Complex Numbers - Absolute Values
The absolute value of a number is often viewed as the "distance" a number is away from 0, the origin. For real numbers, this is just the magnitude of the number without considering its sign. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.
For a complex number \(z = a + bi\) represented on the complex plane by the pair \((a, b)\), the "distance" from the origin is found using the Pythagorean theorem. The absolute value of \(z\) is defined as
\[|a+bi| = \sqrt{a^2 +b^2}.\]
For example, the absolute value of the complex number \(3+4i\) is equal to
\[|3+4i| = \sqrt{3^2 +4^2} = 5.\]
The absolute value can also be written as
\[|z| = \sqrt{z \bar{z}},\]
where\(\bar{z}\) is the complex conjugate of \(z.\)
What is the absolute value of the complex number \(-5+12i?\)
We have \(\lvert -5+12i \rvert=\sqrt{(-5)^2+12^2}=\sqrt{25+144}=\sqrt{169}=\sqrt{13^2}=13.\) \( _\square\)
The absolute value of the complex number \(7+bi\) is \(\sqrt{170}.\) What is the negative number \(b?\)
The absolute value of \(7+bi\) is \(\lvert 7+bi \rvert =\sqrt{7^2+b^2}=\sqrt{170},\) implying \(7^2+b^2=170,\) or \(b^2=121.\) This implies \(b=\pm 11.\) Since \(b\) is a negative number by assumption, thus \(b=-11.\) \( _\square\)
The absolute value of the complex number \(-7-8i\) can be expressed as \[\lvert -7-8i \rvert=\sqrt{(-7-8i)(a+bi)},\] where \(a\) and \(b\) are real numbers and \(i\) is the imaginary number. What are \(a\) and \(b?\)
Observe that the absolute value of a complex number \(z\) can be written as \(\lvert z \rvert = \sqrt{z \bar{z}},\) where\(\bar{z}\) is the complex conjugate of \(z.\) Then since \(z=-7-8i,\) thus \(a+bi=-7+8i,\) which implies that \(a=-7\) and \(b=8.\) \( _\square\)
What is the absolute value of the following sum of complex numbers: \[(12+4i)-(9-13i)-3i?\]
We have \(\lvert (12+4i)-(9-13i)-3i \rvert = \lvert 3+14i \rvert =\sqrt{3^2+14^2}=\sqrt{205}.\) \( _\square\)
Let \((6, 8)\) be the coordinates of the complex number \(z\) on the complex plane. Then what is the absolute value of its complex conjugate \(\bar{z}?\)
Since \(z=6+8i,\) it follows that \(\bar{z}=6-8i.\) Then \[\lvert \bar{z} \rvert=\lvert6-8i \rvert=\sqrt{(6)^2+(-8)^2}=\sqrt{36+64}=\sqrt{100}=\sqrt{10^2}=10. \ _\square\]
The complex number \(a+bi\) has the same absolute value as a different complex number \(3+4i.\) If \(a\) and \(b\) are both positive integers, what are \(a\) and \(b?\)
The absolute value of \(3+4i\) is \(\lvert 3+4i \rvert =\sqrt{3^2+4^2}=\sqrt{25}=5.\) Since \(a+bi\) also has an absolute value of \(5,\) it follows that \(\lvert a+bi \rvert =\sqrt{a^2+b^2}=5,\) implying \(a^2+b^2=25.\) Since \(a\) and \(b\) are both positive integers and \(a+bi\ne 3+4i\) by assumption, the only way that \(a^2+b^2=25\) holds is that \(a=4\) and \(b=3.\) \( _\square\)