6. Explain why 25 is rational and 26 is irrational (5 points)
25 and 26 are rational numbers.
There is nothing to explain.
However, √25 is rational because 25 is a perfect square: 5^2
√26 is irrational for an analogous reason.
To determine whether a number is rational or irrational, we analyze its decimal representation.
1. Rational number: A rational number can be expressed as a fraction of two integers, where the denominator is not zero.
Let's analyze the number 25:
- The decimal representation of 25 is 25.000000...
- As we can see, the decimal representation terminates with no repeating pattern.
- Therefore, 25 can be expressed as a fraction: 25 = 25/1.
- Since 25 can be expressed as a fraction of two integers, it is considered a rational number.
Now let's analyze the number 26:
- The decimal representation of 26 is 26.000000...
- As we can see, the decimal representation also terminates with no repeating pattern.
- However, 26 cannot be expressed as a fraction of two integers.
- Thus, 26 is not considered a rational number.
- In fact, any number that cannot be expressed as a fraction is considered irrational.
In conclusion, 25 is a rational number since it can be expressed as a fraction, whereas 26 is an irrational number since it cannot be expressed as a fraction.
To understand why 25 is rational and 26 is irrational, we need to understand the definitions of rational and irrational numbers.
Rational numbers are those that can be expressed as a fraction, where the numerator and denominator are both integers. In simpler terms, rational numbers can be written as a quotient or division of two whole numbers.
On the other hand, irrational numbers cannot be expressed as a fraction or a quotient of two integers. They are non-repeating, non-terminating decimals, which means their decimal representation goes on infinitely without any recurring pattern.
Now let's examine both numbers:
1. 25: To determine if 25 is rational, we can express it as a fraction. In this case, 25 can be written as 25/1, where both the numerator and denominator are integers. Hence, 25 is a rational number.
2. 26: To determine if 26 is rational or irrational, we need to check if it can be expressed as a fraction. However, when we attempt to write 26 as a fraction, we quickly realize that we can't find two integers that can divide and simplify to give us 26. Therefore, 26 cannot be expressed as a fraction and is considered an irrational number.
In summary, 25 is rational because it can be written as a fraction, while 26 is irrational because it cannot be expressed as a fraction.