I am reading the book Algebra for the practical man, chapter 1, article 6.
Uptil article-6 the author has explained the operations of addition on positive and negative numbers. Now in article-6 he is explaining(not defining) the multiplication using the concept of negative time.
- In the 4th point the author presumes that the multiplication of rate with time will give the distance moved.
In the light of this concept he concludes that $−ve×−ve=+ve $. The author states in the beginning of article-6:
We shall consider motion to the right of the starting point as positive ($+$) and motion to the left of the starting point as negative ($-$).
Then in the 4th point he says:
If the train is now at the starting point and has been traveling to the left, where was it 5 hours ago? Motion to the left $\times$ Past time $(- 40) \times (-5) = +200 $
Here ($-40$) is the rate at which the train moves and ($-5$) is time elapsed.
My question is:
- As the time changes from $0$ to $-5$ the train moves from the starting point towards right in the past, shouldn't we assume the rate of train positive and conclude$(+40) \times (-5) = +200 $?.
- If my interpretation is correct then does it mean that the author's assumption that
rate $\times$ time = distance is incorrect?
NOTE: Since division has not been defined uptil article 6 we cannot use the definition of division.