Narration: Let’s first look at the sum of two rational numbers. What do you think will be the result? Recall that a rational number is any number that can be written as a fraction, where the numerator and denominator are integers. Here are some examples.
two thirds plus one half
negative three-fourths plus four-fifths
two-sevenths minus 2
25 hundredths plus two-fifths
a over b plus c over d
Image description: Text appears on screen of the sentences that are spoken. The expressions appear on screen as they are spoken.
Narration: Will the sum of any two rational numbers always be rational? Two-thirds plus one-half equals four-sixths plus 3 sixths, equals seven-sixths, a rational number. Negative three-fourths plus four-fifths equals negative fifteen-twentieths plus sixteen-twentieths, equals one –twentieth, a rational number. Two-sevenths minus 2 equals two-sevenths minus fourteen-sevenths, equals negative twelve-sevenths, a rational number. Twenty five hundredths plus two-fifths equals one-fourth plus two-fifths equals thirteen-twentieths, a rational number. Does a over b plus c over d equal a rational number?
Image description: The equations appear on screen as they are spoken. Each sum is labeled a rational number when spoken.
Narration: Our examples show rational solutions, but will this situation always be true? Let’s find the sum of two rational numbers a over b and c over d, where a, b , c and d are integers and b is not equal to zero and d is not equal to zero.
Image description: The expressions appear onscreen as they are spoken and described.
Narration: The sum of two rational numbers, a over b and c over d, will always result in a fraction of the form ad plus bc over bd. ad plus bc is an integer and bd is an integer. Therefore, ad plus bc over bd is a rational number! That’s because a rational number by definition is a number that can be made by dividing one integer by another integer.
Image description: The equation is shown as it is spoken. Ad plus bc is labeled integer. Bd in the denominator of ad plus bc is also labeled integer. A over b plus c over d is labeled rational.
Narration: So, we can conclude that the sum of two rational numbers is always rational. Now let’s look at what results from the product of two rational numbers.
Here are some examples.
Image description: The following equations appear. Negative three fourths times four fifths equals negative twelve twentieths equals negative three fifths. This product results in a rational number.
Twenty five hundredths times two fifths equals one fourth times two fifths equals two twentieths equals one tenth. This product results in a rational number.
A over b times c over d equals an unknown product. Will this result be a rational number?
The answer to each equation is circled and labeled rational, except for the final equation, a over b times c over d, which has a question mark in the answer area.
Narration: It appears that the product of two rational numbers will always be rational. Pause the video now to study these equations. Continue when you are ready.
Image description: The answer to each equation is circled and labeled rational, except for the final equation, a over b times c over d, which has a question mark in the answer area.
Narration: Let’s see if our examples correctly demonstrate that the product of two rational numbers is always rational. Consider the product of two rational numbers a over b and c over d, where a,b , c and d are integers and b is not equal to zero and d is not equal to zero. The product of two rational numbers a over b and c over d will always result in a fraction of the form ac over bd. ac over bd is a rational number because ac is an integer and bd is an integer!
Image description: The equation appears as it is spoken.
Narration: Here are some examples showing how the product of two rational numbers results in a rational number.
Image description: The following equations are shown. Two thirds times four fifths equals eight fifteenths. 8 is an integer. Fifteen is an integer. Five eighths times three tenths equals three sixteenths. Three is an integer and sixteen is an integer. Three fourths times four thirds equals one which equals one over one. One is an integer.
Narration: Notice how each product results in a fraction in which the numerator is an integer and the denominator is an integer. We can conclude that the product of two rational numbers is always rational.