Question

NO LINKS!!! NOT MULTIPLE CHOICE!!!

Find a counterexample to show that the following conjectures are false:

a. The sum of two numbers is always greater than zero.
b. The sum of a positive number and a negative number is always positive.
c. If a figure has four sides, then it is a rectangle.

Answer 1

Answer: a. The sum of two numbers is always greater than zero.

Explanation:

A counterexample to this conjecture would be when the two numbers are both negative. For example, the sum of -5 and -3 is -8, which is less than zero.

b. The sum of a positive number and a negative number is always positive.

A counterexample to this conjecture would be when the negative number is greater than the positive number. For example, the sum of 5 and -6 is -1, which is less than zero.

c. If a figure has four sides, then it is a rectangle.

A counterexample to this conjecture would be a square. A square has four sides, but it is not a rectangle because all of its angles are 90 degrees while a rectangle has four right angles.

It's good to keep in mind that a conjecture is a statement that is made based on observations and that it hasn't been proven yet, therefore a counterexample can be used to show that the conjecture is false.

Answer 2

a) -5 + 1 = -4

b) 2 + (-10) = -8

c) A scalene trapezium.

Explanation:

A counterexample is proof that a conjecture is false.

Part a

Given conjecture:

• The sum of two numbers is always greater than zero.

If one of the numbers is negative, and its absolute value is equal to or greater than the value of the positive number, the result will always be equal to or less than zero.

Counterexample:

• -5 + 1 = -4

As -4 is less than zero, this proves that the conjecture is false.

Part b

Given conjecture:

• The sum of a positive number and a negative number is always positive.

If the absolute value of the negative number is greater than the value of the positive number, the result will always be negative.

Counterexample:

• 2 + (-10) = -8

As -8 is negative, this proves that the conjecture is false.

Part c

Given conjecture:

• If a figure has four sides, then it is a rectangle.

Counterexample:

• A scalene trapezium.

The interior angles of a rectangle are congruent (90°).

A scalene trapezium has four sides, but as its interior angles are all different, it is not a rectangle. This proves that the conjecture is false.