going to visit my friend)”.

This is like stating that “It is NOT true (that I am NOT going to visit my friend)”.

The first NOT is included within the parenthesis and it states that “I am NOT going to visit my friend”. The second NOT (left of the parenthesis) negates or cancels out the first NOT. The result is a positive statement: “I am going to visit my friend”.

Numerical Examples for Negative –-> Negative is Positive:

Example 1:

15 – (–3) = 15 + 3 = 18

The minus sign inside the parenthesis indicates negative number. The second minus to the left of the parenthesis negates the negative sign inside the parenthesis and make the operation positive. The second minus is like a statement “It is NOT true that the number inside the parenthesis is negative” and therefore – (–3) = +3 = 3. (Positive number).

Example 2:

–7*(– 4) = – ( – (7*4) ) = – (–28) = 28

The second minus to the left of the parenthesis negates the first minus sign inside the parenthesis and thus – (–28) = +28 = 28. (Positive result).

The minus sign to the left of the parenthesis negates the minus sign inside the parenthesis. Putting this in words: The minus sign to the left of parenthesis states that: “It is NOT true that the number inside the parenthesis is negative” and therefore – (–28) = +28 = 28. (Positive number).

Note again the use of parenthesis to make the mathematical operation unambiguous.

Example 3:

–35/–5 = – (35)/(–5) = – ( – (35/5) ) = – (–7) = +7 = 7

The second minus to the left of the parenthesis negates the minus sign inside the parenthesis and thus – (–7) = +7 = 7 (Positive number).

Again, note the use of parenthesis to make the mathematical operation unambiguous.

More about Positive and Negative operational rules:

The mathematical rules for the positive and negative signs operations are the following:

Positive –-> Negative Rules: Part 1

The Two Signs Rules:

"Two like signs make a positive sign,

two unlike signs make a negative sign"

“Like Signs” “Unlike Signs”

++ –-> + + – –-> +

– – –-> + – + –-> –

Positive –-> Negative Rules: Part 2

positive + positive = positive 2+3 = 5

negative + negative = negative – 2 + (– 3) = – 2 – 3 = – (2+3) = – 5

negative + positive = positive or negative depending on how big the positive or negative number is – 6 + 8 = 2, – 8 + 6 = – 2

positive * positive = positive 4*9 = 36

positive / positive = positive 16/8 = 2

positive / negative = negative 16/–8 = 16/(–8) = – (16/8) = – 2

negative / positive = negative – 24/6 = – (24/6) = – 4

negative / negative = positive – 29/–3 = (–29)/(–3) = – (– (29/3) ) = +( 29/3) = 7

negative * negative = positive –5*–6 = (–5)*(–6) = – (– (5*6) ) = – (–30) = +30 = 30

positive * negative = negative 28/ –7 = 28/ (–7) = – (28/7) = – 4

Positive –-> Negative Rules: Part 3

When you add two positives, you get a positive

when you multiply two positives you get a positive

when you multiply a positive with a negative you get a negative

when you multiply a negative and negative you get a positive

when you add two negatives you get negative.

when you divide two positives you get a positive

when you divide two negatives you get a positive

when you divide a negative and a positive you get a negative

Real life example of a negative number: Debt

Debt is a good example of a negative number. One common form of debt is a mortgage in which you owe the bank money because the bank paid for your house. It is also common for an employer to deduct a mortgage payment from an employee's paycheck to help the employee keep on schedule with the payments.

Yossi