One important distinction between 0 and many other numbers is that it is impossible to divide by 0. Can you determine why this is impossible?

We can consider two cases, one where x equals 0 and one where x does not equal 0. Note 4

Case 1: x = 0

•  If x divided by 0 gives the quotient q, •  then q times 0 equals x. •  Remember that x equals 0 in this case. •  However, we know q times 0 equals 0 for any value of q.

•  If x0 = q, •  then q • 0 = x. •  So q • 0 = x = 0. •  So q • 0 = 0.

The equation q • 0 = 0 will be satisfied for any value q. Thus, there is no unique answer.

Case 2: x 0

•  If x divided by 0 gives the quotient q, •  then q times 0 equals x. •  Remember that x does not equal 0 in this case. •  However, we know q times 0 equals 0 for any value of q.

•  If x0 = q, •  then q • 0 = x. •  So q • 0 = x 0. •  But q • 0 = 0.

The equation q • 0 = x 0 will not be satisfied for any value of q. We can never multiply a number by 0 and get a non-zero answer, regardless of the value q.

In both cases, we were unable to find a unique value of q that could be the quotient; thus, we say that division by 0 is undefined.

Problem C2

Based on your own experience and on your reading of excerpts from Seife's Zero: The Biography of a Dangerous Idea from Session 1, write down whether you think 0 is the most important number in our positional system, and the reasons why or why not.

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