Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Yes.
It can be proved by contradiction.
Let:
a - a rational number
b - an irrational number
c - the sum of a and b
[tex]a+b=c[/tex]
Let assume that c is a rational number. Then a and c can be expressed as fractions with integer numerator and denominator:
[tex]a=\dfrac{d}{e}\\ c=\dfrac{f}{g}\\[/tex] where [tex]d,e,f,g \in \mathbb{Z}[/tex]
[tex]\dfrac{d}{e}+b=\dfrac{f}{g}\\ b=\dfrac{f}{g}-\dfrac{d}{e}\\ b=\dfrac{ef}{eg}-\dfrac{dg}{eg}\\ b=\dfrac{ef-dg}{eg}[/tex]
Since [tex]d,e,f,g[/tex] are all integers, then the products [tex]ef,dg,eg[/tex] and the difference [tex]ef-dg[/tex] are integers as well. It means that the number [tex]b[/tex] is a rational number, but this on the other hand contradicts the earlier assumption that [tex]b[/tex] is an irrational number. Therefore [tex]c[/tex] must be an irrational number.
It can be proved by contradiction.
Let:
a - a rational number
b - an irrational number
c - the sum of a and b
[tex]a+b=c[/tex]
Let assume that c is a rational number. Then a and c can be expressed as fractions with integer numerator and denominator:
[tex]a=\dfrac{d}{e}\\ c=\dfrac{f}{g}\\[/tex] where [tex]d,e,f,g \in \mathbb{Z}[/tex]
[tex]\dfrac{d}{e}+b=\dfrac{f}{g}\\ b=\dfrac{f}{g}-\dfrac{d}{e}\\ b=\dfrac{ef}{eg}-\dfrac{dg}{eg}\\ b=\dfrac{ef-dg}{eg}[/tex]
Since [tex]d,e,f,g[/tex] are all integers, then the products [tex]ef,dg,eg[/tex] and the difference [tex]ef-dg[/tex] are integers as well. It means that the number [tex]b[/tex] is a rational number, but this on the other hand contradicts the earlier assumption that [tex]b[/tex] is an irrational number. Therefore [tex]c[/tex] must be an irrational number.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.