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A rational number can be written as the ratio of two integers.
An irrational number cannot be written as the ratio of two integers.
26, -3/2, 0, and 9
5.737737773... and sqrt(45)
We will first use a calculator to find the value of sqrt(45) as a decimal. Then with the definition of rational numbers in mind, we will consider whether we can write the numbers as a ratio of two integers.
| Number | Ratio of Two Integer |
|---|---|
| 5.737737773... | - |
| 26 | 26/1 |
| sqrt(45)=6.708203... | - |
| - 3/2 | - 3/2 |
| 0 | 0/1 |
| 9 | 9/1 |
Now we will look at the following numbers. 5.737737773... sqrt(45)=6.708203... These numbers have infinitely many digits after the decimal point and these digits do not follow a specific pattern. This means that they neither terminate nor repeat. Consequently, the number cannot be written as the ratio of two integers. Conversely, we were able to write the other numbers as a ratio of two integers. Rational Numbers [0.5em] 26, -3/2, 0, and9
This time we will identify the numbers that are irrational. Let's recall the definition of irrational numbers.
By using the definition of irrational numbers, we will examine the table we made in Part A. Remember that irrational numbers have non-terminating and non-repeating decimal expansions.
| Number | Ratio of Two Integer |
|---|---|
| 5.737737773... | - |
| 26 | 26/1 |
| sqrt(45)=6.708203... | - |
| - 3/2 | - 3/2 |
| 0 | 0/1 |
| 9 | 9/1 |
From the table we can see that 5.737737773... and sqrt(45) cannot be written as a ratio of two integers, so they are irrational numbers. Irrational Numbers 5.737737773... sqrt(45)=6.708203...
