乘积：奇数个负数

(Video) Integer Multiplication and Division

by Math Antics

Here’s a YouTube video by Math Antics that covers multiplying positives or zero, multiplying a negative by a negative, dividing positive integers, dividing negative integers, dividing a negative and a positive integer, multiplying a negative by a positive or zero, product of an even number of negatives, and product of an odd number of negatives.

$\large{\text{Summary}}$

The hardest part of any integer multiplication problem is deciding the sign of the final answer. Remember that negatives change the sign of a problem. For instance, a single negative in a problem will change the answer from a positive to a negative. He uses an example with $3 \times -1$ multiple times.

$$\begin{align} 3 \times -1 &= -3\\ 3 \times -1 \times -1 &=3 \\ 3 \times -1 \times -1 \times -1 &=-3\\ 3 \times -1 \times -1 \times -1 \times -1 &= 3\\ 3 \times -1 \times -1 \times -1 \times -1 \times -1 &=-3 \end{align}$$

Notice that all of the lines with an odd number of negatives have a negative answer. An odd amount of negatives in a multiplication or division problem will always yield a negative number.

However, the product a positive number and a positive number will always be positive. In fact, it doesn't matter how many you multiply together. For instance, the product of fifty positive numbers will be positive! This is the main difference from multiplying negative numbers together.

The product will always be negative.

Every time you multiply something by a negative number, you "flip" the sign of that number. For example, if you multiply a positive number by a negative number, the answer will be a negative number. If you multiply a negative number by a negative number, the answer will be a positive number.

You can think of this like flipping a light switch. The light always starts out on, and every time you see a negative number as you read an expression, you flip the switch (even if it's the very first number). If the light ends up on when you've gotten to the end of the expression, the result should be positive. If it's off, it should be negative.

Whenever there's an odd number of negatives, the switch will end up in a different position than it started in—off—so the answer will be negative.

Let's say you have $-1\times-1$ what does that equal? It equals $1$. Now what if you have $-1\times-1\times-1$? Well that's essentially just a $1\times-1$ which equals $-1$.

So two $-1$'s gave us a positive one, $1$

And three $-1$'s gave us a negative one, $-1$

And four $-1$'s will give us a positive one, $1$

And five $-1$'s will give us a negative one, $-1$

Do you see the pattern? An even number of $-1$'s gives us a positive one, $1$. This can be applied to any set of products. The product of an even number of negatives give us a positive number.

An odd number of $-1$'s gives us a negative one, $-1$. This can be applied to any set of products. The product of an odd number of negatives give us a negative number.

Example $$2\times-3\times-1\times-4\times-1\times-5\\$$

There are $5$ negative signs, so our answer is going to be negative.