Question

1. Is the following statement true or false? Explain your reasoning. All integers are rational numbers.

A. The statement is false because 1/2 is a rational number but not an integer.
B. The statement is true because any integer x can be written in the form x/1 which is the ratio of two integers.
C. The statement is false because 0 is an integer but cannot be written as the ratio of two integers.
D. The statement is true because all numbers are rational numbers.
2. Which of the following must describe an irrational number?
A. a number with a nonrepeating decimal expansion
B. a number with a nonrepeating and nonterminating decimal expansion
C. a number with a nonterminating decimal expansion
D. a number with a repeating or terminating decimal expansion
3. Which axiom is used to prove that the product of two rational numbers is rational?
A. Natural numbers are closed under multiplication.
B. Whole numbers are closed under division.
C. Integers are closed under division.
D. Integers are closed under multiplication.
4. What rational number, when multiplied by an irrational number, has a product that is a rational number?
A. 1/10
B. 1/2
C. 1
D. 0

Answer 1

1. B, 2. B, 3. D, 4. A.

Explanation:

1. The statement is true because any integer x can be written in the form x/1 which is the ratio of two integers. In mathematics, a rational number is defined as any number that can be expressed as the quotient or fraction p/q, where p and q are integers and q is not equal to 0. Since any integer can be expressed in this form with q = 1, all integers are indeed rational numbers.

2. An irrational number is a number with a nonrepeating and nonterminating decimal expansion. Irrational numbers cannot be expressed as fractions of two integers, and their decimal expansions go on forever without repeating. For example, the square root of 2 (√2) is an irrational number because its decimal expansion is nonrepeating and nonterminating.

3. Integers are closed under multiplication. This property means that when you multiply two integers together, the result is always an integer. Since rational numbers can be expressed as the quotient of two integers, when you multiply two rational numbers, you're essentially multiplying two fractions (integers), and the result is still a rational number.

4. When you multiply an irrational number by 1/10, the product is a rational number. This is because any number multiplied by a rational number remains rational. In this case, the irrational number gets "scaled down" by a rational factor, resulting in a rational product.