Exercise · ZW-EUCLIDEAN-DOMAIN-003 · ★☆☆ Basic
In the Gaussian integers \(\mathbb{Z}[i]\) the norm \(N(a+bi)=a^2+b^2\) is used. What is \(N(3+4i)\)?
Choose one option.
Show answer
- B. \(25\)
Explanation
\(N(3+4i)=3^2+4^2=25\). The value 5 is the modulus \(|3+4i|\), not the norm; 7 and 12 add or multiply without squaring.
Review the concept: Euclidean domain
An integral domain \(D\) with a norm \(N\) (a function from \(D\setminus\{0\}\) to the nonnegative integers) allowing division of \(a\) by \(b\neq0\) with quotient \(q\) and remainder \(r\) zero or of smaller norm than \(b\).
\[a=bq+r,\qquad r=0\ \text{ or }\ N(r)
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