This answer is adapted from an entry in the sci.math Frequently Asked Questions file, which is Copyright (c) 1994 Hans de Vreught (hdev@cp.tn.tudelft.nl).

The first thing to realize about the system of notation that we use (decimal notation) is that things like the number 357.9 really mean "3*100 + 5*10 + 7*1 + 9/10". So whenever you write a number in decimal notation and it has more than one digit, you're really implying a sum.

So in modern mathematics, the string of symbols 0.9999... = 1 is understood to mean "the infinite sum 9/10 + 9/100 + 9/1000 + ...". This in turn is shorthand for "the limit of the sequence of numbers

    9/10,
    9/10 + 9/100,
    9/10 + 9/100 + 9/1000,
    ...."

One can show that this limit is 9/10 + 9/100 + 9/1000 ... using Analysis, and a proof really isn't all that hard (we all believe it intuitively anyway); a reference can be found in any of the Analysis texts referenced at the end of this message. Then all we have left to do is show that this sum really does equal 1:

   Proof: 0.9999... =     Sum         9/10^n 
                     (n=1 -> Infinity)

                    =  lim               sum      9/10^n
                     (m -> Infinity) (n=1 -> m)

                    =  lim           .9(1-10^-(m+1))/(1-1/10)
                     (m -> Infinity) 

                    =  lim           .9(1-10^-(m+1))/(9/10)
                     (m -> Infinity) 

                    = .9/(9/10)
                     
                    = 1

Not formal enough? In that case you need to go back to the construction of the number system. After you have constructed the reals (Cauchy sequences are well suited for this case, see [Shapiro75]), you can indeed verify that the preceding proof correctly shows

lim_(m --> oo) sum_(n = 1)^m (9)/(10^n) = 1

   0.9999... = 1

   Thus     x = 0.9999...  
          10x = 9.9999... 
      10x - x = 9.9999... - 0.9999... 
           9x = 9 
            x = 1.

Another informal argument is to notice that all periodic numbers such as

People have suggested many ways of picturing what is going on when a negative number is multiplied by a negative number. It's not easy to do, however, and there doesn't seem to be a visualization that works for everyone.

Debt

Debt is a good example of a negative number. One common form of debt is a mortgage in which you owe the bank money because the bank paid for your house. It is also common for an employer to deduct a mortgage payment from an employee's paycheck to help the employee keep on schedule with the payments.

Suppose $700 is being deducted each month to pay the mortgage. After six months, how much money has been taken out of the pay for the mortgage? We can figure out the answer by doing multiplication.

6 * -$700 = -$4,200

This is an illustration of a positive times a negative resulting in a negative.

Now suppose that, as a bonus, the employer decides to pay the mortgage for one year. The employer removes the mortgage deduction from the monthly paychecks. How much money is gained by the employee in our example? We can represent "removes" by a negative number and figure out the answer by multiplying.

-12 * -$700 = $8,400

This is an illustration of a negative times a negative resulting in a positive.

If one thinks of multiplication as grouping, then we have made a positive group by taking away a negative number twelve times.

This example may not work for you, and you might want to read others by following the related links below.

Visualizing isn't the same as understanding. Let's see how a mathematician might understand what's going on when a negative number is multiplied by a negative number.

A Mathematical Explanation

If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way:

   (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab 

For example,

    -2 * -3 = (-1)(2)(-1)(3)
 
            = (-1)(-1)(2)(3)

            = (-1)(-1) * 6

So the real question is,

   (-1)(-1) = ?

and the answer is that the following convention has been adopted:

   (-1)(-1) = +1

This convention has been adopted for the simple reason that any other convention would cause something to break.

For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn't work for negative numbers:

   (-1)(1 + -1) = (-1)(1) + (-1)(-1)
	
        (-1)(0) = -1 + -1

              0 = -2

As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth."

Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.