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Learning Math: Number and Operations

Number Sets, Infinity, and Zero Part C: Examining Zero (50 minutes)

In This Part: The Behavior of Zero
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?

 


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. See Note 4 below.

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.


In This Part: Positional Number Systems

One of the most important roles 0 serves in our number system is as a placeholder. Without 0 or an equivalent placeholder, we would not be able to tell the difference between 102, 12, and 1,002. In this positional number system, we use zeros to indicate that there are no tens in the case of 102, and, similarly, that there are no tens or hundreds in 1,002.

To get a better understanding of what simple operations would be like without 0, try to solve the following problems using Roman numerals!

The values of Roman numerals are as follows: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, and M = 1,000. The numerals are written from largest to smallest and then added, with one exception: Writing a smaller number before a larger one means the smaller should be subtracted from the larger; this happens because four of the same numeral cannot occur consecutively in Roman numerals. In other words, IV (not IIII) represents 4; IX (not VIIII) represents 9; and XL (not XXXX) represents 40. The year 1066 is represented as MLXVI, while 1492 is MCDXCII.

You can quickly see that performing the above computations with Roman numerals is a nearly impossible task!


In This Part: Exploring Zero and Infinity on a Graph
We can further explore the number line and its elements through a graphic representation of an equation. For example, on such a graph we can visually locate an irrational number or demonstrate what happens when we try to divide with 0.

Let’s explore the graph of the equation x • y = 12.

Examine this graph. Answer Problems C3-C8 and corroborate your answers on the graph.

Here is the graph of the equation x • y = 12 in the first quadrant:


Problem C3
a. 
Would the point (2, 6) be on the graph? How do you know?
b. 
What about the point (24, 0.5)?
c. 
What about the point (-3, 4)?
d. 
Experiment with putting different numbers from the number line into the equation. What happens?


Problem C4
What y value would be paired with x = 4? How do you know?


Problem C5
a. 
What is the significance of the point of intersection of this graph and the line y = x?
b. 
Estimate the coordinates of this point.


Problem C6
Will the graph ever touch either the x- or y-axis? Explain.


Problem C7
What happens on the graph when x = 0 or y = 0? How does this demonstrate why you cannot divide a number by 0?

Video Segment
In this video segment, Andrea and L.J. explore whether the curve will ever touch one of the two axes. Professor Findell helps them resolve their dilemma. Watch this video segment after you’ve completed Problem C7.

Think about whether the graph will behave in the same way for all four quadrants.

You can find this segment on the session video approximately 21 minutes and 1 second after the Annenberg Media logo.

 


Problem C8
For a non-interactive version of this problem, look at the graph of x • y = 12 above. This graph shows only the first quadrant. Would there be points in any other quadrant? Does this change your answer to Problem C5? Explain.

For a non-interactive version of this problem, look at the graph of x • y = 12 above. This graph shows only the first quadrant. Would there be points in any other quadrant? Does this change your answer to Problem C5? Explain.

Notes

Notes 4
For division, we can say that 123 = 4 because 4 • 3 = 12. We use this reasoning to show why we cannot divide by 0, because anything multiplied by 0 will always result in 0.

Solutions

Problem C1
Some of the distinctive properties of 0 are as follows:

  • Any number added to 0 returns the original number (7 + 0 = 0 + 7 = 7). Zero is the only number that behaves this way.
  • Any number multiplied by 0 results in 0 (7 • 0 = 0 • 7 = 0). Again, 0 is the only number that behaves this way.
  • No number can be divided by 0 to return a unique real answer.
  • If two different numbers multiplied together make 0, one or the other must also be 0.

Problem C2
Answers will vary. Some important attributes may be that 0 is a placeholder that enables us to distinguish between different numbers and their place values, even if there is no value at a particular place; for example, 1,002 and 102. Without those zeros, we wouldn’t be able to tell which number is one hundred and two and which is one thousand and two. Zero also separates the positive from the negative numbers.

Problem C3
a. Yes. This point must be on the graph, since it satisfies the equation x • y = 12 (x = 2 and y = 6).
b. Yes. This point also satisfies x • y = 12.
c. No, this point is not on the graph because the product, x • y, yields -12 rather than 12. Similarly, any point in the coordinate system whose product of the x- and y- coordinates does not yield 12 will not be on the graph.
d. For each value on the x-axis that you pick, you would have a corresponding y value such that the product of the two always equals 12. As the x values on the x-axis get larger and larger, the y values on the y-axis get smaller and smaller. And vice versa, as the x values get smaller and smaller (i.e., approach 0), the y values get larger and larger. So if the x value is infinitely large, the corresponding y value will be infinitely small, and the product will still be 12. For this reason, the curve will never touch either of the axes. If it did touch one of the axes — let’s say the x-axis — it would mean that the y value is 0. Consequently, the product would equal 0, and that does not satisfy this equation.

Problem C4
The unique value is y = 3. We are looking to solve the equation 4 • y = 12. Since 4 has the inverse 1/4, we can multiply both sides by 1/4 to produce y. The solution is then y = 1/4 • 12 = 12/4 = 3.

Problem C5
a. This is the point at which x and y are equivalent while still solving x • y = 12. Since x and y are equivalent, we can change the equation to x • x = 12, or x2 = 12.
b. According to the graph, x and y must be between 3 and 4, since (3, 4) and (4, 3) are both on
the graph. An estimate of (3.5,3.5) would be a good guess. From the equation above (x2 = 12), we know that the x- and y-coordinates of this point are both. The actual coordinates are, to three decimal places, (3.464,3.464). A decimal for the exact coordinates can never be fully written, since  is an irrational number.

Problem C6
No, it cannot touch either axis. If it did, then x or y would be 0. In this case, it would be impossible to have x • y = 12, since multiplying by 0 always produces 0.

Problem C7
Look back at the solution for Problem C4; this solution depended on our ability to find a multiplicative inverse for the number 4. Zero is the only real number without a multiplicative inverse, so it is the only number where we cannot “divide” to find the other coordinate.

Problem C8
Yes, it should, as there are now two intersection points for these graphs. The other is a point that still satisfies x2 = 12. Since a negative number multiplied by itself equals a positive number, this suggests that the other x-coordinate is the opposite of the coordinate. To three decimal places, the coordinates are (-3.464,-3.464).

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Learning Math: Number and Operations

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Produced by WGBH Educational Foundation. 2003.
  • ISBN: 1-57680-678-2

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