Positive and Negative Numbers - I
- Introduction to basic arithmetic operations with positive and negative numbers, emphasizing the importance of mental math skills for the SAT.
- Explanation of the concept that subtraction can be rewritten as the addition of a number with the opposite sign, offering a simplification strategy.
- Detailed walkthrough of handling double negatives and the reversal of subtraction order when factoring out a negative sign, to simplify calculations.
- Practical exercises and examples to reinforce understanding and application of these arithmetic concepts in various scenarios.
- Highlight on the universality of these techniques, applicable to integers, decimals, and fractions, and their significance in building a strong mathematical foundation.
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