Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 2, Part C:
Examining Zero (50 minutes)

In This Part: The Behavior of Zero | Positional Number Systems
Exploring Zero and Infinity on a Graph

 The real number system is a positional system. In such a system, the position of a number within a string of numbers -- its place value -- is meaningful. One of the significant elements of this positional system is the number 0. Problem C1 Make a list of the characteristics of the number 0 that make it different or significant in relation to other numbers.

 Does it behave like other numbers in relation to operations?    Close Tip Does it behave like other numbers in relation to operations?

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.

 Session 2: Index | Notes | Solutions | Video