Induction

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INDUCTIVE REASONING<BR>

INDUCTIVE REASONING

Inductive reasoning is the method of drawing general conclusions from a limited set of observations. It is reasoning from the particular to the general.

QUESTIONS

 
1             = 1 x 1

1 + 3         = 2 x 2

1 + 3 + 5     = 3 x 3

1 + 3 + 5 + 7 = 4 x 4

Write the next equation in this pattern.
Is the equation true?
Write what you think the seventh equation of the pattern should be.
Is it true?
Write what you think the tenth equation of the pattern should be.
Is it true?
Do you think that the pattern goes on indefinitely?
What kind of reasoning are you using?

GALILEO'S LAW

In the sixteenth century, the great Italian scientist Galileo used inductive reasoning to make several discoveries about the behavior of swinging weights—discoveries that led to the invention of the pendulum. One of these discoveries was of a relation between the length of the pendulum and the time of the swing.

This table lists the swing times of a series of pendulums having different lengths.
```
Length of Pendulum  Time of Swing

1 unit              1 second

4 units             2 seconds

9 units             3 seconds

16 units            4 seconds
```
From the pattern in the table, how does the length of the pendulum seem to be related to the time of the swing?
What do you think the length of a pendulum with a swing time of 5 seconds would be?
What do you think the length of a pendulum with a swing time of 10 seconds would be?

BOYLE'S LAW

In 1661, the English chemist Robert Boyle did a series of experiments with the pressure of the air. One of his discoveries, through inductive reasoning, was a relation between the volume of a gas and the pressure it exerts. The table below lists some volumes and pressures for a gas at a given temperature.
```
Volume      Pressure

1 unit      120 units

2 units     60 units

4 units     30 units

8 units     15 units
```
What happens to the pressure as the volume of the gas gets larger?
What happens to the pressure when the volume is multiplied by 2?
What happens to the pressure when the volume is multiplied by 4?
On the basis of your answers to exercises 5 and 6,
what do you think would happen to the pressure if the volume is multiplied by 3?
What do you think the pressure would be when the volume is 3 units?
What do you think it would be when the volume is 16 units?

BODE'S LAW

In 1772, the German astronomer Johann Elert Bode used inductive reasoning to find a pattern in the distances of the planets from the sun. At that time, only six planets were known. The actual relative distances of the planets from the sun and his pattern are shown in the table below. Distances are based on taking the distance from the earth to the sun to be 10 units.


Planet  Actual Distance     Bode�s Pattern



Mercury 4                    0 + 4 =  4

Venus   7                    3 + 4 =  7

Earth   10                   6 + 4 = 10

Mars    15                  12 + 4 = 16

                             ? + ? =  ?

Jupiter 52                  48 + 4 = 52

Saturn  96                  96 + 4 = 100

                             ? + ? =  ?

Notice that there is a pretty good match between the two sets of numbers.

What equation do you think belongs between Bode�s equations for Mars and Jupiter?
What equation do you think belongs after Bode�s equation for Saturn?

In 1781, William Herschel discovered Uranus, the next planet beyond Saturn.
Because its distance of 192 units comes remarkably close to the number predicted by this equation, astronomers came to the conclusion that the equation between Bode�s equations for Mars and Jupiter also must mean something.
What do you suppose they thought it meant?

In 1801, the asteroid Ceres was discovered at a distance of 28 units from the Sun.

Sun

THE LIMITS OF INDUCTIVE REASONING

Inductive thinking, of course, does not always lead to true conclusions.
All Arabs are NOT terrorists, but tell that to airline security.

The first modern Olympics Games were held in Athens in 1896.
The following games were held in 1900, 1904, 1908, and 1912.
1. What pattern do you see in these dates?
2. What method of reasoning did you use in arriving at this pattern?
3. According to the pattern, what year would seem to be the date of the next Olympics after 1912?
4. Do you think this date is correct? Why?
Do the indicated calculations to find the numbers that will make the following equations true.
1. 1 x 9 + 2 =
2. 12 x 9 + 3 =
3. 123 x 9 + 4 =
4. 1,234 x 9 + 5 =
5. Guess the next equation in this pattern.
6. Check to see whether it is true.
7. Write what you think the ninth equation in this pattern should be.
8. Check to see whether it is true.
9. Do you think that the pattern goes on indefinitely?
The numbers in this list are all prime.
That is, each number is evenly divisible by only itself and 1.

31
331
3,331
33,331
333,331
3,333,331
33,333,331
1. What is the next number in this list?
2. If you conclude that it is also prime, what kind of reasoning are you using?
3. Divide the number that you chose in exercise 14 by 17.
4. What does this tell you about the number you chose?
5. What does your result show about inductive reasoning?
Here is a set of three circles. Two points have been chosen on the first circle and a straight line segment drawn between them. The circle is separated into two regions as a result. Three points were chosen on the second circle and connected with three line segments to form four regions. Four points were chosen on the third circle and, after being connected in all possible ways, eight regions resulted.
1. Guess from the pattern in the second line of numbers what the missing numbers are.
2. Choose five points on one circle and six points on the other.
  Click on the circle to choose a point.
  Click on the circle again to make a line.
  Join the points of each circle in every possible way.
  Then number each area by pointing to the area and typing a number for that area.
  To delete a digit, point at it and press the BACKSPACE or ESC key.
3. How many regions are formed in each circle?
  Check your drawing by asking yourself
  "How many lines should be drawn from each point?"
4. Do both results agree with your guesses?

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Last Modified 12/10/08 4:53 PM

Induction

QUESTIONS

GALILEO'S LAW

BOYLE'S LAW

BODE'S LAW

THE LIMITS OF INDUCTIVE REASONING