| Math Mojo - Making Math Meaningful |
This was the question:
What is the expanded form of a number, and can you give me an example of what it might be used for?
Professor Homunculus' answer:
Very good question. It is important
to know this one.
Expanded form (or "expanded
notation") simply means the number written out in terms of the digits multiplied
by their "columns."
Let’s see what that means:
The number 5 has only one column
(it is a one-digit number), and that column is the units, (or "ones")
column. So, 5 in expanded notation is 5 x 1. That means there are 5 "ones"
in the number.
So far, cake, right?
Let’s take a two digit number,
like 27. It has a 2 in the tens column, and a 7 in the ones column,
so, in expanded notation, it becomes 2 x 10 + 7 x 1. Sometimes, for clarity,
it is written is (2 x 10) + (7 x 1), but technically, it makes no difference.
It still means there are 2 groups of tens and 7 groups of ones in the number.
One more - a harder one.
398 = (3 x 100) + (9 X 10) + (8 x 1).
Make sense?
Expanded notation can be confusing, especially the way they may teach it in your school. For some clear information on what may be confusing to you, check out"
http://mathmojo.com/chronicles/2008/06/19/standard-and-expanded-notation/
Now the greater question is, "What is it good for?"
There are lots of things it is good for. I use it for speedmath (among other things). If you want to multiply, say 27 x 64, you can realize that that is the same thing as {(2 x 10) + (7 x 1)} x {(6 x 10) + (4 x 1)}. Of course I don't have to realize this every time I do a problem like this. I know expanded notation better than you know your multiplication tables, because I forced myself to learn it and get good at it. It didn’t take long (and I am not a mathematician or anything). So now it just comes naturally.
OK, so what does that big mathematical expression up there mean? Well, if you
also know your distributive
law, you can see that {(2 x 10) + (7 x 1)} x {(6 x 10) + (4 x 1)}
is the same thing as:
(2 x 6 x 100) + (2 x 10 x 4 x 1) + (7 x 1 x 6 x 10) + (7 x 1 x 4 x 1)
Let's break it down:
(2 x 6 x 100) (which you can easily do in your head and get 1200)
+
(2 x 10 x 4 x 1) (which is also cake to do in your head and get 80. Add that
to the 1200 and get 1280)
+
(7 x 1 x 6 x 10) (which, of course, you can also easily do in your head and
get 420, which you add, in your head to 1280 and get 1700)
+
(7 x 1 x 4 x 1) (which you can instantly see is 28, which added to 1,700 is
obviously 1,728.*
Have you noticed that it is also
done from left-to-right instead of doing the chump-change first?
This method of multiplication may not be interesting to you, but understanding expanded notation may be the single most important thing to understanding how our (and other) base systems work, and is the basic building block of understanding numbers and how to use them.
If you want to know a little about the method for multiplying above, check
out the lesson on left-to-right
multiplication. You never actually think about the steps above as
you do it. That is just used to explain why it works. The actual steps
are a lot easier than the lame way they teach you multiplication in school.
And faster. And most often quicker than a calculator.
* By the way, that happens to be just
one less than Ramanujan’s number. Do you know what it is, or who Ramanujan
was? Look it up! It is cool. (Well, for a nerd, maybe).
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