positive and negative numbers - i
- 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.
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
