Math Core

In the last unit you measured segments and angles and noticed how they fit together. Now you'll learn how mathematicians decide that something is always true. There are two very different ways to reason: you can spot a pattern and guess, or you can start from facts you already accept and follow the logic to a conclusion that can't be wrong.

Inductive reasoning: looking for patterns

Inductive reasoning means looking at specific examples, finding a pattern, and making a general guess. That guess is called a conjecture.

Suppose you add pairs of odd numbers:

1 + 3 = 4 \qquad 5 + 7 = 12 \qquad 11 + 9 = 20 \qquad 15 + 21 = 36

Every sum is even. A reasonable conjecture is: the sum of two odd numbers is always even.

Inductive reasoning is how most discoveries start. Scientists, detectives and mathematicians all use it. But it has one big weakness: no matter how many examples agree with a conjecture, the next example might not.

Definition

Conjecture

A conjecture is an unproven statement based on observations. Inductive reasoning produces conjectures, not certainties.

Counterexamples

To show a conjecture is false, you only need one example where it fails. That example is called a counterexample.

Conjecture: "For every number $x$ , $x^2 \ge x$ ." Try a few values: $3^2 = 9 \ge 3$ , $(-2)^2 = 4 \ge -2$ , $1^2 = 1 \ge 1$ . It looks true. But try $x = \tfrac{1}{2}$ : $\left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4}$ , and $\tfrac{1}{4}$ is less than $\tfrac{1}{2}$ . One counterexample, and the conjecture is dead.

Common mistake

A pile of examples does not prove a conjecture. A classic trap: the expression $n^2 + n + 41$ gives a prime number for $n = 0, 1, 2, \dots, 39$ . That's forty examples in a row! But at $n = 40$ you get $40^2 + 40 + 41 = 1681 = 41^2$ , which is not prime. Examples can suggest; only a proof can guarantee.

Deductive reasoning: following the logic

Deductive reasoning starts from facts that are accepted as true (definitions, properties, postulates and theorems already proven) and uses logic to reach a conclusion. If the starting facts are true and the logic is valid, the conclusion must be true.

Here's a deductive argument:

Fact: Vertical angles are congruent.
Fact: $\angle 1$ and $\angle 2$ are vertical angles.
Conclusion: $\angle 1 \cong \angle 2$ .

You didn't measure anything, and you didn't look at examples. The conclusion follows from the facts. Every proof in geometry works this way.

Two patterns of deductive logic come up constantly.

Law of Detachment

If "if $p$ , then $q$ " is true, and $p$ is true, then $q$ is true.

If an angle measures $90^\circ$ , then it is a right angle. $\angle B$ measures $90^\circ$ . So $\angle B$ is a right angle.

Law of Syllogism

If "if $p$ , then $q$ " and "if $q$ , then $r$ " are both true, then "if $p$ , then $r$ " is true. The statements link like a chain.

If it snows, then school is canceled. If school is canceled, then you sleep in. So: if it snows, then you sleep in.

Two kinds of reasoning

	inductive	deductive
starts from	specific examples	accepted facts and rules
produces	a conjecture (probably true)	a conclusion (certainly true, if the facts are)
can be disproved by	one counterexample	nothing, if the logic is valid

Common mistake

The Law of Detachment only works forward. From "If an animal is a dog, then it has four legs" and "Rex has four legs," you cannot conclude Rex is a dog. Rex might be a cat. Knowing the conclusion is true tells you nothing about the hypothesis.

Worked example: Making and testing a conjecture

Draw several triangles, measure their angles, and add. You might get $180^\circ$ , $179^\circ$ , $181^\circ$ (small measuring errors) each time. What conjecture fits? Is this inductive or deductive?

Solution. A good conjecture is "the angles of any triangle add to $180^\circ$ ." It's based on measured examples, so it's inductive. (Later in the course you'll prove it deductively, which is what makes it a theorem.)

Worked example: Finding a counterexample

Conjecture: "If two angles are supplementary, then one of them is obtuse." Find a counterexample.

Solution. Supplementary angles add to $180^\circ$ . Take two angles of $90^\circ$ each: they're supplementary, but neither is obtuse (both are right). So the conjecture is false.

Worked example: Using the laws of logic

What can you conclude, if anything?

If two angles form a linear pair, then they are supplementary. $\angle 3$ and $\angle 4$ form a linear pair.
If a figure is a square, then it is a rectangle. If a figure is a rectangle, then it has four right angles.
If a number is divisible by $6$ , then it is even. The number $14$ is even.

Solutions.

Law of Detachment: $\angle 3$ and $\angle 4$ are supplementary.
Law of Syllogism: if a figure is a square, then it has four right angles.
No conclusion. The given fact matches the conclusion of the conditional, not the hypothesis. (Indeed, $14$ is not divisible by $6$ .)

Tip

To check a deductive argument, ask: "Is the given fact the if part?" If yes, detachment applies. If the fact matches the then part, stop: you can't conclude anything.

Practice

Practice 1

Use inductive reasoning to find the next number: $3, 6, 12, 24, \dots$

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Find the next number: $1, 4, 9, 16, 25, \dots$

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which statement is an example of deductive reasoning?

Practice 4

Which value of $n$ is a counterexample to the conjecture " $n^2$ is always greater than $n$ "?

Practice 5

Given: "If a polygon is a hexagon, then it has six sides" and "Polygon $P$ has six sides." What can you conclude?

Practice 6

Use the Law of Syllogism. "If $x = 4$ , then $3x = 12$ ." "If $3x = 12$ , then $3x + 1 = 13$ ." Which statement follows?

Practice 7

Mark $n$ points on a circle and connect every pair. For $n = 1, 2, 3, 4, 5$ the circle is cut into $1, 2, 4, 8, 16$ regions. With the points placed so no three chords meet at one point, $6$ points give the most regions possible. That number breaks the "doubling" pattern. How many regions do $6$ points give? (Its value is $1 + \binom{6}{2} + \binom{6}{4}$ .)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.