Why can't I divide by zero?

We tell students they can't do it, but do they really understand why?

Objective: Students will use various methods to demonstrate why division by zero can't be done.

Reformatted: January 2010

Method 1: Word problems with diagrams

What does division mean it terms of every day experiences?

Write simple word problems to demonstrate what division means. Build from easy to understand examples. Be sure to show examples in both orders so students understand the meaning of the order. Have the students try to draw a diagram for each.

 a: You have 2 pizzas and you want to split them between 6 people. How much does each person get? 2 / 6 = 1/3 : each person gets 1/3 pizzas. (pictured here ->) b: You have 6 pizzas and you split them between 2 people. How much does each person get? 6 / 2 = 3 : each person gets 3 c: You have 0 pizzas and you split them between 2 people. How much does each person get? 0 / 2 = 0 : each person gets nothing d: You have 2 pizzas and you split them between 0 people. How much does each person get? 2 / 0 = undefined : where are the people to eat the pizza?
Example 2:
 You have a 3 meter long board and you want to cut it into pieces that are X cm long. How many pieces will you have? 3 / 0 = ?? How many 0 cm pieces will be needed to make 3 meters. What will those pieces look like? You have a 3 meter long board and you want to cut it into X equal pieces. How long will each piece be? 3 / 0 = ?? How can you take an existing item and reduce it to zero pieces? If these questions are unanswerable, then the formula 3 / 0 will also be unanswerable or undefined.

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Method 2: The limit using a simple pattern

Vocabulary: Limit is a term used in calculus meaning to approach - get real close to - with out going the whole way to that value.

What does a pattern imply if we approach division by zero?

Use a simple pattern where the numbers do not confuse the students. Let them determine the answers and then try to interpret where that pattern will take them as they approach division by zero.

 100 / 100 = 1 100 / 10 = 10 100 / 1 = 100 100 / 0.1 = 1000 100 / 0.01 = 10000 100 / 0.001 = 100000 In this pattern the numerator is staying the same while the denominator is getting closer to 0. The answer is increasing rapidly. Where will this result go and we approach division by zero?

Method 3: The limit using a graph If your students are familiar with graphs, generate the graph of y=24/x. Ask them to determine where does the y-value go as x approaches 0? How is the approach from the negative side different from the approach on the positive side? Are the negative and positive answers the same? Are they finite?

Method 4: Inverse Operations

Division is the inverse of multiplication. If we can multiply, then we can divide by reversing the numbers. What does this relationship imply for us? Show easy relationships to set up the pattern first:

 2 * 7 = 14 14 / 7 = 2 14 / 2 = 7 10*52 = 520 520 / 52 = 10 520 / 10 = 52 0 * 5 = 0 0 / 0 = ?? 0 / 5 = 0 8 * X = 32 32 / 8 = X = 4 X can only be 4 0 * X = 0 0 / 0 = X X can be any answer

Special Case Challenge:

1. Pick two numbers and perform an impossible operation with the result being a variable.
2. Show the inverse operation
3. Define an X to make your statements consistent

Examples:

 Creating Negative Numbers 5 - 8 = X you were told you can't subtract a large from a small. X + 8 = 5 show the inverse -3 + 8 = 5 define into existence negative numbers to make these operations work
 Creating Imaginary Numbers X = sqrt(-1) you were told you can't take the square root of a negative X2 = -1 show the inverse i2 = -1 define imaginary numbers to make this work
 Creating Division by Zero 5 / 0 = X show division by zero 0 * X = 5 show the inverse  can you define a system of numbers that will make these equations make sense? How? or Why not? Readings: Dual Numbers chapter of A Mathematical Mosaic 