Tell whether each statement is true or false. If false, provide a counterexample.
The set of whole numbers contains the set of rational numbers. *true
Every terminating decimal is a rational number. *false
Every square root is a rational number. *false
The integers are closed under addition. *true
am i right??
The first and fourth statements are true. The second and third statements are false.
For the second statement, a counterexample would be the decimal number 0.333... which goes on forever. This decimal is not a rational number.
For the third statement, a counterexample would be the square root of 2 (√2). It is an irrational number, meaning it cannot be expressed as a fraction or ratio of two integers.
Let's go through each statement one by one:
1. The set of whole numbers contains the set of rational numbers. - True
To verify this statement, we need to understand the definitions of whole numbers and rational numbers. The set of whole numbers, denoted by "W", includes all positive numbers starting from zero (0), along with their negatives. On the other hand, the set of rational numbers, denoted by "Q", includes all numbers that can be written as a fraction (ratio) of two integers, where the denominator is not zero.
Since every whole number can be expressed as a fraction with a denominator of 1, every whole number is also a rational number. Therefore, the set of whole numbers indeed contains the set of rational numbers.
2. Every terminating decimal is a rational number. - False (Counterexample: √2)
Counterexample: The square root of 2 (√2)
A terminating decimal is a decimal number that ends after a finite number of digits. Rational numbers are numbers that can be expressed as a fraction of two integers.
However, the square root of 2 (√2) is an example of a non-terminating decimal that is not a rational number. The decimal representation of √2 goes on indefinitely without repeating or terminating. Since √2 cannot be expressed as a fraction of two integers, it is an irrational number. Therefore, not every terminating decimal is a rational number.
3. Every square root is a rational number. - False (Counterexample: √2)
Counterexample: The square root of 2 (√2)
Again, we consider the square root of 2 (√2), which is an example of a square root that is not a rational number. The decimal representation of √2 does not terminate or repeat, making it an irrational number. Therefore, not every square root is a rational number.
4. The integers are closed under addition. - True
To demonstrate this statement, we need to understand the closure property of addition. The closure property states that if you add any two numbers in a set, the result will also be in that set.
The integers include all positive and negative whole numbers (including zero) and their opposites. When you add two integers together, the sum will always be an integer. For example, adding any two whole numbers, such as 5 and -3, will result in another whole number, like 2. Therefore, the integers are closed under addition.
Based on this explanation, your answers are mostly correct. The only two statements that are false are:
- Every terminating decimal is a rational number.
- Every square root is a rational number.
false -- 2/3 is not a whole number
true -- a terminating decimal is like 0.35 which is 35/100
false
true
so, 2 out of 4.