Absolute Value of a Complex Number: Complex Plane, Magnitude, Examples

Absolute Value of a Complex Number: Complex numbers play an important role in the fields of engineering and science. They are widely used in control theory, relativity, and fluid dynamics. The absolute value of a number is the distance of the number from zero on a number line. This distance is independent of the direction. The absolute value of a number is the distance that is always expressed as a positive number. It is never negative.

The symbol to represent absolute value is a set of vertical bars. The same concept is applicable for complex numbers too. Let us learn more about this in detail here.

Calculation of Absolute Value of a Number

1. The absolute value of \(5\) is \(5.\) This can be written as  \(|5|=5\)

The distance between \(0\) and \(5\) is \(5\) units.

2. \(|-5|=5\)

The distance between \(0\) and \(-5\) is \(5\) units.

Observe that the absolute value of a positive number is the number itself, and that of a negative number is its negation. The absolute value function deprives the real number of its sign.

Learn All About Complex Numbers

Complex Number

A complex number is a combination of a real number and an imaginary number. The general form of a complex number is \(z = a+bi.\) Here,\( a\) is the real part, and \(b\) is the imaginary part. i is called iota, and it is the imaginary unit. The value of \(i=\sqrt{-1}\)

Observe that the imaginary part does not include the imaginary unit \(i.\) That is, in \(z,\) the imaginary part is \(b\) and not \(bi.\) The set of all complex numbers is denoted by \(C.\)

Unlike a real number, this cannot be plotted on a number line. We use a complex plane to plot a complex number. 

Complex Plane

A complex plane has a number line of real numbers running horizontally from left to right and a number line of complex numbers running vertically from top to bottom. These are called the real axis and imaginary axis, respectively. The complex plane is also called the s-plane.

Let us now put this complex plane to use, similar to a Cartesian plane. Here, the real part of a complex number will act as the \(x\)-coordinate, and the imaginary part will be the \(y\)-coordinate.

To plot the complex number \(3+2i\) on a plane, we mark it as an ordered pair \((3,2).\) Here, observe that the real axis corresponds to the \(x\)-axis, and the imaginary axis corresponds to the \(y\)-axis. 

Complex Number as a Vector

Consider a complex number to be a vector. A vector is defined as a quantity that has both direction and magnitude. It is a directed line segment whose length is the magnitude, and orientation is the direction in space.

The complex number \(z=a+bi\) is plotted as shown in the figure. The magnitude of \(z\) is the distance of the point \((a,b)\) from the origin \(O.\) This distance of \(z\) from \(O\) is calculated using the Pythagoras Theorem.

Pythagoras Theorem

For the complex number, \(z = a + bi\), consider the right triangle with the right angle at \(O.\) The other vertices are \(z\) and a lie on the real axis. The length of the horizontal side is \(|a|,\) and that of the vertical side is \(|b|.\) According to Pythagoras, the square of the hypotenuse length is the sum of squares of the other two sides. 

\(z^{2}=a^{2}+b^{2}\)

We know that the distance of \(z\) from the origin \(O\) is the magnitude and is an absolute value. Therefore,

\(|z|=\sqrt{a^{2}+b^{2}}\)

Absolute Value of a Complex Number

The absolute value of a complex number, \(z = a+bi,\) is defined as the distance between the origin \(O\) and the point \((a,b)\) in the complex plane. In other words, it is the length of the hypotenuse of the right triangle formed. 

\(|z|=|a+b i|=\sqrt{a^{2}+b^{2}}\)

This is also called the modulus of a complex number.

Formula to Calculate the Absolute Value of a Complex Number

We know that the absolute value of a complex number is the magnitude of the vector it represents. The formula to calculate the absolute value of a complex number is given by:

\(|a+b i|=\sqrt{a^{2}+b^{2}}\)

Here,

\(a →\) real part

\(b →\) imaginary part

\(i →\) imaginary unit

Unit Circle

The unit circle is a circle of radius \(1\) with the centre at the origin \(O.\)

Consider these complex numbers:

1. \(1+i\)
2. \(1-i\)
3. \(-1+i\)
4. \(-1-i\)

These are a few complex numbers whose absolute value is \(1.\) Observe that \(1\) is the absolute value of both \(1\) and \(-1\). It is also the absolute value of \(i\). This is because they are one unit away from the origin \((0,0)\) on the real axis and imaginary axis. 

The unit circle includes all the complex numbers whose absolute value is \(1.\)

Definition of Unit Circle: The locus of a point at a distance of \(1\) unit from the origin (of the s-plane) is called a unit circle.

Pythagorean Triples and the Unit Circle

One other example of such a complex number is \(\pm \frac{\sqrt{2}}{2} \pm \frac{\sqrt{2}}{2} i\), taken in any order of pluses and minuses. The other such complex numbers that lie on the unit circle can be found from Pythagorean triples.

Pythagorean triples are three positive integers \(a, b,\) and \(c\) such that \(a^{2}+b^{2}=c^{2}\). Diving on both sides by \(c^{2}\), we get \(\frac{a^{2}}{c^{2}}+\frac{b^{2}}{c^{2}}=1\) This can be written as \(\left(\frac{a}{c}\right)^{2}+\left(\frac{b}{c}\right)^{2}=1 .\). From this, we can say that \(\frac{a}{c}+\frac{b}{c} i\) is a complex number that lies on the unit circle. For example, the best known Pythagorean triples are \((3, 4, 5)\). Here, \(a=3, b=4\), and \(c=5.\) The complex number that has an absolute value of one is \(\frac{3}{5}+\frac{4}{5} i\). This list does not end here. There are infinitely many such complex numbers whose absolute value is one.

Calculate Absolute Value of a Complex Number – Solved Examples

Q.1. Find the absolute value of \(3+2i.\)
Ans: \(|3+2 i|=\sqrt{3^{2}+2^{2}}\)
\(=\sqrt{9+4}\)
\(=\sqrt{13}\)
The absolute value of \(3+2 i=\sqrt{13}.\)

Q.2. Find \(\left| {1 – 3i} \right|.\)
Ans: \(|1-3 i|=\sqrt{1^{2}+3^{2}}\)
\(=\sqrt{1+9}\)
\(=\sqrt{10}\)
Therefore, \(|1-3 i|=\sqrt{10}.\)

Q.3. Calculate the magnitude of \(-3+5i.\)
Ans: \(|-3+5 i|=\sqrt{(-3)^{2}+5^{2}}\)
\(=\sqrt{9+25}\)
\(=\sqrt{34}\)
The magnitude of \(-3+5 i=\sqrt{34}.\)

Q.4. If \(z\) is a complex number of magnitude \(\sqrt{45}\) and its real part is \(3.\) Find the imaginary part and \(z.\)
Ans: \(\sqrt{a^{2}+b^{2}}=\sqrt{45}\)
\(\sqrt{3^{2}+b^{2}}=\sqrt{45}\)
\(\sqrt{9+b^{2}}=\sqrt{45}\)
\(9+b^{2}=45\)
\(b^{2}=45-9\)
\(b^{2}=36\)
\(b=\sqrt{36}\)
\(b=6\)
The complex number \(z=3+6i.\)

Q.5. Solve for \(\mathrm{x}:\left|x+\frac{63}{25} i\right|=\frac{13}{5}\)
Ans: \(\left|x+\frac{63}{25} i\right|=\frac{13}{5} \rightarrow\) Given
\(\sqrt{x^{2}+\left(\frac{63}{25}\right)^{2}}=\frac{13}{5}\)
\(x^{2}+\left(\frac{63}{25}\right)^{2}=\left(\frac{13}{5}\right)^{2}\)
\(x^{2}+\left(\frac{63}{25}\right)^{2}=\frac{169}{25}\)
\(x^{2}=\frac{169}{25}-\frac{3969}{625}\)
\(x^{2}=\frac{4225}{625}-\frac{3969}{625}\)
\(x^{2}=\frac{256}{625}\)
\(x=\sqrt{\frac{256}{625}}\)
\(x=\frac{16}{25}\)

Summary

In this article, we learnt that the concept of absolute value is applicable to complex numbers, similar to real numbers. We have learnt to plot a complex number on a complex plane and to calculate its absolute value. The idea of a unit circle was also explained with Pythagorean triples that form complex numbers whose absolute value is always one. Lastly, the calculations of the absolute value of complex numbers were also familiarised with the help of solved problems.

Frequently Asked Questions – Absolute Value of a Complex Number

Q.1. How do you find the absolute value of a complex number?
Ans: A complex number, \(z=a+bi,\) has two parts – real and imaginary. The absolute value of a complex number is the length of the hypotenuse in the triangle thus formed on the complex plane.  It is given by the formula:
\(|z|=|a+b i|=\sqrt{a^{2}+b^{2}}\)

Q.2. What is an example of absolute value?
Ans: In mathematics, the absolute value is defined as the non-negative value of a real number. The absolute value of a positive number is the number itself, and that of a negative number is its negation. The absolute value function deprives the real number of its sign, making it a positive value. It is also called modulus and is represented by vertical bars.

Q.3. What is the value of i in a complex number?
Ans: Complex numbers have two parts – real and imaginary. The imaginary part is defined in terms of \(i.\) This is called iota and is the imaginary unit. The value of \(i\) is \(\sqrt{-1}\).The presence of a negative one inside the square root represents the imaginary value.

Q.4. Is zero (0) a complex number?
Ans: The general form of a complex number is \(z=a+bi.\) Here, \(a\) and \(b\) are real numbers, and \(i\) is equal to \(\sqrt{-1}\) Any number that can be represented in this form is a complex number.
\(0\) can be written as a complex number as \(0+0i.\)
Hence, we can say that: 
i. \(0\) is a complex number whose imaginary part is zero. This means that zero is a real number. 
ii. \(0\) is a complex number whose real part is zero. This means that zero is an imaginary number, thus making it complex. 
Hence, we can say that zero is both real and complex.

Q.5. Are complex numbers positive or negative?
Ans: Complex numbers are two dimensional, with each dimension (real and imaginary) being either positive or negative. But complex numbers, as a whole, are all neutral. They are neither positive nor negative.

We hope this detailed article on the absolute value of a complex number helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!