Narration: What type of number is the sum of a rational number and an irrational number?
Recall that a rational number is a number that can be written as a fraction with an integer numerator and an integer denominator.
Image description: Examples of rational numbers appear showing negative one third, eight, five sixths, two and five tenths, two and three tenths repeated.
Narration: An irrational number cannot be written in this form. Some examples of each type of number are shown.
Image description: Examples of irrational numbers appear showing the square root of three, pi, negative two square root of five divided by three.
Narration: For the moment, let’s assume the sum of a rational number and an irrational number is rational.
Image description: Rational plus irrational equals rational is shown on screen.
Narration: Let a over b and c over d be rational numbers. Let’s use “I” to represent an irrational number. Note that “I” is not an accepted notation for irrational numbers. There actually is no symbol for irrational numbers. We are using it here to simplify our explanations. a over b plus “I” equals c over d.
Solve for “I” by subtracting a over b from both sides of the equation. We know now that the sum of two rational numbers is always rational.
Image description: The equations appear as they are spoken and described.
Narration: By definition, an irrational number cannot equal a rational number. Therefore, we are forced to conclude that the sum of a rational number and an irrational number must be irrational. Now let’s explore what results from the product of a nonzero rational number and an irrational number. We will follow the same logical reasoning as before.
Image description: Rational times irrational equals rational appears on screen.
Narration: Assume the product of a (nonzero) rational number and an irrational number is rational.
Let a over b and c over d be rational numbers. Let’s use “I” to represent an irrational number.
Solve for “I” by multiplying both sides of the equation by b over a. We know now that the product of two rational numbers is always rational. By definition, an irrational number cannot equal a rational number. Therefore, we are forced to conclude that the product of a nonzero rational number and an irrational number must be irrational.
Image description: The equations appear as they are spoken and described.
Narration: Something else to think about: What type of number will result from the sum or product of two irrational numbers? That question is left up to you to explore.