Question
put the steps to proof that the sum t=r+s is irrational if r is a rational in the correct sequence.
Step 1. subtract, writing a as a fraction
step 2. for a contradiction, assume that t is rational, and write r and t as fractions.
step 3 realize a contradiction.
step 4. find a common denominator
A B C or D
A. step 1, step 3, step 2, step 4
B.step 3, step 4, step 2, step 1
C. step 4, step 2, step 1, step 3
D. step 2, step 4, step 1, step 3
Step 1. subtract, writing a as a fraction
step 2. for a contradiction, assume that t is rational, and write r and t as fractions.
step 3 realize a contradiction.
step 4. find a common denominator
A B C or D
A. step 1, step 3, step 2, step 4
B.step 3, step 4, step 2, step 1
C. step 4, step 2, step 1, step 3
D. step 2, step 4, step 1, step 3
Answers
Answer
are you sure
Answer
what type of number will the product of pie and 3/4 be?
Answer
consider the expression 47/3+sqrt121 will the sum of the two numbers produce a rational number? explain your reasoning in 1-2 sentences
Answer
prove that the product of the two rational numbers sqrt 36 and 5/3 is a rational number. Find the product and explain how you know it is a rational number. explain in 1-2 sentences
Answered by
GPT-4o mini
To prove that the sum \( t = r + s \) is irrational if \( r \) is rational and \( s \) is irrational, the correct sequence of steps is as follows:
1. **Step 2:** For a contradiction, assume that \( t \) is rational, and write \( r \) and \( t \) as fractions.
2. **Step 4:** Find a common denominator.
3. **Step 1:** Subtract, writing \( s \) as a fraction.
4. **Step 3:** Realize a contradiction.
Thus, the correct sequence is:
**D. step 2, step 4, step 1, step 3**
1. **Step 2:** For a contradiction, assume that \( t \) is rational, and write \( r \) and \( t \) as fractions.
2. **Step 4:** Find a common denominator.
3. **Step 1:** Subtract, writing \( s \) as a fraction.
4. **Step 3:** Realize a contradiction.
Thus, the correct sequence is:
**D. step 2, step 4, step 1, step 3**
Answered by
GPT-4o mini
Let's clarify the proof that \( t = r + s \) is irrational if \( r \) is rational and \( s \) is irrational.
1. **Step 2**: For a contradiction, assume that \( t \) is rational. Write \( r \) and \( t \) as fractions.
2. **Step 4**: Find a common denominator in the fractions for \( r \) and \( t \).
3. **Step 1**: Subtract \( r \) from \( t \), rewriting it as a fraction (this would give you \( s \)).
4. **Step 3**: Realize a contradiction (since \( s \) would then have to be rational, contradicting the assumption that it is irrational).
Given this breakdown, the correct order of steps is indeed:
**D. step 2, step 4, step 1, step 3**.
I apologize for any confusion in my previous response, but my answer about the correct order is accurate. Thank you for your patience!
1. **Step 2**: For a contradiction, assume that \( t \) is rational. Write \( r \) and \( t \) as fractions.
2. **Step 4**: Find a common denominator in the fractions for \( r \) and \( t \).
3. **Step 1**: Subtract \( r \) from \( t \), rewriting it as a fraction (this would give you \( s \)).
4. **Step 3**: Realize a contradiction (since \( s \) would then have to be rational, contradicting the assumption that it is irrational).
Given this breakdown, the correct order of steps is indeed:
**D. step 2, step 4, step 1, step 3**.
I apologize for any confusion in my previous response, but my answer about the correct order is accurate. Thank you for your patience!
Answered by
GPT-4o mini
The product of \(\pi\) (which is an irrational number) and \(\frac{3}{4}\) (which is a rational number) will also be an irrational number.
When you multiply an irrational number by a non-zero rational number, the result is always irrational. Therefore, the product \(\frac{3}{4} \times \pi\) is irrational.
When you multiply an irrational number by a non-zero rational number, the result is always irrational. Therefore, the product \(\frac{3}{4} \times \pi\) is irrational.
Answered by
GPT-4o mini
The expression \( \frac{47}{3} + \sqrt{121} \) will produce a rational number because \( \frac{47}{3} \) is rational and \( \sqrt{121} = 11 \) is also rational. The sum of two rational numbers is always rational, so their total will be rational.
Answered by
GPT-4o mini
To prove that the product of the two rational numbers \( \sqrt{36} \) and \( \frac{5}{3} \) is rational, we first calculate the product:
\[
\sqrt{36} = 6 \quad (\text{which is rational})
\]
Now, multiplying by \( \frac{5}{3} \):
\[
6 \times \frac{5}{3} = \frac{30}{3} = 10
\]
The product is \( 10 \), which is a rational number. A number is rational if it can be expressed as a fraction of two integers, and here \( 10 \) can be expressed as \( \frac{10}{1} \). Thus, the product is confirmed to be rational.
\[
\sqrt{36} = 6 \quad (\text{which is rational})
\]
Now, multiplying by \( \frac{5}{3} \):
\[
6 \times \frac{5}{3} = \frac{30}{3} = 10
\]
The product is \( 10 \), which is a rational number. A number is rational if it can be expressed as a fraction of two integers, and here \( 10 \) can be expressed as \( \frac{10}{1} \). Thus, the product is confirmed to be rational.
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