Inductive Reasoning

Learning Objective

Introduction

Inductive reasoningA form of logical thinking that makes general conclusions based on specific situations, inductive reasoning takes the path of observation to generalization to conjecture. is a sophisticated mathematical tool, yet we've all been using it since we were babies! When we use inductive reasoning, we use our experiences and observations to draw conclusions about what will happen in the future. The first several times we dropped something as a child, the object fell to the floor. Eventually, we decided this pattern would continue, no matter what the object was: things fall down. Inductive reasoning is a great way to discover new things in mathematics, too. But is it still that simple?

Generalizing and Making Conjectures

We'll start by looking for patterns in a diagram. Let's try to predict what the next three figures will be in this sequence:

The diagram shows figures in a pattern: red triangle, blue triangle; inverted red triangle, inverted blue triangle; red triangle, blue triangle; inverted red triangle.

To answer this, we have to take several steps, steps which make up the inductive reasoning process.

`1`.  First, observe the figures, looking for similarities and differences. In the example, there are two colors, red and blue, and they alternate. Also, all of the figures are triangles that appear to be the same size and shape, just turned differently.

`2`.  Next, generalizeThe process of using observations of specific events to make statements or conjectures about more general situations. these observations. When we generalize, we take observations about a few examples and assume that every other example will work the same. In this case, generalizing means we assume that the patterns will continue—for example, that the colors will continue to be red and blue, and that they will continue to alternate.

`3`.  Then, we form a conjectureA statement that attempts to make a conclusion but has not been proved true or false.. A conjecture is an attempt to make a conclusion about examples based on our generalization. Conjectures haven’t been proven to be right or wrong, and as long as we’re not just making a wild guess, all conjectures are worth thinking about. In this example, we can conjecture about the color and orientation of triangles we can't yet see.

`4`.  Finally, in some situations, we can apply our conjecture to make a prediction about the next few figures. We might predict the next triangles will be blue, then red, then blue again. If we did, we'd be right. Here are the next three figures in the pattern:

The diagram shows figures in a pattern: inverted blue triangle; red triangle, blue triangle.

We use the same steps when finding patterns in a table of data or a sequence, which is an ordered list of objects such as numbers or diagrams. Here's a numerical example:

Example

Problem

Write an expression for the `n`th term in the sequence `3, 5, 7, 9, 11`,... and find the value of the `100`th term.

 

- The terms are all odd numbers.

- Each term is greater than the one before it.

- The difference between one term and the next one is `2`.

 

First look for similarities and differences between the terms.

 

Assume that each term is going to be `2` more than the one before it. (They will all continue to be odd.)

 

Generalize the observations.

 

The next terms will be `13`, `15`, and `17`.

 

Term `1` is `3`.

Term `2` is `5`.

Term `3` is `7`.

Term `4` is `9`.

 

As the term position goes up by `1`, the term value goes up by `2`.

 

Try multiplying the term position by `2`:

Term `1` gives `2`.

Term `2` gives `4`.

Term `3` gives `6`.

 

Each of these is too low by `1`, so add `1`: Term `4` is `2*4 + 1 = 9`. That worked!

 

The `n`th term has a value of `2n+1`.

 

Write a conjecture. Since the problem asks for the `n`th term, we want an algebraic expression that connects the position of the term in the sequence to the term’s value.

 

 

 

 

 

 

 

 

 

The `100`th term has a value of `2(100) + 1`, or `201`.

 

Make the prediction requested by the problem. (This step may not always be needed.)

Answer

The `n`th term is `2n + 1`; the `100`th term is `201`.

 

 

 

Consider this sequence of numbers:

 

`100, 97, 94, 91, 88`,...

 

Make a conjecture: What is the `10`th term of this sequence? What is the `n`th term?

 

Testing and Revising Conjectures

An important thing to remember about conjectures is that they may or may not be right. As with any logical statement, take the time to look for counterexamplesA situation that provides evidence that a logical statement is false., and also verify that the examples on which the conjecture was based actually fit it, too.

Take a look at this example of a conjecture gone wrong:

Helen evaluated the polynomial `n^2 + 19n-19` for several values of `n` and noticed something about the results:

`N` 

`n^2+19n-19` 

`1`

`1`

`2`

`23`

`3`

`47`

`4`

`73`

`5`

`101`

Except for `1`, all the values in the `n^2 + 19n-19` column are prime. Helen generalized that this will happen for all values of `n` from `2` up, and wrote this conjecture: For any whole number `n` where `n>=2`the value `n^2 + 19n-19` is prime.

To test her idea, Helen checked another ten whole numbers, from `6` to `15`and the results were all prime! But a friend looked at the equation and said, You've got `19`'s in the expression. What if `n = 19`? Then you have `19^2 + 19(19)-19`.” Can you see that this expression would have a factor of `19`? The friend provided a counterexample to Helen's conjecture because when `n = 19` the expression has a value of  `703`which is not prime because `703 = 19*37`

This is one of the disadvantages of inductive reasoning: A conjecture found by inductive reasoning may or may not always be true. This is common when we overgeneralizeA logical mistake caused by basing a generalization on inadequate evidence or observation or by making too broad a conjecture, such as generalizing a pattern seen only in whole numbers to all real numbers., that is, when we use a small number of observations and try to apply them to a much wider situation. If all we can do is test individual examples, it's difficult, if not impossible, to say that there may not be a counterexample that we haven't found yet. For example, remember that childhood conjecture that objects will always fall when dropped? What if the object is a balloon filled with helium? Oops.

Finding a counterexample doesn't call for despair—we may be able to use the new information to revise our conjecture.

Example

Problem

Deshawn looked at the `x``x^2`, and `x^3` values below and decided that `x<=x^2<=x^3` for all real numbers. Find counterexamples and refine Deshawn's conjecture.

`x`

`x^2`

`x^3`

`1`

`1`

`1`

`2`

`4`

`8`

`3`

`9`

`27`

`4`

`16`

`64`

`5`

`25`

`125`

 

 

`x`

`x^2`

`x^3`

`x<=x^2<=x^3`?

`0.5`

`0.25`

`0.125`

No

`1.5`

`2.25`

`3.375`

Yes

`-1`

`1`

`-1`

No

`-1.5`

`2.25`

`-3.375`

No

`-0.5`

`0.25`

`-0.125`

No

 

 

 

 

 

 

The table only includes natural numbers for `x`, but Deshawn overgeneralized the results to all real numbers. The conjecture is true for some rational numbers (such as `x=1.5`), but is not true for all of them (such as `x = 0.5`). It doesn't appear to be true for any negative numbers.

 

On a coordinate plane, 3 equations are graphed: Y equals X, Y equals X squared, and Y equals X cubed. Y equals X is a red line with a positive slope that passes through (negative 1, negative 1), the origin, and (1, 1). Y equals X squared is a blue parabola that opens upward and passes through (negative 1, 1), the origin, and (1, 1). Y equals X cubed is a green curve that passes through (negative 1, negative 1), the origin, and (1, 1). It opens downward between (negative 1, negative 1) and the origin.  It opens upward between the origin and (1, 1).

 

To get a good look at when this inequality is true, graph the functions, `y = x` (in red), `y = x^2` (in blue), and `y = x^3` (in green).

 

 

 

A coordinate plane shows the same 3 equations and highlights the interval from X equals negative 1 to 1. On the interval form X is greater than negative 1 to X is less than 0, the X squared curve (blue) runs above the X cubed curve (green) and the X cubed curve runs above the X curve (red). All three curves pass through the origin and then the positioning changes. On the interval from X is greater than 0 to X is less than 1, the X curve (red) is on top, the X squared curve (blue) is in the middle, and the X cubed curve (green) is on the bottom. Then all three curves pass through (1, 1).

 

The inequality says the red should be below both the blue and green, and the blue should be below the green. Find the areas where this is true.

 

The red is below the blue when `x<0` and `x>1`. However, the green is below (or touches) the blue when `x<1`. So the only values when `x<x^2<x^3` are `x>1`

 

Answer

Counterexamples are `x = 0.5``x = -0.5`, and `x = -1`.

 

Revised conjecture is `x<=x^2<=x^3` for all real numbers of `x>=1`.

 

Since Deshawn's conjecture included `x = x^2 = x^3`, the value `x = 1` can be included in the revision.

         

Identifying Inductive Reasoning

Because conjectures based on inductive reasoning may or may not be true, it's important to recognize when inductive reasoning is being used to make a conjecture. Then we can think more about whether the conjecture is reasonable based on the observations, whether the conjecture overgeneralizes those observations, and whether there may be counterexamples that we should look for.

Example

Problem

Jeanne's birthday was on a Wednesday one year. She noticed that the next year it would fall on a Thursday, and it would fall on a Friday in two years. Without checking, she said, My birthday will be on a Wednesday again in seven years.

 

Is she using inductive reasoning? Why or why not?

 

Whether or not Jeanne is using inductive reasoning depends on the process she uses, not whether she is correct or not.

 

Jeanne has:

- Noticed similarities and differences in the day her birthday falls each year. For the three years she observed, she noticed that the day of the week was one later each year.

- Generalized her observations, assuming that the pattern would continue.

- Formed a conjecture. Although she didn't state it, exactly, her conjecture was: If my birthday is `x` years after my last birthday, it will fall `x` days later in the week than my last birthday did.

Notice that she even applied the conjecture to make a prediction (though that's not a necessary part of the inductive reasoning process).

 

 

Answer

Yes, she used inductive reasoning. She made a conjecture about a longer pattern by generalizing some specific observations.

 

       

(By the way, Jeanne's conjecture is incorrect. A leap year occurs every four years and throws the pattern off.)

When trying to graph `y = |-x|`, Benny reasoned this way:

 

Absolute value just makes the value positive or zero.

So, the result of ` |-x| ` is the positive version of `x`, or `0` if `x=0`.

But the result of ` |x|` is also the positive version of `x`, or `0` if `x=0`.

That means `|-x| = |x|`

So the graph of `y = |-x|` is the same as the graph of `y = |x|`.

 

Is Benny using inductive reasoning?

 

A) No

B) Yes

 

Summary

Inductive reasoning is a kind of logical reasoning that involves drawing a general conclusion, called a conjecture, based on a specific set of observations. In this process, specific examples are examined for a pattern, and then the pattern is generalized by assuming it will continue in unseen examples. Conjectures and predictions can then be made. Conjectures may not be true, especially if a pattern has been overgeneralized, that is, applied to a larger set of circumstances than the observations support. If counterexamples to a conjecture are found, it may be possible to revise the conjecture so that it becomes always true.