Tell whether the statement "All integers are rational numbers" is true or false. Complete the explanation.
The statement is (true)(falese) , because integers can be written as the
(product)(quotient) of two integers.
true - quotient
integer x = x/1
- 3
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4
The statement "All integers are rational numbers" is true.
Explanation:
An integer is any whole number, positive, negative, or zero. A rational number is any number that can be expressed as a quotient or fraction of two integers, where the denominator is not zero. As integers can be written as the quotient of two integers (for example, 5 can be written as 5/1 or -7 can be written as -7/1), all integers are rational numbers.
The statement "All integers are rational numbers" is true. To understand why, let's first define what integers and rational numbers are:
- Integers are the set of whole numbers and their negatives, including zero. Examples of integers are -3, -2, -1, 0, 1, 2, 3, and so on.
- Rational numbers are numbers that can be expressed as a fraction, where both the numerator and denominator are integers, and the denominator is not zero. Examples of rational numbers are 1/2, -3/4, 2/5, and so on.
Now, to determine whether all integers are rational numbers, we need to check if every integer can be expressed as a fraction of two integers. In other words, we need to see if we can write any integer as a numerator divided by a non-zero integer denominator.
When we consider any integer, we can express it as itself divided by 1. Since both the numerator (the integer itself) and the denominator (1) are integers, the integer can be written as a fraction. Hence, all integers can be expressed as a fraction with a denominator of 1.
Therefore, since all integers can be expressed as fractions, and rational numbers are defined as fractions where both numerator and denominator are integers, we can conclude that all integers are indeed rational numbers.