1. Is the following sum rational or irrational? Why? 7.3+2.010010001...
A. Irrational because 2.010010001... has an infinite decimal expansion which means it is irrational. We know that a rational added to an irrational will be irrational.
B. Irrational because 2.010010001... is irrational since we see an infinite pattern in the decimal expansion and it is not a repeating pattern. We know that a rational added to an irrational will be irrational.
C. Rational because both numbers are rational since their decimal expansion either terminates or follows a pattern and rational numbers are closed under addition.
D. Rational because when we add these we get 9.31 after rounding to two decimal places, and now the decimal expansion terminates so the sum is rational.
2. Is the following product rational or irrational? Why? 4.3¯¯¯×9/4
A. Rational because both numbers are rational since 4.3¯¯¯ has an infinitely repeating decimal expansion and 9/4 is a fraction of integers. We know that the rational numbers are closed under multiplication.
B. Irrational because both numbers are rational since 4.3¯¯¯ has an infinitely repeating decimal expansion and 9/4 is a fraction of integers. We know that two rational numbers multiply to make irrational numbers.
C. Rational because 4.3¯¯¯ has a terminating decimal expansion and 9/4 is a fraction of integers. We know that the rational numbers are closed under multiplication.
D. Rational because 4.3¯¯¯ has a terminating decimal expansion and 9/4 is a fraction of integers. We know that the rational numbers are closed under multiplication.
3. Two students are trying to prove that the sum of two rational numbers is still rational. Which student correctly proves this fact and why?
Julia says that the sum of two rational numbers is rational because if you take two rational numbers like a/b and c/d , where a, b, c, and d must be integers by the definition of a rational number, and add them you get ad+bc/bd which will also be rational because integers are closed under addition and multiplication.
A. Julia proved this fact because she proved this for all possible rational numbers since she used variables to represent the integers and all rational numbers can all be written as a fraction of integers.
B. Juan proved this fact because he gave an example that showed adding two rational numbers gave us a rational back.
C. Julia proved this fact because she had more to say than Juan.
D. Juan proved this fact because he was more concise than Julia.
1. B. Irrational because 2.010010001... is irrational since we see an infinite pattern in the decimal expansion and it is not a repeating pattern. We know that a rational added to an irrational will be irrational.
2. A. Rational because both numbers are rational since 4.3 has an infinitely repeating decimal expansion and 9/4 is a fraction of integers. We know that the rational numbers are closed under multiplication.
3. A. Julia proved this fact because she proved this for all possible rational numbers since she used variables to represent the integers and all rational numbers can all be written as a fraction of integers.
A. Julia proved this fact because she proved this for all possible rational numbers since she used variables to represent the integers and all rational numbers can all be written as a fraction of integers.
The correct answers are:
1. B. Irrational because 2.010010001... is irrational since we see an infinite pattern in the decimal expansion and it is not a repeating pattern. We know that a rational added to an irrational will be irrational.
To determine if the sum of 7.3 and 2.010010001... is rational or irrational, we need to analyze the decimal expansion of the second number. Since it has an infinite non-repeating pattern, it is considered irrational. Adding a rational number (7.3) to an irrational number will always result in an irrational number.
2. A. Rational because both numbers are rational since 4.3¯¯¯ has an infinitely repeating decimal expansion and 9/4 is a fraction of integers. We know that the rational numbers are closed under multiplication.
To determine if the product of 4.3¯¯¯ and 9/4 is rational or irrational, we analyze the nature of both numbers. 4.3¯¯¯ has a repeating decimal expansion, which is still considered rational. 9/4 is a fraction of integers, which is also rational. Multiplying two rational numbers will always result in a rational number.
3. A. Julia proved this fact because she proved this for all possible rational numbers since she used variables to represent the integers and all rational numbers can all be written as a fraction of integers.
Julia correctly proves that the sum of two rational numbers is still rational. She uses variables (a, b, c, d) to represent the integers in a/b and c/d, which are the rational numbers. By adding them and simplifying, she shows that the sum (ad+bc/bd) is also a fraction of integers. Her proof covers all possible rational numbers since any rational number can be written as a fraction of integers.