The sum of two squares is zero
1,768
Solution 1
Clearly if $ a = 0 $ and $ b = 0 $ then $ a^2 + b^2 = 0 $.
Conversely, suppose that $ a^2 + b^2 = 0 $ . If $ a \neq 0 $ then $ a^2 > 0 $. But we always have $ b^2 \geq 0 $ so we deduce that $ a^2 + b^2 > 0 $ a contradiction. Likewise we cannot have $ b \neq 0 $. So we must have $ a = 0 $ and $ b = 0 $.
Solution 2
$$\begin{array}{ccc}a^2+b^2&\color{green}{a=0}&a>0\\\hline\color{green}{b=0}&\color{green}{=0}&>0\\b>0&>0&>0\end{array}$$
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Comments

B. David about 3 years
Prove: If $a,b \in \mathbb{R}: a^2+b^2= 0\Longleftrightarrow a=0$ and $b=0$
My work:
If $a = 0$, then $a^2 >0$, since $b^2 ≥0$, it follows that $a^2 +b^2 >0$
How can I continue my proof?

Stella Biderman about 6 yearsThis problem seems to be missing something. Did you mean to ask about $a^2+b^2=0$? Also, it's generally considered best to put the full question in the body of the post, even if you might need to repeat something in the title. Think of the title like the title of a book.

MCCCS about 6 yearsif $a=0$, then how can $a^2$ be bigger than 0?

B. David about 6 years@StellaBiderman Now the title is correct

Stella Biderman about 6 years@Daniel I've added a few tags and rewritten your question to abide by the community guidelines for titles that I mentioned.

B. David about 6 yearsThank you @StellaBiderman
