Absolute value of a complex number: complex level, size, examples (2023)

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 number's distance from zero on a number line. This distance is independent of direction. The absolute value of a number is the distance which is always expressed as a positive number. It's never negative.

The symbol to represent the absolute value is a set of vertical bars. The same concept also applies to complex numbers. Let's learn more about it in detail here.

Calculating the 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.

Note 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 \(i=\sqrt{-1}\)

Note 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 graph a complex number.

complex plan

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

Now let's use this complex plane, similar to a Cartesian plane. Here, the real part of a complex number will act as the coordinate \(x\), and the imaginary part will be the coordinate \(y\).

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

complex number as vector

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Consider a complex number as a vector. A vector is defined as a quantity that has direction and magnitude. It is a directed line segment whose length is the magnitude and the 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 Theorem of Pythagoras.

Theorem of Pythagoras

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

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

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 for calculating 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 for calculating the absolute value of a complex number is given by:

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

Here,

\(the →\) real part

\(b →\) imaginary part

\(i →\) imaginary unit

unit circle

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

Consider these complex numbers:

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

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

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

Definition of the 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

Another example of such a complex number is \(\pm \frac{\sqrt{2}}{2} \pm \frac{\sqrt{2}}{2} i\), taken in any order of plus and minus Os other similar complex numbers found on the unit circle can be found in Pythagorean triples.

Pythagorean triples are three positive integers \(a, b,\) and \(c\) such that \(a^{2}+b^{2}=c^{2}\). Plunging 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 complex numbers whose absolute value is one.

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Calculate the absolute value of a complex number - worked examples

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

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

P.3. Magnitude calculations of \(-3+5i.\)
Responder:\(|-3+5 i|=\sqrt{(-3)^{2}+5^{2}}\)
\(=\square root{9+25}\)
\(=\square root{34}\)
A magnitude de \(-3+5 i=\sqrt{34}.\)

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

P.5. Resolvedor para \(\mathrm{x}:\left|x+\frac{63}{25} i\right|=\frac{13}{5}\)
Responder:\(\left|x+\frac{63}{25} i\right|=\frac{13}{5} \rightarrow\) Dado
\(\sqrt{x^{2}+\left(\frac{63}{25}\right)^{2}}=\frac{13}{5}\)
\(x^{2}+\left(\frac{63}{25}\right)^{2}=\left(\frac{13}{5}\right)^{2}\)
\(x^{2}+\left(\frac{63}{25}\right)^{2}=\frac{169}{25}\)
\(x^{2}=\frac{169}{25}-\frac{3969}{625}\)
\(x^{2}=\frac{4225}{625}-\frac{3969}{625}\)
\(x^{2}=\frac{256}{625}\)
\(x=\square root{\frac{256}{625}}\)
\(x=\frac{16}{25}\)

Summary

In this article, we learned that the concept of absolute value is applicable to complex numbers, similar to real numbers. We learned how to graph a complex number on a complex plane and how to calculate its absolute value. The unit circle idea was also explained with the Pythagorean triples that form complex numbers whose absolute value is always one. Finally, you also got acquainted with the calculations of the absolute value of complex numbers with the help of solved problems.

FAQ: Absolute value of a complex number

Q.1. How do you find the absolute value of a complex number?
Responder: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 in 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?
Responder:In mathematics, 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 strips the real number of its sign, making it a positive value. It is also called a module and is represented by vertical bars.

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

Q.4. Is zero (0) a complex number?
Responder:The general form of a complex number is \(z=a+bi.\) Here, \(a\) and \(b\) are real numbers, and \(i\) equals \(\sqrt{- 1}\) Any number that can be represented this way is a complex number.
\(0\) can be written as a complex number like \(0+0i.\)
Therefore, we can say that:
Yo. \(0\) 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, which makes it complex.
Therefore, we can say that zero is real and complex.

Q.5. Are complex numbers positive or negative?
Responder:Complex numbers are two-dimensional, and each dimension (real and imaginary) is either positive or negative. But the 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 has helped you in your studies. If you have any doubts, questions or suggestions regarding this article, feel free to ask us in the comments section and we will be happy to help you. Happy learning!