Positive and Negative Numbers - I

Summary
The content provides a comprehensive guide on handling arithmetic operations involving positive and negative numbers, focusing on addition and subtraction techniques to simplify complex calculations.
• Introduction to basic arithmetic operations with a focus on adding and subtracting positive and negative numbers.
• Explanation of the concept that subtraction can be re-written as the addition of a number of the opposite sign, offering a simplification strategy.
• Detailed strategies for dealing with tricky arithmetic scenarios, such as double negatives and reversing the order of subtraction for easier calculation.
• Emphasis on the importance of practicing these arithmetic operations for proficiency, especially in the context of mental math to aid in smoother test-taking.
• The principles discussed apply not only to integers but also to positive and negative decimals and fractions, highlighting the universal applicability of these strategies.
Chapters
00:02
Understanding Basic Arithmetic Operations
01:07
Subtraction as Addition of Opposite Sign
03:01
Simplifying Complex Calculations
03:51
Practical Applications and Mental Math

Q: When we factor out the negative sign, do we multiply the whole equation by a negative?

A: Short answer: We are multiplying the *side of the equation that we factored the negative out of by (-1). (Not the whole equation; the examples in the video were not factoring out a (-1) from the whole equation.

We are of course not multiplying the whole side of the equation by a negative that has come from nowhere. We are taking out a negative sign from each term in the equation, and then we leave that negative sign outside of the terms as a coefficient. If we distribute the negative sign back to each of the terms, we should get our original expression. We can't forget to leave that negative that we've taken out of each term; otherwise we'll be changing the value of the equation, and we'll get the wrong answer.

We can do this with equations with numbers and with equations with variables:

• -12 - 37 = -49

Take out a negative from each term on the left-hand side, leave the right-hand side as it is:

• (-1) * [12 + 37] = -49

---> (-1) * 49 = -49

Check :-) Both sides of the equation are equal.  If we were solving it of course, we wouldn't know from the beginning that -12 - 37 = -49. This was to check that distributing out the negative from one side of the equation didn't change the equation.

Here's a case from the video with 2 positives, let's leave off the answer at first:

• 62 - 74 = ?

Pull out a negative from each term on the left-hand side, the right-hand side is unaffected:

• (-1) * [-62 + 74] = ?

---> (-1) * [12] = -12

So 62 - 74 = (-1) * [-62 + 74] = -12

And here's an example with a variable (I'm just using this as an example, it doesn't really make it easier to take out the -1 in this case):

• -10x - 24 = 66

---> -10x = 90

---> x = 90/-10  x = -9

• -10x - 24 = 66

--> -1(10x + 24) = 66

---> (10x + 24) = -66

---> 10x = -66 - 24

---> x = -90/10  x = -9

The mathematical reason why we have to put the (-1) as a coefficient for the terms we pulled it out of is because what we are actually doing when we pull out a (-1) from each term on one side is dividing that side of the equation by -1. In order to keep the value of that equation unchanged, we need to also multiply that side of the equation by (-1) to cancel out the division:

• -10x - 24 = 66
• (-1/-1) = 1

If we multiply one side of an equation by 1, that doesn't change its value. So we can do that here :-)

• (-1) * [(-10x/-1) - (24/-1)] = 66

--->* (-1) * [(10x) - (-24)] = 66

---> (-1) * [10x + 24] = 66

Same thing as -10x - 24 = 66