Prove that if x is rational and x\ne0, then 1/x is rationa

skeexerxo175o

skeexerxo175o

Answered question

2021-11-16

Prove that if x is rational and x0, then 1x is rational.

Answer & Explanation

Unpled

Unpled

Beginner2021-11-17Added 23 answers

Step 1
Given: x is rational x0
TO PROOF: 1x is rational
DIRECT PROOF
Property rational numbers:
If a is a rational number, then there exists two integers y and z such that a=yz(with z0).
x is an rational number and thus there exists integers y and z such that:
x=yz
Since x¬0, we then also know that the numerator cannot be zero
y0
We are interested in 1x ::
1x=1yz=zy
Since z and y are integers (with y0), we then know that 1x is rational.
B
If x is rational and x0, then 1x is rational.

Vasquez

Vasquez

Expert2023-06-15Added 669 answers

To prove that if x is rational and x0, then 1x is rational, we can start by assuming that x is a rational number. By definition, a rational number can be expressed as the ratio of two integers, where the denominator is not zero. So, we can write x as:
x=pq where p and q are integers, and q0. Since x0, we know that p0 as well.
Now, let's consider 1x. Using the definition of division, we have:
1x=1pq
To divide by a fraction, we can multiply by its reciprocal. Therefore, we can rewrite the above expression as:
1x=1·qp
Simplifying further, we get:
1x=qp
Here, qp is the ratio of two integers, where p and q are integers and p0. Thus, 1x is also a rational number.
Hence, we have proved that if x is rational and x0, then 1x is rational.
In conclusion, we used the definition of rational numbers, along with the properties of division and multiplication, to show that if x is a non-zero rational number, then its reciprocal 1x is also a rational number.
RizerMix

RizerMix

Expert2023-06-15Added 656 answers

Let's assume that x is a rational number. By definition, a rational number can be expressed as the ratio of two integers: x=ab, where a and b are integers and b0.
We need to prove that 1x is also a rational number. To do that, we need to find integers c and d such that 1x=cd, where d0.
To find c and d, we can take the reciprocal of x:
1x=1ab=ba
Since a and b are integers and b0, ba is also a fraction with integers in the numerator and denominator. Therefore, we can express 1x as the ratio of two integers, which means 1x is a rational number.
Hence, if x is a rational number (x0), then 1x is also a rational number.
nick1337

nick1337

Expert2023-06-15Added 777 answers

Step 1:
Now, we need to show that 1x is also a rational number. We can express 1x as 1pq. To simplify this expression, we can multiply the numerator and denominator by the reciprocal of pq, which is qp. Thus, we have:
1x=1pq=1pq·qpqp=1·qppq·qp
Step 2:
Simplifying further, we get:
1x=qp·qp=q2p2
Since q2 and p2 are both integers (since q and p are integers), we can conclude that 1x is a rational number.
Therefore, we have proven that if x is rational and x0, then 1x is also a rational number.

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