Activity
In a study done by Margariete Montague Wheeler and Issa Feghali in 1983, preservice elementary teachers were given a test to see how well they understood the concept of zero. Before proceeding through this website, check your understanding by participating in the Desmos activity below. This activity contains 4 of the basic questions about zero that these preservice teachers were tested on (Wheeler, 1983). Keep your answers in mind as you proceed.
Desomos ActivityAddition, Subtraction, and Multiplication
Understanding the concept of zero as a number is crucial in understanding how to use operations containing zero. These operations include addition, multiplication, subtraction, and division. Although the idea seems simple to those that have already memorized the rules of working with zero, for children, using these basic operations with the number zero has proven to be confusing at first. Glenda J. Anthony and Margaret A. Walshaw asked children some simple questions to help determine whether it was the concept of zero that the students were struggling with or just the operations in general. They found that those that think of addition and subtraction of zero as “doing nothing to the number” it is being added to or subtracted from, have greater difficulty when it comes to multiplication by zero (Anthony, 2004). It appears that they don’t understand why zero does nothing to a number when it is added and subtracted (i.e. 1+0=1 or 1-0 =1), but then when multiplied by that number it changes it to zero (i.e. 1*0=0).
Division by Zero:
Although recognizing zero as a number helps with understanding how to add, subtract, and multiply by zero, there are extreme problems that even adults don’t fully understand when it comes to division by zero. Let’s take a look how we go about dividing a number by zero:
When we divide any number by zero, as we have seen, the answer is clearly undefined. As was mentioned in the video though, we are expected to divide by zero when we use differential calculus invented by Newton to determine the area underneath curves. It is fascinating to note that even though we cannot divide by zero, as we look at the numbers that are within epsilon of dividing by zero, it becomes possible to take the limit at any given point. We will explore a similar idea to this more in the significance and applications tab.
Now what would happen if we divided zero by zero. It may seem logical at first to say that the answer should be one because we have learned the “rule” that anything divided by itself is one. However, zero is the only number for which this rule does not make any sense. Since division is the opposite of multiplication, by saying that zero divided by zero is one we are also saying that zero times one is zero. This is certainly true, but what would happen if we set zero divided by zero equal to 2? 5? -42? N? Would they not also make true statements (i.e. 0*2=0, 0*5=5, 0*-42=0, and 0*N=0). Since these are all true statements, we call zero divided by zero an indeterminate. There is undetermined amount of solutions possible for zero divided by zero.
Exploring Other Weird Properties of Zero:
Dividing by zero is just one of the properties of zero that destroys the logical nature of mathematics. Zero seems to be the exception to the “rules” that we are taught in school. To get a better understanding of what I mean by this, let’s explore two seemingly simple rules: 1) any number multiplied by zero is zero and 2) any number raised to the zero power is 1. Recognizing that raising a number to a power is the same as multiplying that number by itself as many times as the power to which it is raised, brings up the question: What about zero raised to the zero power?
Use the applet below along with the questions that follow to explore why any number to the zero power is 1 and understand the seemingly simple question: “What is zero to the zero power?”
Part 1: Exploring the limits of n^0:
Make sure the function f(x)=a^x is on. If the sequence b^x is on, turn it off for now. Set the slider a equal to 1.5
1. What happens to the function as x approaches zero from the left?
2. What happens to the function as x approaches zero from the right?
Move the slider so a changes value.
3. How do the functions differ?
4. How are they the same?
Set the slider a equal to 0.
5. What is the limit as x approaches zero?
Part 2: Exploring using sequences of n^x:
Turn off the f(x)=a^x and turn on the sequence b^x. Move the slider b so that b changes value.
1. Set a number for b, and write down the sequence of number produced by b^x:
2. What happens to the numbers in the sequence as x increases (moves to the right)? What happens to the numbers in the sequence as x decreases (moves to the left)?
3. What do you notice when x=0 in the sequence? What can we infer about any number to the zero power from this observation?
4. Is this the same or different from the answer you got in the last section question 5? Write a conjecture for why you think that may be.
Part 3: Exploring n^0 algebraically:
1. What is 34/34 in its most simplified terms?
2. Now simplify 34/34 using exponential rules and show your steps. What conjecture would you make about any number to the zero power?
What about 0^0?
Change a so that it equals zero, if it’s not already. Click the “set b=a” button.
1. What does this say about 0^0?
2. Based on what you discovered today, what conclusions can you make about 0^0?
Conclusion:
Clearly zero is an exception to the “rules” that we have learned in school. To find more puzzling questions about zero and other mathematical topics, check out the following link: Frequently Asked Questions in Math