Well, of course, having done Addition and Subtraction, Multiplication and Division has to be next. I've said it before, and I'll say it again. Mathematicians are among the laziest people on the planet. Personally, I have always had a good handle on the lazy part. Still working on the mathematician part. Anyway, most of the rules in mathematics involve making things easier. So, I've you're ever wondering - "Why is it done this way?" The answer is easier to find if you think in terms of how did this save them time and/or effort.
Multiplication and Division are two sides of the same coin. 20 times 3 is 60 and 60 divided by 3 is 20, and 60 divided by 20 is 3. Any two numbers you can multiply together, can always be divided apart. And life would be just wonderful if that were always true. So, of course there's exceptions.
Actually, there's only one exception - but, it's a doozy. In 773 some guy named Mohammed ibn-Musa al-Khowarizmi started using zero as something more than a place holder. There are people in this world who get violently angry when I talk about history in what's supposed to be a math lesson. So, if you want elaboration click here - zero.
Anyway, now that we have zero - things will never be the same. For instance, you can't divide by 0. Sorry, it's just not allowed. 3/0 has no answer. I like to take three marker and set them on a desk. Then I show the class that I can divide those 3 markers by 3. I take the 3 markers away 1 time. I put the markers back and divide by 1. I take 1 marker away, I take another marker away, and I take the third marker away. Taking 1 marker away at a time gets me all the markers in three attempts. Now, put the three markers back on the table. How many attempts will it take to get all the markers if I take none away each time? Hopefully you notice that even if I take infinite attempts - the three markers remain.
You can multiply by zero but you are not allowed to divide. This is the cardinal rule of Multiplication and Division.
Multiplication was invented because people were tired of adding the same number over and over again. You 20 guys owe me 3 coins for taxes. So, that means I will have collected 3 + 3 + 3 + 3 + 3 + 3 + 3 = 3 ..., oh the heck with it, lets call it 20 x 3! And multiplication was born. In grade school we used an x for multiply. Most calculators also use an x. But, we've been pushed into using something else now that they've started using an x to signify a variable. Let's look at the expression 3 times x. We can write that as 3 times x (obviously), or 3 * x. We use 3 * x because most keyboards won't let us put a dot between the 3 and the x without it looking like a decimal point. But 3 dot x is another way of writing a multiplication. We could also write (3)(x), or 3(x), or (3)x, or even 3x and everyone is supposed to know that all of those mean 3 times x.
Wait a minute! Did you catch the part where I said 3x is the same as 3 times x? So does that mean that 3 1/2 means 3 times one half? No - that would be too easy. When a fraction is next to a number it is considered added to that number. So, 3 1/2 is the same as 3 + 1/2 or 3 plus one half, or just 3 and a half. I don't like it either, but like it appears to be with so many things, they acted without consulting with me first.
So, then there's division - the opposite of multiplication. Division is also written a whole bunch of ways. You've probably seen the minus sign with a dot above and below it. I don't know how to type it, but here's a picture...
So, we'll call the symbol between the 6 and 3 the division sign. We could have written that as 6/3. We call it division, or a fraction, or a ratio or one of several other terms depending on the application.
I may be a bit repetitive, but so many people have a problem with this that I'm going to explain it again. What would a discussion of Multiplication and Division be without talking about Dividing the Divided? Division in a problem like (1/3)/(1/5) which would read one third divided by one fifth. All you do is invert or flip the second fraction and then multiply it by the first. So (1/3)/(1/5) becomes (1/3)*(1/5) or 1 times 1 for the numerator and 3 times 5 for the denominator. The end result is the fraction 1/15.