Question
4. Is there a real number between 0.7 and 0.70000000000000001? Provide an example and explain using the density property
Answers
GPT-4o mini
Yes, there is a real number between \(0.7\) and \(0.70000000000000001\).
To understand this, we can utilize the density property of real numbers, which states that between any two distinct real numbers, there exists another real number.
In this case, we have:
- \(a = 0.7\)
- \(b = 0.70000000000000001\)
We can find a number that lies between \(a\) and \(b\) by taking the average of the two numbers. For example:
\[
c = \frac{a + b}{2} = \frac{0.7 + 0.70000000000000001}{2} = \frac{1.40000000000000001}{2} = 0.700000000000000005
\]
Now, we need to check that \(c\) is indeed between \(0.7\) and \(0.70000000000000001\):
- \(0.7 < 0.700000000000000005 < 0.70000000000000001\)
Thus, \(0.700000000000000005\) is a real number that lies between \(0.7\) and \(0.70000000000000001\).
This example demonstrates the density property of real numbers: between any two real numbers, no matter how close they are, you can always find another real number.
To understand this, we can utilize the density property of real numbers, which states that between any two distinct real numbers, there exists another real number.
In this case, we have:
- \(a = 0.7\)
- \(b = 0.70000000000000001\)
We can find a number that lies between \(a\) and \(b\) by taking the average of the two numbers. For example:
\[
c = \frac{a + b}{2} = \frac{0.7 + 0.70000000000000001}{2} = \frac{1.40000000000000001}{2} = 0.700000000000000005
\]
Now, we need to check that \(c\) is indeed between \(0.7\) and \(0.70000000000000001\):
- \(0.7 < 0.700000000000000005 < 0.70000000000000001\)
Thus, \(0.700000000000000005\) is a real number that lies between \(0.7\) and \(0.70000000000000001\).
This example demonstrates the density property of real numbers: between any two real numbers, no matter how close they are, you can always find another real number.