A student memorized the statement the product of any rational number and an irrational number is irrational holds true for non-zero rational number
Solution:
A rational number is a number that can be written as a fraction.
When a rational number fraction is divided to form a decimal value,
it becomes a terminating or repeating decimal.
An Irrational Number is a real number that cannot be written as a simple fraction. It cannot be expressed as a fraction with integer values in the numerator and denominator.
When an irrational number is expressed in decimal form, it goes on forever without repeating.
"The product of a rational number and an irrational number is SOMETIMES irrational."
If you multiply any irrational number by the rational number zero, the result will be zero, which is rational.
Any other situation, however, of a product of rational and irrational will be irrational.
A better statement would be:
"The product of a non-zero rational number and an irrational number is irrational."
Let us understand this with a example
[tex]\text { non - zero rational number } \times \text { irrational number }=\text { irrational number }[/tex]
Consider [tex]\frac{1}{2} \rightarrow rational number[/tex]
[tex]\frac{\sqrt{3}}{4} \rightarrow irrational number[/tex]
Now multiply both,
[tex]\frac{1}{2} \times \frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}=0.433012701[/tex]
We got 0.433012701 which is a irrational number because in decimal form, it goes on forever without repeating.
Thus we can frame two statements:
Answer:
Step-by-step explanation:
"The product of any rational number and an irrational number is irrational"
To know if the statement is true or false, we can test an example. Lets use the rational number [tex]\frac{3}{2}[/tex] and the irrational number [tex]\pi[/tex]. Id we calculate the product of these two we would have:
[tex]\frac{3}{2}\pi= 4.71238898038469...[/tex]
As you can observe, the product is an irrational number, because it's infinite and not periodic, it doesn't have any pattern.
Therefore, the statement is true.
