Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer. (4 points)
The relationship between rational numbers and whole numbers is that whole numbers are a subset of rational numbers. Therefore, not all rational numbers are whole numbers, but all whole numbers are rational numbers.
A rational number is a number that can be expressed as the ratio of two integers. This means that it can be written in the form p/q, where p and q are integers and q is not equal to zero. Whole numbers, on the other hand, are non-negative integers including zero.
Examples:
- The number 3 is a whole number because it is a non-negative integer. It can also be expressed as the rational number 3/1, where both the numerator and denominator are integers.
- The number 1/2 is a rational number because it can be expressed as the ratio of two integers (1 and 2). However, it is not a whole number because it is not an integer.
- The number 0 is both a whole number and a rational number. It can be expressed as the rational number 0/1, where 0 is the numerator and 1 is the denominator.
The relationship between rational numbers and whole numbers is that every whole number is a rational number, but not every rational number is a whole number.
To understand this, let's first define rational numbers and whole numbers:
- Rational numbers: These are numbers that can be expressed as a fraction or a ratio of two integers. They can be written in the form of p/q, where p and q are integers, and q is not equal to zero.
- Whole numbers: These are non-negative integers (0, 1, 2, 3, ...).
Now, let's address the questions:
1. Are all rational numbers also whole numbers?
No, not all rational numbers are whole numbers. A rational number can be a fraction with a numerator and denominator that are integers, where the numerator and denominator are not necessarily whole numbers. For example, 3/2 is a rational number, but it is not a whole number because it is not an integer.
2. Are all whole numbers also rational numbers?
Yes, all whole numbers are also rational numbers. This is because every whole number can be expressed as a fraction with a denominator of 1. For example, the whole number 4 can be written as 4/1, which is a fraction and therefore a rational number.
In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers. The key distinction is that whole numbers are a subset of rational numbers, where the denominator is always 1.
The relationship between rational numbers and whole numbers is that every whole number can be considered a rational number, but not every rational number can be considered a whole number.
A rational number is any number that can be expressed as a fraction where the numerator and denominator are both integers. This includes whole numbers since they can be written as a fraction with a denominator of 1.
For example, the whole number 3 can be expressed as the fraction 3/1, making it a rational number.
However, not all rational numbers are whole numbers. Any fraction that is not a whole number, such as 2/3 or -4/5, is a rational number but not a whole number.
To summarize, all whole numbers are rational numbers, but not all rational numbers are whole numbers.