a puzzle: I get 7/(sqrt{2}-7) equalling (-7 sqrt{2} + 49)/47, but calc says "no" ??

[allegansveritatem]

a puzzle: I get 7/(sqrt{2}-7) equalling (-7 sqrt{2} + 49)/47, but calc says "no" ??

What is happening here?

---

90.. . .Is the following true?

. . . . .A.. .\(\displaystyle \dfrac{7}{\sqrt{2\,}\, -\, 7}\, \mbox{ and }\, \dfrac{-7\, \sqrt{2\,}\, +\, 49}{47}\) have the same

. . . . . . .decimal approximations when a calculator is used.

---

When I simplify the first expression I get one thing and when I use the calculator I get something else. I don't know what to make of this. Here is my dilemma in graphic form:

. . . . .\(\displaystyle \begin{align}\dfrac{7}{\sqrt{2\,}\, -\, 7}\, &=\, \dfrac{7\, \left(\sqrt{2\,}\, +\, 7\right)}{\left(\sqrt{2\,}\, -\, 7\right)\, \left(\sqrt{2\,}\, +\, 7\right)}\\
\\
&=\, \dfrac{7\, \sqrt{2\,}\, +\, 49}{2\, -\, 49}\\
\\
&=\, \dfrac{-7\sqrt{2\,}\, +\, 49}{47}\end{align}\)

But my calculator says:

. . . . .\(\displaystyle \dfrac{-49\, -\, 7\, \sqrt{2\,}}{47}\)

?

 

[topsquark]

What is happening here?

---

90.. . .Is the following true?

. . . . .A.. .\(\displaystyle \dfrac{7}{\sqrt{2\,}\, -\, 7}\, \mbox{ and }\, \dfrac{-7\, \sqrt{2\,}\, +\, 49}{47}\) have the same

. . . . . . .decimal approximations when a calculator is used.

---

When I simplify the first expression I get one thing and when I use the calculator I get something else. I don't know what to make of this. Here is my dilemma in graphic form:

. . . . .\(\displaystyle \begin{align}\dfrac{7}{\sqrt{2\,}\, -\, 7}\, &=\, \dfrac{7\, \left(\sqrt{2\,}\, +\, 7\right)}{\left(\sqrt{2\,}\, -\, 7\right)\, \left(\sqrt{2\,}\, +\, 7\right)}\\
\\
&=\, \dfrac{7\, \sqrt{2\,}\, +\, 49}{2\, -\, 49}\\
\\
&=\, \dfrac{-7\sqrt{2\,}\, +\, 49}{47}\end{align}\)

But my calculator says:

. . . . .\(\displaystyle \dfrac{-49\, -\, 7\, \sqrt{2\,}}{47}\)

?

You didn't distribute the negative sign correctly in your work. \(\displaystyle \dfrac{7 \sqrt{2} + 49}{-47} = - \dfrac{7 \sqrt{2} + 49}{7} = \dfrac{-7 \sqrt{2} - 49}{47}\)

So do they have the same decimal expansion?

-Dan

 

[allegansveritatem]

You didn't distribute the negative sign correctly in your work. \(\displaystyle \dfrac{7 \sqrt{2} + 49}{-47} = - \dfrac{7 \sqrt{2} + 49}{7} = \dfrac{-7 \sqrt{2} - 49}{47}\)

So do they have the same decimal expansion?

-Dan

I see it. I was treating the elements (or terms) of the numerator as though they were factors. But, of course, they are not, they are addends (?)and can't conduct the charge of the negative sign like factors can. Thanks for pointing this out.