why is a negative multiplied by a negative a positive
It's best to think, perhaps, of the negative sign as a "change in direction". The default direction is to the positive end of the number line (to the right). So $3times 2$ moves us six units to the right, to land at $6$. $-3times 2$ changes direction and moves us $6$ units to the left of zero, to land at $-6$. $-3 times -2 = -(-3 times 2)$ reverses our direction once again, taking us back to the right six units from zero, the opposite direction than does $-3 times 2$ in the amount of $6$ units to the right. So in the end, we are at the "same position" whether we use $3times 2$ or $-3times -2$. Note that our use of arithmetic, in a practical sense, depends on the context in which we are applying it. So your example is just a poor context to apply multiplication of negative numbers, since, of course, it seems absurd to think of having $-3$ cows. But negative numbers can represent position, with respect to some point of origin, as I noted in the number line example above. And time can be represented in terms of the past (negative time), now (the point of reference), and the future (positive values for time). Similarly, in finance, negative numbers can represent debt or loss, while positive numbers represent profit, or gain. I'm sure if you put your thinking cap on, you'll see that mutltiplication of negative numbers certainly can, and does make sense.
times a negative number a positive number? This is a common question by many students taking mathematics courses in middle school and even high school.
It is not surprising to most that this is difficult concept for students to understand. Negative numbers are not easily understood by most people. The difficulty with understanding a negative times a negative is that this is not something we do in our everyday lives. Below are several methods that may be used to help students better the meaning of a negative times a negative. We can start of explaining negative multiplication by helping the student understand the easier concepts such as a positive times a positive or a positive times a negative. For example in the case of money, we can represent a positive times a negative by saying $700 is being deducted each month to pay ones mortgage payment. After six months, how much money has been taken out of the pay for the mortgage? We can figure out the answer by doing multiplication. 6 * -$700 = -$4,200. This is an illustration of a positive times a negative resulting in a negative. Now suppose that, as a bonus, the employer decides to pay the mortgage for one year. The employer removes the mortgage deduction from the monthly paychecks. How much money is gained by the employee in our example? We can represent removes by a negative number and figure out the answer by multiplying. -12 * -$700 = $8,400 This is an illustration of a negative times a negative resulting in a positive.
If one thinks of multiplication as grouping, then we have made a positive group by taking away a negative number twelve times. Most students are quick to agree that a negative number can be represented as a number times -1. a positive is easier to understand. = -x. (-x) (-y) = (-1) (x) (-1) (y) = (-1) (-1) (x) (y), So what is (-1 -1)? First we will start with things that we know. For example, we know that -1(0) = 0. 0) = (-1)(-1 + 1), then using the distributive property on the right side of the equation we get: Now we know that (-1)(1) = -1, but we arent sure what (-1)(-1) is, but we do know that whatever it is must be the equation is equal to zero, so since it cant be -1 for that would make the equation equal to -2, then it must be +1. See the math below 0 = (-1) (0) = (-1)(-1 + 1) = (-1) (-1) + (-1) (1) =? + (-1) therefore =? + (-1), from our statement above (-1 -1) must be +1 to complete the statement: 0 =? + (-1). Which may help to conclude that a negative times a negative equals a positive. b be any two real numbers. Consider the number x defined by b) + (-a)(-b). x = ab + (-a)[ (b) + (-b) ] (factor out -a) = ab + (-a)(0) = ab + 0 = ab. Also, x = [ a + (-a) ]b + (-a)(-b) (factor out b) = 0 * b + (-a)(-b) = 0 + (-a)(-b) = (-a)(-b). Hence, by the transitivity of equality, we have = (-a)(-b). Some people think of the word NOT as a negative meaning. One might say I am NOT going to my friends house.
This seems like a negative version of saying I AM going to my friends house. So what if I said this with two NOTS. I am not going to not go to my friends house. It seems the two NOTS cancel each other out and I am going to my friends house is derived. This seems that a double negative statement really derives a positive statement. look at the sequence below. notice that the numbers are going down by 5. -4 x 5 = -20. do this same kind of sequence replacing 5 with -5. notice that the numbers are going up by 5. -4 x -5 = 20. a negative is again a positive. We can use the coordinate system in Geometers Sketchpad to create a line. Remember that the slope of a line is rise over run. If we plot a point in the coordinate plane and use a slope to plot a second point we will look at the slope to determine if the line has a positive slope of a negative slope. In the following diagram we plotted a point (2, 2): Lets use a slope of 1/2 to find the next three points. Now if we construct a line through the points we will see that the result is a line with a positive slope. Lets try this same thing starting at the point (2, 2) and using the slope -1/-2. See the diagram below for results. Notice that the line is the same as before, a positive slope. Therefore this could be a graphical approach to showing the relationship of two negative numbers. www. ncsu. edu plato. stanford. edu
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