Tell whether each statement is true or false.
Every terminating decimal is a rational number.
Every square root is a rational number.
The integers are closed under addition.
To determine if each statement is true or false, we need to understand the concepts involved.
1. Every terminating decimal is a rational number:
A terminating decimal is a decimal that has a finite number of digits after the decimal point. A rational number is a number that can be expressed as a fraction (ratio) of two integers. To determine if this statement is true or false, we need to know that terminating decimals can be written in the form of a fraction. If we can express the decimal as a fraction, it is rational.
To check if a decimal is rational, we can follow these steps:
- Take the decimal and write it as the fraction numerator.
- The denominator will be a power of 10, depending on the number of decimal places.
For example, the decimal 0.75 can be written as 75/100, which simplifies to 3/4. Since we can express it as a fraction, it is rational.
2. Every square root is a rational number:
A square root is an operation that gives you the value that, when multiplied by itself, gives the original number. A rational number, as mentioned before, can be expressed as a fraction. To determine if this statement is true or false, we need to consider if every square root can be written as a fraction.
To check if a square root is rational, we can square the possible fraction form of the number and see if it equals the original number. If it does, the square root is rational. For example, the square root of 4 is 2, which can be written as 2/1. If we square 2/1, we get (2/1) * (2/1) = 4. Since it equals the original number, the square root of 4 is rational.
3. The integers are closed under addition:
The integers are the set of whole numbers (positive, negative, and zero) and their opposites. Addition is closed if, when we perform an addition operation on two integers, the result is always an integer.
To check if the integers are closed under addition, we can take any two integers and add them together. If the result is still an integer, then the integers are closed under addition. For example, let's take the integers 5 and -3 and add them: 5 + (-3) = 2, which is an integer. This demonstrates that the integers are closed under addition.
Based on these explanations:
- The statement "Every terminating decimal is a rational number" is true.
- The statement "Every square root is a rational number" is false.
- The statement "The integers are closed under addition" is true.