Multiplying Billions

Date: 12/17/97 at 09:38:22
From: Emily
Subject: Multiplication

How do you multiply 50 billion times 55 septillion?

Date: 12/19/97 at 10:58:30
From: Doctor Mark
Subject: Re: Multiplication

Hi Emily,

Well, the (very) short answer to your question is that you do it the 
same way you do any other multiplication. ;<)

But that's sort of a pain in this case, since you have *so* many 
zeros to keep track of, so let's see if there is another way. The 
explanation takes a little while, but if you understand it, you will 
be able to multiply lots of other big numbers very easily.

Suppose you wanted to multiply 100 by 1000.  You know that that is 
equal to 100,000, right? But notice something. 100 ends in 2 zeros, 
and 1000 ends in 3 zeros, and 100,000 ends in 2 + 3 = 5 zeros. Maybe 
there is a pattern here: let's try another product.

What's 100 times 100? Each of the 100's ends in 2 zeros, and 2 + 2 = 
4, so we might expect that 100 times 100 is equal to a 1 followed by 
4 zeros, 10000 = 10,000.  Is that right?  You could check, either by 
hand, or using a calculator, that it is.

So maybe we have a general rule here:

To multiply a number which is a 1 followed by m zeros by a number 
which is a 1 followed by n zeros (where m and n are some integers), 
just write down a 1, and then write down m + n zeros.

Let's see if it works for another example. Let's try 10,000 times 
100,000.

Our rule would say that this is a 1 followed by 4 zeros times a 1 
followed by 5 zeros, so the answer should be a 1 followed by 4 + 5 = 
9 zeros, i.e., 1000000000 = 1,000,000,000 (one billion). If your 
calculator allows you to multiply numbers that big, you can check 
that this is correct.

Let's look again at your original problem: 50 billion times 55 
septillion.

Hmmm.... neither of these is a 1 followed by a bunch of zeros, but 
maybe we can make it look like something like that. Let's look at a 
simpler example.

Suppose you wanted to multiply 20,000 by 1,400. You could just do it 
of course, but let's see if there is a simpler way.

You have to remember how to multiply an integer times a power of 10, 
like 100 or 10000000.  To multiply 73 time 100,000, for instance, just 
put 5 zeros (the number of zeros in 100,000) after the 73:  
73 x 100,000 = 73(00000) = 7,300,000.

We note, first, that 20,000 is the same thing as 20 x 1000, or (what 
turns out to be a better way of thinking of it) 2 times 10,000, and 
1400 is the same thing as 14 x 100.

So if we want 20,000 x 1400, we could do it like this:

20,000 x 1400 = (2 x 10000) x (14 x 100)

              = 2 x 10000 x 14 x 100 = 2 x 14 x 10000 x 100

Where we remembered that when you multiply numbers together, the order
doesn't matter (that's expressed by saying that "multiplication is
commutative"), so we could put the 2 and the 14 next to each other.

But 2 x 14 is just 28, so we have

20,000 x 1400 = [ 2 x 14 ] x 10000 x 100 = [ 28 ] x 10000 x 100

But now, using the rule we discovered, this is a piece of cake!

10000 has 4 zeros, and 100 has 2 zeros, so 10000 x 100 must be a 1 
followed by 4 + 2 = 6 zeros. So 10000 x 100 = 1(000000) = 1000000 = 
1,000,000. Just put our two results together, and we have the answer:

20,000 x 1400 = 28 x 10000 x 100 = 28 x 1000000.

But we find that product by writing down 28, then writing 6 zeros:

= 28(000000) = 28000000 = 28,000,000. (Whew!)

Let's summarize: to find 20000 x 1400, multiply the 2 and the 14 (to 
get 28), then add the number of zeros after the 2 (that would be 4, 
right?) to the number of zeros after the 14 (that would be 2, right?) 
to get 6 (since 4 + 2 = 6), and write those 6 zeros after the 28, to 
get 28000000, the answer.

So now, FINALLY, we are ready to answer your question: "What is 50 
billion times 55 septillion? "

Since you asked the question, I assume you know that 50 billion is:

  50 billion = 50,000,000,000; this is a 5 followed by 10 zeros

and that 55 septillion is:

  55 septillion = 55,000,000,000,000,000,000,000,000; this is a 55 
  followed by 24 zeros.

So, then, to get the product of 50 billion and 55 septillion, we 
multiply 5 by 55 to get 275, and then we add the number of zeros in 50 
billion (that's 10, right?) to the number of zeros in 55 septillion 
(that's 24, right?) to get 10 + 24 = 34, and write that many zeros 
after the 275:

50 billion times 55 septillion 

   = 2750000000000000000000000000000000000

   = 2,750,000,000,000,000,000,000,000,000,000,000,000

In words, that is

   2 undecillion 750 decillion.

What a huge number!

To test that you understand this, why don't you try to multiply 
6 billion times 300,000?  You should get 1800000000000000 = 
1,800,000,000,000,000, which is, in words, 1 quadrillion 800 trillion.

As you can see, it takes a while to figure out the answer to this even
after you know the trick. That's why, a long time ago, people got 
together and decided to use something called "Scientific Notation" for 
really big (and also, really small) numbers. You will learn about that 
when you study exponents in pre-Algebra or Algebra. Then you will find 
that the rule we discovered about adding the number of zeros together 
is related to the rule for multiplying exponentials.  Once you 
understand how to do that, you can figure out the answer to your 
question in about two LINES, which is a lot less than it took here.

Good luck, Emily, and write back if you have any other questions.

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/