Multiplying integers is just like the multiplication of whole numbers, except that with integers, you have to keep tract of your signs.
Recall that 6 + 6 + 6 = 6 × 3
Instead of adding 6 three times, you can multiply 6 by 3 and get 18, the same answer.
Similarly,
6 + 6 + 6 + 6 + 6 + 6 + 6 = 6 × 7 = 42
Still by the same token,
2 + 2 + 2 + 2 = 2 × 4
In algebra, 2 × 4 can be written as (2)(4)
You can think of this as four groups of 2
This situation is shown in the number line below.
You basically start at 0 and count by 2's until you have put four 2's on the number line. You end up at 8 and 8 is positive.
Representation of 2 times 4

Notice that 2 and 4 are positive.
In general,when multiplying integers, remember the followings:
Positive × Positive = Positive
For example,
7 × 6 = 42
2 × 5 = 10
3 × 10 = 30
8 × 2 = 16
Now, try adding 3 to 3
 3 + 3 = 3 × 2
The reasoning is the same; Instead of adding 3 two times, you can just multiply 3 by 2.
To model this on the number line, just start at 0 and put 2 groups of 3 of the number line. You end up at 6 and 6 is negative.
Representation of 3 times 2

Notice that 3 is negative and 2 is positive
In general,
Negative × Positive = Negative
For example:
8 × 5 = 40
6 × 5 = 30
4 × 2 =  8
3 × 4 =  12
Now what about 3 × 2 ?
Start at 0 and put three groups of  2 on the number line and you will end up at 6 as shown below:
Representation of 3 times 2
Notice that this time 3 is positive and 2 is negative, yet we still get 6. 
2 × 1 = 2
2 = 2
Since 2 is never equal to 2, we have found a contradiction.
Therefore, The asumption that 2 × 5 = 10 is false
We must have 2 × 5 = 10.
Notice that 5 divided by 5 is 1 because 5 × 1 = 5 or 5 goes into 5 once.
10 divided by 5 is 2 because 5 × 2 = 10 or 5 goes into 10 twice.
The proof above is not a real proof because it takes into consideration specific numbers. A good proof is always generic. A generic proof is beyond the scope of this topic, so let us be happy with that at least for now.
To sum up, here is the big picture when multiplying integers :
The product of two numbers with the same signs will be positive.
The product of two numbers with different signs will be negative.
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Oct 24, 16 11:10 AM
Explanation of the ambiguous case of the law of sines.