Question: Which of the following statements is the most logical way to prove that root 5 is irrational?
- Root 5 is an integer.
- Root 5 is a rational number.
- Root 5 is an irrational number.
- Root 5 can be expressed as a fraction of two integers, where the denominator is not equal to 0.
Answer: √5 is an irrational number.
Prove that root 5 is an Irrational Solution:
- A rational number is a number that can be expressed as a fraction of two integers, where the denominator is not equal to 0. An irrational number is a number that cannot be expressed as a fraction of two integers.
- To prove that root 5 is irrational, we can use a proof by contradiction. We start by assuming that root 5 is rational. We can then square both sides of this equation to get a^2 = 5b^2.
- Since 5 is prime, this means that 5 must divide a^2. But 5 also divides 5b^2. Therefore, 5 must divide both a and b. This contradicts our assumption that a and b are integers with no common factors other than 1.
So, we have √5 = p/q.
Now, let’s square both sides of the equation to eliminate the square root:
(√5)^2 = (p/q)^2 5 = p^2/q^2
Now, rearrange the equation:
5q^2 = p^2
From this equation, we can see that p^2 is a multiple of 5. This implies that p must also be a multiple of 5 since squaring an integer will result in a multiple of 5. So, we can write p as p = 5k, where k is an integer.
Substituting p = 5k back into our equation:
5q^2 = (5k)^2 5q^2 = 25k^2 q^2 = 5k^2
Now, we see that q^2 is also a multiple of 5, which means q must also be a multiple of 5.
However, this contradicts our initial assumption that p and q have no common factors other than 1 because p and q are divisible by 5. This contradiction shows that our assumption that √5 is a rational number must be false.
170 CM in Feet
If you’re wondering how tall 170 cm is in feet, the conversion is simple. 170 cm is approximately equal to 5 feet 7 inches. To convert centimeters to feet, divide the number of centimeters by 30.48, as 1 foot equals 30.48 cm. Therefore, 170 cm in feet is a common height measurement used in countries that primarily use the metric system. Understanding this conversion can be helpful when comparing heights between metric and imperial systems. So, if you are 170 cm tall, you stand at around 5 feet 7 inches.
