Complex Numbers

A Complex Number is a combination of a Real Number and an Imaginary Number

Real Numbers are numbers like:

Nearly any number you can think of is a Real Number!

Imaginary Numbers when squared give a negative result.

Normally this doesn't happen, because:

  • when we square a positive number we get a positive result, and
  • when we square a negative number we also get a positive result (because a negative times a negative gives a positive ), for example −2 × −2 = +4

But just imagine such numbers exist, because we want them.

Let's talk some more about imaginary numbers ... 

The "unit" imaginary number (like 1 for Real Numbers) is i , which is the square root of −1

When we square i we get −1

Examples of Imaginary Numbers:

And we keep that little "i" there to remind us we still need to multiply by √−1

When we combine a Real Number and an Imaginary Number we get a Complex Number :

Can a Number be a Combination of Two Numbers?

Can we make a number from two other numbers? Sure we can!

We do it with fractions all the time. The fraction 3 / 8 is a number made up of a 3 and an 8. We know it means "3 of 8 equal parts".

Well, a Complex Number is just two numbers added together (a Real and an Imaginary Number).

Either Part Can Be Zero

So, a Complex Number has a real part and an imaginary part.

But either part can be 0 , so all Real Numbers and Imaginary Numbers are also Complex Numbers.

Complicated?

building complex

Complex does not mean complicated.

It means the two types of numbers, real and imaginary, together form a complex , just like a building complex (buildings joined together).

A Visual Explanation

You know how the number line goes left-right ?

Well let's have the imaginary numbers go up-down :

A complex number can now be shown as a point:

The letter  z is often used for a complex number:

  • z is a Complex Number
  • a and b are Real Numbers
  • i is the unit imaginary number = √−1

we refer to the real part and imaginary part using Re and Im like this:

Re(z) = a, Im(z) = b

The conjugate  (it changes the sign in the middle) of z is shown with a star:

z * = a − bi

We can also use angle and distance like this (called polar form ):

So the complex number 3 + 4i can also be shown as distance 5 and angle 0.927 radians . To convert from one form to the other use Cartesian to Polar conversion .

The magnitude of z is:

|z| = √(a 2 + b 2 )

And the angle of z , also called is:

Arg(z) = tan -1 (b/a)   (for a>0)

Example: z = 3 + 4i

  • 3 and 4 are Real Numbers
  • z * = 3 − 4i
  • |z| = √(3 2 + 4 2 ) = 5
  • Arg(z) = tan -1 (4/3) = 0.927... radians

To add two complex numbers we add each part separately:

(a+b i ) + (c+d i ) = (a+c) + (b+d) i

Example: add the complex numbers 3 + 2 i and 1 + 7 i

  • add the real numbers, and
  • add the imaginary numbers:

(3 + 2 i ) + (1 + 7 i ) = 3 + 1 + (2 + 7) i = 4 + 9 i

Let's try another:

Example: add the complex numbers 3 + 5 i and 4 − 3 i

(3 + 5 i ) + (4 − 3 i ) = 3 + 4 + (5 − 3) i = 7 + 2 i

On the complex plane it is:

Multiplying

To multiply complex numbers:

Each part of the first complex number gets multiplied by each part of the second complex number

Just use "FOIL", which stands for " F irsts, O uters, I nners, L asts" (see Binomial Multiplication for more details):

Example: (3 + 2 i )(1 + 7 i )

Example: (1 + i ) 2, but there is a quicker way.

Use this rule:

(a+b i )(c+d i ) = (ac−bd) + (ad+bc) i

Example: (3 + 2 i )(1 + 7 i ) = (3×1 − 2×7) + (3×7 + 2×1) i = −11 + 23 i

Why Does That Rule Work?

It is just the "FOIL" method after a little work:

And there we have the (ac − bd) + (ad + bc) i  pattern.

This rule is certainly faster, but if you forget it, just remember the FOIL method.

Let us try i 2

Just for fun, let's use the method to calculate i 2

Example: i 2

We can write i with a real and imaginary part as 0 + i

And that agrees nicely with the definition that i 2 = −1

So it all works wonderfully!

Learn more at Complex Number Multiplication .

We will need to use conjugates in a minute!

A conjugate is where we change the sign in the middle like this:

A conjugate can be shown with a little star, or with a bar over it:

5 − 3 i   =   5 + 3 i

The conjugate is used to help complex division.

The trick is to multiply both top and bottom by the conjugate of the bottom .

Example: Do this Division:

2 + 3 i 4 − 5 i

Multiply top and bottom by the conjugate of 4 − 5 i :

2 + 3 i 4 − 5 i × 4 + 5 i 4 + 5 i   =   8 + 10 i + 12 i + 15 i 2 16 + 20 i − 20 i − 25 i 2

Now remember that i 2 = −1 , so:

=   8 + 10 i + 12 i − 15 16 + 20 i − 20 i + 25

Add Like Terms (and notice how on the bottom 20 i − 20 i cancels out!):

=   −7 + 22 i 41

Lastly we should put the answer back into a + b i form:

=   −7 41 + 22 41 i

Yes, there is a bit of calculation to do. But it can be done.

Multiplying By the Conjugate

There is a faster way though.

In the previous example, what happened on the bottom was interesting:

(4 − 5 i )(4 + 5 i ) = 16 + 20 i − 20 i − 25 i 2

The middle terms (20 i − 20 i ) cancel out:

(4 − 5 i )(4 + 5 i ) = 16 − 25 i 2

Also i 2 = −1 :

(4 − 5 i )(4 + 5 i ) = 16 + 25

And 16 and 25 are (magically) squares of the 4 and 5:

(4 − 5 i )(4 + 5 i ) = 4 2 + 5 2

Which is really quite a simple result. The general rule is:

(a + b i )(a − b i ) = a 2 + b 2

That can save us time when we do division, like this:

Example: Let's try this again

2 + 3 i 4 − 5 i × 4 + 5 i 4 + 5 i   =   8 + 10 i + 12 i + 15 i 2 16 + 25

And then back into a + b i form:

We often use z for a complex number. And Re() for the real part and Im() for the imaginary part, like this:

Which looks like this on the complex plane:

The Mandelbrot Set

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Complex Numbers

Overview: This article covers the definition of complex numbers of the form $$ a+ bi $$ and how to graph complex numbers.

What are complex numbers?

A complex number can be written in the form a + b i where a and b are real numbers (including 0) and i is an imaginary number .

Therefore a complex number contains two 'parts':

  • one that is real
  • and another part that is imaginary

note: Even though complex have an imaginary part, there are actually many real life applications of these "imaginary" numbers including oscillating springs and electronics.

Examples of a complex number

$$ \begin{array}{c|c} \blue 3 + \red 5 i & \\\hline \blue{12} + \red{\sqrt{-3}} & \red{\sqrt{-3}} \text{ is the } \blue{imaginary} \text{ part} \\\hline \blue 9 - \red i & \\\hline \blue{12} - \red{\sqrt{-25}} & \red{\sqrt{-25}} \text{ is the } \blue{imaginary} \text{ part} \\\hline \end{array} $$

How do you graph complex numbers?

Complex numbers are often represented on a complex number plane (which looks very similar to a Cartesian plane) .

  • On this plane, the imaginary part of the complex number is measured on the 'y-axis' , the vertical axis;
  • the real part of the complex number goes on the 'x-axis' , the horizontal axis;

picture of complex plan and cartesian plane

Practice Problems of complex number

Identify the coordinates of all complex numbers represented in the graph on the right.

solving problems using complex number

In what quadrant, is the complex number $$ 2- i $$ ?

This complex number is in the fourth quadrant.

picture of graph of two minus i

In what quadrant, is the complex number $$ 2i - 1 $$ ?

This complex number is in the 2nd quadrant.

picture of graph of two i minus 1

In what quadrant, is the complex number $$ -i - 1 $$ ?

This complex number is in the 3rd quadrant.

picture of graph of -i minus 1

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  • To multiply two complex numbers z1 = a + bi and z2 = c + di, use the formula: z1 * z2 = (ac - bd) + (ad + bc)i.
  • What is a complex number?
  • A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. The number a is called the real part of the complex number, and the number bi is called the imaginary part.
  • Is 0 is a complex number?
  • 0 is a complex number, it can be expressed as 0+0i
  • How do you add complex numbers?
  • To add two complex numbers, z1 = a + bi and z2 = c + di, add the real parts together and add the imaginary parts together: z1 + z2 = (a + c) + (b + d)i
  • How do you subtract complex numbers?
  • To subtract two complex numbers, z1 = a + bi and z2 = c + di, subtract the real parts and the imaginary parts separately: z1 - z2 = (a - c) + (b - d)i

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Mathematics LibreTexts

5.1: The Complex Number System

  • Last updated
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  • Page ID 7124

  • Ted Sundstrom & Steven Schlicker
  • Grand Valley State University via ScholarWorks @Grand Valley State University

Focus Questions

The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions.

  • What is a complex number?
  • What does it mean for two complex numbers to be equal?
  • How do we add two complex numbers together?
  • How do we multiply two complex numbers together?
  • What is the conjugate of a complex number?
  • What is the modulus of a complex number?
  • How are the conjugate and modulus of a complex number related?
  • How do we divide one complex number by another?

The quadratic formula \(x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) allows us to find solutions to the quadratic equation \(ax^2+ bx + c = 0\). For example, the solutions to the equation \(x^{2} + x + 1 = 0\) are

\[x = \dfrac{-1 \pm \sqrt{1 - 4}}{2} = \dfrac{-1 \pm \sqrt{-3}}{2}. \nonumber\]

A problem arises immediately with this solution since there is no real number \(t\) with the property that \(t^{2} = -3\) or \(t = \sqrt{-3}\). To make sense of solutions like this we introduce complex numbers . Although complex numbers arise naturally when solving quadratic equations, their introduction into mathematics came about from the problem of solving cubic equations.

If we use the quadratic formula to solve an equation such as \(x^{2} + x + 1 = 0\),

we obtain the solutions \(x = \dfrac{-1 + \sqrt{-3}}{2}\) and \(x = \dfrac{-1 - \sqrt{-3}}{2}\). These numbers are complex numbers and we have a special form for writing these numbers. We write them in a way that that isolates the square root of \(-1\). To illustrate, the number

\[\dfrac{-1 + \sqrt{-3}}{2} \nonumber\]

can be written as follows:

\[\dfrac{-1 + \sqrt{-3}}{2} = -\dfrac{1}{2} + \dfrac{\sqrt{-3}}{2} = -\dfrac{1}{2} + \dfrac{\sqrt{3}\sqrt{-1}}{2} = -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}\sqrt{-1} \nonumber\]

Since there is no real number \(t\) satisfying \(t^{2} = -1\), the number \(\sqrt{-1}\) is not a real number. We call \(\sqrt{-1}\) an imaginary number and give it a special label \(i\). Thus, \(i = \sqrt{-1}\) or \(i^{2} = -1\). With this in mind we can write

\[\dfrac{-1 + \sqrt{-3}}{2} = -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i \nonumber\]

and every complex number has this special form.

Definition: Complex Numbers

A complex number is an object of the form

where \(a\) and \(b\) are real numbers and \(i^{2} = -1\).

The form \(a + bi\), where a and b are real numbers is called the standard form for a complex number. When we have a complex number of the form \(z = a + bi\), the number \(a\) is called the real part of the complex number \(z\) and the number \(b\) is called the imaginary part of \(z\). Since i is not a real number, two complex numbers \(a + bi\) and \(c + di\) are equal if and only if \(a = c\) and \(b = d\).

There is an arithmetic of complex numbers that is determined by an addition and multiplication of complex numbers. Adding and subtracting complex numbers is natural:

\[(a + bi) + (c + di) = (a + c) + (b + d)i\]

\[(a + bi) - (c + di) = (a - c) + (b - d)i\]

That is, to add (or subtract) two complex numbers we add (subtract)their real parts and add (subtract) their imaginary parts. Multiplication is also done in a natural way – to multiply two complex numbers, we simply expand the product as usual and exploit the fact that \(i^{2} = -1\). So the product of two complex number is

\[\begin{align*} (a + bi) + (c + di) &= ac + (ad)i + (bc)i + (bd)i^{2} \\[4pt] &= (ac - bd) + (ad + bc)i \end{align*}\]

Complex Number Properties

It can be shown that the complex numbers satisfy many useful and familiar properties, which are similar to properties of the real numbers. If \(u\), \(w\), and \(z\), are complex numbers, then

  • \(w + z = z + w\)
  • \(u + (w + z) = (u + w) + z\)
  • The complex number \(0 = 0 + 0i\) is an additive identity, that is \(z + 0 = z\).
  • If \(z = a + bi\), then the additive inverse of \(z\) is \(-z = (-a) + (-b)i\). That is, \(z + (-z) = 0\).
  • \(wz = zw\)
  • \(u(wz) = (uw)z\)
  • \(u(w + z) = uw + uz\)
  • If \(wz = 0\), then \(w = 0\) or \(z = 0\).

We will use these properties as needed. For example, to write the complex product \((1 + i)i\) in the form \(a + bi\) with \(a\) and \(b\) real numbers, we distribute multiplication over addition and use the fact that \(i^{2} = -1\) to see that

\[(1 + i)i = i + i^{2} = i + (-1) = (-1) + i.\]

For another example, if \(w = 2 + i\) and \(z = 3 - 2i\), we can use these properties to write \(wz\) in the standard \(a + bi\) form as follows:

\[wz = (2 + i)z = 2z + iz = 2(3 - 2i) + i(3 - 2i) = (6 - 4i) + (3i - 2i^{2}) = 6 - 4i + 3i - 2(-1) = 8 - i\]

Exercise \(\PageIndex{1A}\)

Write each of the sums or products as a complex number in standard form.

  • \((2 + 3i) + (7 - 4i)\)
  • \((4 - 2i)(3 + i)\)
  • \((2 + i)i - (3 + 4i)\)

(a) \((2 + 3i) + (7 - 4i) = 9 - i\)

(b) \((4 - 2i)(3 + i) = (4 - 2i)3 + (4 - 2i)i = 14 - 2i\)

(c) \((2 + i)i - (3 + 4i) = (2i - 1) - 3 - 4i = -4 - 2i\)

Exercise \(\PageIndex{1B}\)

Use the quadratic formula to write the two solutions to the quadratic equation \(x^{2} - x +2 = 0\) as complex numbers of the form \(r + si\) and \(u + vi\) for some real numbers \(r\), \(s\), \(u\), and \(v\).

( Hint : Remember: \(i = \sqrt{-1}\). So we can rewrite something like \(\sqrt{-4}\) as \(\sqrt{-4} = \sqrt{4}\sqrt{-1} = 2i\).)

We use the quadratic formula to solve the equation and obtain \[x = \dfrac{1 \pm \sqrt{-7}}{2}.\]

We can then write \(\sqrt{-7} = i\sqrt{7}\). So the two solutions of the quadratic equation are:

\[\begin{align*} x &= \dfrac{1 \pm i\sqrt{7}}{2} \\[4pt] &= \dfrac{1}{2} \pm \dfrac{\sqrt{7}}{2}i \\[4pt] \end{align*}\]

Division of Complex Numbers

We can add, subtract, and multiply complex numbers, so it is natural to ask if we can divide complex numbers. We illustrate with an example.

Example \(\PageIndex{2}\): Dividing by a Complex Number

Write the quotient \(\dfrac{2 + i}{3 + i}\) as a complex number in the form \(a + bi\).

This problem is rationalizing a denominator since \(i = \sqrt{-1}\). So in this case we need to “remove” the imaginary part from the denominator. Recall that the product of a complex number with its conjugate is a real number, so if we multiply the numerator and denominator of \(\dfrac{2 + i}{3 + i}\) by the complex conjugate of the denominator, we can rewrite the denominator as a real number. The steps are as follows. Multiplying the numerator and denominator by the conjugate \(3 - i\) or \(3 + i\) gives us

\[\dfrac{2 + i}{3 + i} = \left(\dfrac{2 + i}{3 + i}\right)\left(\dfrac{3 - i}{3 - i}\right) = \dfrac{(2 + i)(3 - i)}{(3 + i)(3 - i)} = \dfrac{(6 - i^{2}) + (-2 + 3)i}{9 - i^{2}} = \dfrac{7 + i}{10} \nonumber\]

Now we can write the final result in standard form as

\[\dfrac{7 + i}{10} = \dfrac{7}{10} + \dfrac{1}{10}i. \nonumber\]

Example \(\PageIndex{2}\) illustrates the general process for dividing one complex number by another. In general, we can write the quotient \(\dfrac{a + bi}{c + di}\) in the form \(r + si\) by multiplying numerator and denominator of our fraction by the conjugate \(c - di\) of \(c + di\) to see that

\[\dfrac{a + bi}{c + di} = \left(\dfrac{a + bi}{c + di}\right)\left(\dfrac{c - di}{c - di}\right) = \dfrac{(ac + bd) + (bc - ad)i}{c^{2} + d^{2}} = \dfrac{ac + bd}{c^{2} + d^{2}} + \dfrac{bc - ad}{c^{2} + d^{2}}i\]

Therefore, we have the formula for the quotient of two complex numbers.

Definition: Quotient of Complex Numbers

The quotient \(\dfrac{a + bi}{c + di}\) of the complex numbers \(a + bi\) and \(c + di\) is the complex number

\[\dfrac{a + bi}{c + di} = \dfrac{ac + bd}{c^{2} + d^{2}} + \dfrac{bc - ad}{c^{2} + d^{2}}i\]

provided \(c + di \neq 0\).

Exercise \(\PageIndex{3}\)

Let \(z = 3 + 4i\) and \(w = 5 - i\).

  • Write \(\dfrac{w}{z} = \dfrac{5 - i}{3 + 4i}\) as a complex number in the form \(r + si\) where \(r\) and \(s\) are some real numbers. Check the result by multiplying the quotient by \(3 + 4i\). Is this product equal to \(5 - i\)?
  • Find the solution to the equation \((3 + 4i)x = 5 - i\) as a complex number in the form \(x = u + vi\) where \(u\) and \(v\) are some real numbers.
  • Using our formula with \(a = 5, b = -1, c = 3\) and \(d = 4\) gives us \[\dfrac{5 - i}{3 + 4i} = \dfrac{15 - 4}{15} + \dfrac{-3 -20}{25}i = \dfrac{11}{25} - \dfrac{23}{25}i\] As a check, we see that \[\left(\dfrac{11}{25} - \dfrac{23}{25}i\right)\left(3 + 4i\right) = \left(\dfrac{33}{25} - \dfrac{69}{25}i\right) + \dfrac{44}{25}i - \dfrac{92}{25}i^{2} = \left(\dfrac{33}{25} + \dfrac{92}{25}\right) + \left(-\dfrac{69}{25}i + \dfrac{44}{25}i\right) = 5 - i\]
  • We can solve for \(x\) by dividing both sides of the equation by \(3 + 4i\) to see that \[x = \dfrac{5 - i}{3 + 4i} = \dfrac{11}{25} - \dfrac{23}{25}i\]

Geometric Representations of Complex Numbers

Each ordered pair \((a , b)\) of real numbers determines:

  • A point in the coordinate plane with coordinates \((a , b)\).
  • A complex number \(a + bi\)
  • A vector \(a\textbf{i} + b\textbf{j} = ( a, b )\)

This means that we can geometrically represent the complex number \(a + bi\) with a vector in standard position with terminal point \((a , b)\). Therefore, we can draw pictures of complex numbers in the plane. When we do this, the horizontal axis is called the real axis , and the vertical axis is called the imaginary axis . In addition, the coordinate plane is then referred to as the complex plane . That is, if \(z = a + b i\), we can think of \(z\) as a directed line segment from the origin to the point (a, b), where the terminal point of the segment is \(a\) units from the imaginary axis and \(b\) units from the real axis. For example, the complex numbers \(3 + 4i\) and \(-8 + 3i\) are shown in Figure 5.1.

5.1.png

Figure \(\PageIndex{1}\): Two complex numbers.

In addition, the sum of two complex numbers can be represented geometrically using the vector forms of the complex numbers. Draw the parallelogram defined by \(w = a + bi\) and \(z = c + di\). The sum of \(w\) and \(z\) is the complex number represented by the vector from the origin to the vertex on the parallelogram opposite the origin as illustrated with the vectors \(w = 3 + 4i\) and \(z = -8 + 3i\) in Figure \(\PageIndex{2}\).

Exercise \(\PageIndex{4}\)

Let \(w = 2 + 3i\) and \(z = -1 + 5i\).

  • Write the complex sum \(w + z\) in standard form.
  • Draw a picture to illustrate the sum using vectors to represent \(w\) and \(z\).

1. The sum is \(w + z = (2 - 1) + (3 + 5)i = 1 + 8i\).

2. A representation of the complex sum using vectors is shown in the figure below.

5.4.png

We now extend our use of the representation of a complex number as a vector in standard position to include the notion of the length of a vector. Recall from Section 3.6 that the length of a vector \(\textbf{v} = a\textbf{i} + b\textbf{j}\) is \(|\textbf{v}| = \sqrt{a^{2} + b^{2}}\).

5.2.png

Figure \(\PageIndex{2}\): The Sum of Two Complex Numbers.

When we use this idea with complex numbers, we call it the norm or modulus of the complex number.

Definition: Norm

The norm (or modulus ) of the complex number \(z = a + bi\) is the distance from the origin to the point \((a, b)\) and is denoted by \(|z|\). We see

that \[|z| = |a + bi| = \sqrt{a^{2} + b^{2}}.\]

There is another concept related to complex number that is based on the following bit of algebra.

\[(a + bi)(a -bi) = a^{2} - (bi)^{2} = a^{2} - b^{2}i^{2} = a^{2} + b^{2}\]

The complex number \(a - bi\) is called the complex conjugate of \(a + bi\). If we let \(z = a + bi\), we denote the complex conjugate of \(z\) as \(\bar{z}\). So \[\bar{z} = \overline{a + bi} = a - bi.\]

We also notice that

\[z\bar{z} = (a + bi)(a - bi) = a^{2} + b^{2},\]

and so the product of a complex number with its conjugate is a real number. In fact,

\[z\bar{z} = a^{2} + b^{2} = |z|^{2},\], and so \[|z| = \sqrt{z\bar{z}}\]

Exercise \(\PageIndex{5}\)

Let \(w = 2 + 3i\) and \(z = -1 + 5i\)

  • Find \(\bar{w}\) and \(\bar{z}\).
  • Compute \(|w|\) and \(|z|\).
  • Compute \(w\bar{w}\) and \(z\bar{z}\).
  • What is \(\bar{z}\) if \(z\) is a real number?

1. Using the definition of the conjugate of a complex number we find that \(\bar{w} = 2 - 3i\) and \(\bar{z} = -1 - 5i\). 2. Using the definition of the norm of a complex number we find that \(|w| = \sqrt{2^{2} + 3^{2}} = \sqrt{13}\) and \(|z| = \sqrt{(-1)^{2} + 5^{2}} = \sqrt{26}\). 3. Using the definition of the product of complex numbers we find that

\[w\bar{w} = (2 + 3i)(2 - 3i) = 4 + 9 = 13\]

\[z\bar{z} = (-1 + 5i)(-1 - 5i) = 1 + 25 = 26\]

4. Let \(z = a + 0i\) for some \(a \in \mathbb{R}\). Then \(\bar{z} = a - 0i\). Thus, \(\bar{z} = z\) when \(z \in \mathbb{R}\).

In this section, we studied the following important concepts and ideas:

  • A complex number is an object of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i^{2} = -1\). When we have a complex number of the form \(z = a + bi\), the number \(a\) is called the real part of the complex number \(z\) and the number \(b\) is called the imaginary part of \(z\).
  • We can add, subtract, multiply, and divide complex numbers as follows:

\[(a + bi) + (c + di) = (a + c) + (b + d)i \nonumber \]

\[(a + bi) - (c + di) = (a - c) + (b - d)i\nonumber\]

\[(a + bi)(c + di) = (ac - bd) + (ad + bc)i\nonumber\]

\[\dfrac{a + bi}{c + di} = \dfrac{ac + bd}{c^{2} + d^{2}} + \dfrac{bc - ad}{c^{2} + d^{2}}i\nonumber\] provided \(c + di \neq 0\)

  • A complex number \(a + bi\) can be represented geometrically with a vector in standard position with terminal point \((a, b)\). When we do this, the horizontal axis is called the real axis , and the vertical axis is called the imaginary axis . In addition, the coordinate plane is then referred to as the complex plane . That is, if \(z = a + bi\) we can think of \(z\) as a directed line segment from the origin to the point \((a, b)\), where the terminal point of the segment is a units from the imaginary axis and \(b\) units from the real axis.
  • The norm (or modulus ) of the complex number \(z = a + bi\) is the distance from the origin to the point \((a, b)\) and is denoted by \(|z|\). We see that \[|z| = |a + bi| = \sqrt{a^{2} + b^{2}} \nonumber\]
  • The complex number \(a - bi\) is called the complex conjugate of \(a + bi\). Note that \[(a + bi)(a - bi) = a^{2} + b^{2} = |a + bi|^{2} \nonumber\]

solving problems using complex number

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Complex Numbers and the Quadratic Formula

The Imaginary Complexes Quadratic Formula

For the time being, you will probably be using complex numbers only in the context of solving quadratic equations for their zeroes. (There are many other practical uses for complexes, but you'll have to wait for more interesting classes like "Engineering 201" to get to the good stuff.) So let's refresh on the topic.

The Quadratic Formula takes the generic quadratic equation, stated as:

a x 2 + b x + c = 0

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...and provides the solution values of the variable x by plugging the values a , b , and c of the numerical coefficients into the following formula:

Previously, when the Formula gave you a negative value inside the square root (that is, inside the "discriminant"), you would have responded that the equation under question had "no solutions". Were you to graph the quadratic equation (setting the quadratic equal to y instead of to 0 ), your graph would not cross the horizontal axis.

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Now, with complex numbers, when the Formula gives you a negative inside the root, you now can simplify that solution by using the imaginary and respond that the equation under question has no real-valued solution, but it does have a complex-valued solution. Were you to graph the quadratic equation, your graph will still not cross the horizontal axis.

The difference now is that you have solutions (from the Quadratic Formula) to the equation which are not graphable. "Solutions", "roots", and "zeroes" of a given quadratic equation are now no longer necessarily also " x -intercepts".

The values on the x , y -plane are real numbers, so the complex-valued solutions of the equation cannot be seen on the x -axis. Your complex-valued answer is still a valid "zero" or "root" or "solution" for that quadratic equation, because, if you plug the answer value back into the quadratic equation, you'll get zero after you simplify. (This is the definition of a "solution"; it is a value that solves the equation, making it true.) However, this complex-valued answer will not be a graphable x -intercept of the graph.

What is the connection between the Quadratic Formula, complex numbers, and graphing?

The connection between the Quadratic Formula, complex numbers, and graphing is illustrated in the table below:

In the discussion above, I repeatedly pointed out that complex solutions to a quadratic equation are not graphable in the x , y -plane, because all points on the x , y are pairs of real numbers. However, if we change what the axes stand for, we can plot complex numbers.

How can you graph complex numbers?

To graph a complex number, you:

  • Draw the regular x , y -plane.
  • using the regular x and y labels
  • using the regular x for the horizontal axis, but use yi for the vertical axis
  • using "Re" for the horizontal axis having real-number values, and using "Im" for the vertical axis having imaginary-number values
  • Convert the complex number from " a  +  bi " summing (that is, additive) form to " ( a ,  b ) coordinate form
  • Plot the point just as you would a coordinate point on the coordinate plane

For instance, you would plot the complex number 3 − 2 i by converting the complex number into coordinate-point form (that is, into (3, −2) form), and then graphing in the usual way: starting at the origin, moving three units to the right along the x -axis, moving two units down parallel to the y -axis, and drawing a dot. Here's what it looks like:

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This graphability of complex numbers leads somewhere interesting.

When you learned about regular (that is, about "real") numbers, you also learned about their order on the number line. But x , y -points don't come in any particular order. You can't say that one point "comes after" another point in the same way that you can say that one number comes after another number; you can't say that (4, 5) "comes after" (4, 3) in the way that you can say that 5 comes after 3 .

Pretty much all you can do is compare the "sizes" of the different points, and, for complex numbers as for regular x , y -points, "size" means "how far from the origin". To find a point's size, you use the Distance Formula . Complexes that are closer to the origin will have smaller sizes; complexes further away will have larger values.

This "size" concept is called "the modulus" of a complex-valued point. For instance, the modulus of the above-graphed point (denoted by absolute-value bars around the absolute-value bars) is computed by using the Distance Formula:

modulus is sqrt(13)

Now, isn't that interesting? 😎

URL: https://www.purplemath.com/modules/complex3.htm

You can use the Mathway widget below to practice finding the magnitude (or modulus or "size") of complex numbers. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's.

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solving problems using complex number

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Complex Numbers Problems with Solutions and Answers - Grade 12

Complex numbers are important in applied mathematics. Problems and questions on complex numbers with detailed solutions are presented.

  • If (x + yi) / i = ( 7 + 9i ) , where x and y are real, what is the value of (x + yi)(x - yi)?
  • P(z) = z 4 + a z 3 + b z 2 + c z + d is a polynomial where a, b, c and d are real numbers. Find a, b, c and d if two zeros of polynomial P are the following complex numbers: 2 - i and 1 - i.

Solutions to the Above Questions

  • a) -5 + 7i b) -6 - 22i c) -4 + 2i d) -7/15 - 4i/15
  • (x + yi) / i = ( 7 + 9i ) (x + yi) = i(7 + 9i) = -9 + 7i (x + yi)(x - yi) = (-9 + 7i)(-9 - 7i) = 81 + 49 = 130
  • Let z = a + bi , z' = a - bi ; a and b real numbers. Substituting z and z' in the given equation obtain a + bi + 3*(a - bi) = 5 - 6i a + 3a + (b - 3b) i = 5 - 6i 4a = 5 and -2b = -6 a = 5/4 and b = 3 z = 5/4 + 3i
  • z z' = (a + bi)(a - bi) = a 2 + b 2 = 25 a + b = 7 gives b = 7 - a Substitute above in the equation a 2 + b 2 = 25 a 2 + (7 - a) 2 = 25 Solve the above quadratic function for a and use b = 7 - a to find b. a = 4 and b = 3 or a = 3 and b = 4 z = 4 + 3i and z = 3 + 4i have the property z z' = 25.
  • a) Substitute solution in equation: (2 + 4i) 2 + b(2 + 4i) + c = 0 Expand terms in equation and rewrite as: (-12 + 2b + c) + (16 + 4b)i = 0 Real part and imaginary part equal zero. -12 + 2b + c = 0 and 16 + 4b = 0 Solve for b: b = -4 , substitute and solve for c: c = 20 b) Since the given equation has real numbers, the second root is the complex conjugate of the given root: 2 - 4i is the second solution. Check: (2 - 4i) 2 - 4 (2 - 4i) + 20 (Expand) = 4 - 16 - 16i - 8 + 16i + 20 = (4 - 16 - 8 + 20) + (-16 + 16)i = 0
  • Let z = a + bi Substitute into given equation: (a + bi) 2 = -1 + 2 sqrt(6) i Expand: a 2 - b 2 + 2 ab i = - 1 + 2 sqrt(6) i Real part and imaginary parts must be equal. a 2 - b 2 = - 1 and 2 ab = 2 sqrt(6) Equation 2 ab = 2 sqrt(6) gives: b = sqrt(6) / a Substitute: a 2 - ( sqrt(6) / a ) 2 ) = - 1 a 4 - 6 = - a 2 Solve above equation and select only real roots: a = sqrt(2) and a = - sqrt(2) Substitute to find b and write the two complex numbers that satisfies the given equation. z1 = sqrt(2) + sqrt(3) i , z2 = - sqrt(2) - sqrt(3) i
  • Let z = a + bi where a and b are real numbers. The complex conjugate z' is written in terms of a and b as follows: z'= a - bi. Substitute z and z' in the given equation (4 + 2i)(a + bi) + (8 - 2i)(a - bi) = -2 + 10i Expand and separate real and imaginary parts. (4a - 2b + 8a - 2b) + (4b + 2a - 8b - 2a )i = -2 + 10i Two complex numbers are equal if their real parts and imaginary parts are equal. Group like terms. 12a - 4b = -2 and - 4b = 10 Solve the system of the unknown a and b to find: b = -5/2 and a = -1 z = -1 - (5/2)i
  • Since z = -2 + 7i is a root to the equation and all the coefficients in the terms of the equation are real numbers, then z' the complex conjugate of z is also a solution. Hence z 3 + 6 z 2 + 61 z + 106 = (z - (-2 + 7i))(z - (-2 - 7i)) q(z) = (z 2 + 4z + 53) q(z) q(z) = [ z 3 + 6 z 2 + 61 z + 106 ] / [ z 2 + 4z + 53 ] = z + 2 Z + 2 is a factor of z 3 + 6 z 2 + 61 z + 106 and therefore z = -2 is the real root of the given equation.
  • a) (2i) 4 + (2i) 3 + 2 (2i) 2 + 4 (2i) - 8 = 16 - 8i - 8 + 8i - 8 = 0 b) 2i is a root -2i is also a root (complex conjugate because all coefficients are real). z 4 + z 3 + 2 z 2 + 4 z - 8 = (z - 2i)(z + 2i) q(z) = (z 2 + 4)q(z) q(z) = z 2 + z - 2 The other two roots of the equation are the roots of q(z): z = 1 and z = -2.
  • Since all coefficients of polynomial P are real, the complex conjugate to the given zeros are also zeros of P. Hence P(z) = (z - (2 - i))(z - (2 + i))(z - (1 - i))(z - (1 + i)) = = z 4 - 6 z 3 + 15 z 2 - 18 z + 10 Hence: a = -6, b = 15, c = -18 and d = 10.

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Problems on Complex Numbers

We will learn step-by-step how to solve different types of problems on complex numbers using the formulas.

1.  Express \((\frac{1 + i}{1 - i})^{3}\) in the form A + iB where A and B are real numbers.

Given \((\frac{1 + i}{1 - i})^{3}\)

Now \(\frac{1 + i}{1 - i}\)

= \(\frac{(1 + i)(1 + i)}{(1 - i)(1 + i)}\)

= \(\frac{(1 + i)^{2}}{(1^{2} - i^{2}}\)

= \(\frac{1 + 2i + iˆ{2}}{1 - (-1)}\)

= \(\frac{1 + 2i - 1}{2}\)

= \(\frac{2i}{2}\)

Therefore, \((\frac{1 + i}{1 - i})^{3}\) = i\(^{3}\)= i\(^{2}\) ∙  i = - i = 0 + i (-1), which is the required form A + iB where A = 0 and B = -1.

2. Find the modulus of the complex quantity (2 - 3i)(-1 + 7i).

The given complex quantity is (2 - 3i)(-1 + 7i)

Let z\(_{1}\) = 2 - 3i and z\(_{2}\) = -1 + 7i

Therefore, |z\(_{1}\)| = \(\sqrt{2^{2} + (-3)^{2}}\) = \(\sqrt{4 + 9}\) = \(\sqrt{13}\)

And |z\(_{2}\)| = \(\sqrt{(-1)^{2} + 7^{2}}\) = \(\sqrt{1 + 49}\) = \(\sqrt{50}\) = 5\(\sqrt{2}\)

Therefore, the required modulus of the given complex quantity = |z\(_{1}\)z\(_{1}\)| = |z\(_{1}\)||z\(_{1}\)| = \(\sqrt{13}\)  ∙ 5\(\sqrt{2}\) = 5\(\sqrt{26}\)

3. Find the modulus and principal amplitude of -4.

Let z = -4 + 0i.

Then, modulus of z = |z| = \(\sqrt{(-4)^{2} + 0^{2}}\) = \(\sqrt{16}\) = 4.

Clearly, the point in the z-plane the point z = - 4 + 0i = (-4, 0) lies on the negative side of real axis.

Therefore, the principle amplitude of z is π.

4. Find the amplitude and modulus of the complex number -2 + 2√3i.

The given complex number is -2 + 2√3i.

The modulus of -2 + 2√3i = \(\sqrt{(-2)^{2} + (2√3)^{2}}\) = \(\sqrt{4 + 12}\) = \(\sqrt{16}\) = 4.

Therefore, the modulus of -2 + 2√3i = 4

Clearly, in the z-plane the point z = -2 + 2√3i = (-2, 2√3) lies in the second quadrant. Hence, if amp z = θ then,

tan θ = \(\frac{2√3}{-2}\) = - √3 where, \(\frac{π}{2}\) < θ ≤ π.

Therefore, tan θ = - √3 = tan (π - \(\frac{π}{3}\)) = tan \(\frac{2π}{3}\)

Therefore, θ = \(\frac{2π}{3}\)

Therefore, the required amplitude of -2 + 2√3i is \(\frac{2π}{3}\).

5. Find the multiplicative inverse of the complex number z = 4 - 5i.

The given complex number is z = 4 - 5i.

We know that every non-zero complex number z = x +iy possesses multiplicative inverse given by

\((\frac{x}{x^{2} + y^{2}}) + i (\frac{-y}{x^{2} + y^{2}})\)

Therefore, using the above formula, we get

z\(^{-1}\) = \((\frac{4}{4^{2} + (-5)^{2}}) + i (\frac{-(-5)}{4^{2} + (-5)^{2}})\)

= \((\frac{4}{16 + 25}) + i (\frac{5)}{16 + 25})\)

= \((\frac{4}{41}) + (\frac{5}{41})\)i

Therefore, the multiplicative inverse of the complex number z = 4 - 5i is \((\frac{4}{41}) + (\frac{5}{41})\)i

6. Factorize: x\(^{2}\) + y\(^{2}\)

x\(^{2}\) - (-1) y\(^{2}\) = x\(^{2}\) - i\(^{2}\)y\(^{2}\) = (x + iy)(x - iy)

11 and 12 Grade Math   From Problems on Complex Numbers to HOME PAGE

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Complex Numbers in Maths

Complex numbers are the numbers that are expressed in the form of a+ib where, a,b are real numbers and  ‘i’ is an imaginary number called “ iota” . The value of i = (√-1). For example, 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im). 

Examples of complex numbers:

  • -1 3 – 3i
  • 0.89 + 1.2 i

An imaginary number is usually represented by ‘i’ or ‘j’, which is equal to √-1. Therefore, the square of the imaginary number gives a negative value. 

The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc., which rely on sine or cosine waves, etc.

The complex number is basically the combination of a real number and an imaginary number. The complex number is in the form of a+ib, where a = real number and ib = imaginary number. Also, a,b belongs to real numbers and i = √-1 . 

Hence, a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number. One part of it is purely real and the other part is purely imaginary.

Complex Numbers in Maths

What are Real Numbers?

Any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. It is represented as Re(). For example: 12, -45, 0, 1/7, 2.8, √5, etc., are all real numbers.

What are Imaginary Numbers?

The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as Im(). Example: √-2, √-7, √-11 are all imaginary numbers.

The complex numbers were introduced to solve the equation x 2 +1 = 0. The roots of the equation are of form x = ±√-1 and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.

We denote √-1 with the symbol ‘i’, which denotes Iota (Imaginary number).

An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = ib.

See the table below to differentiate between a real number and an imaginary number.

Is 0 a complex Number?

As we know, 0 is a real number. And real numbers are part of complex numbers. Therefore, 0 is also a complex number and can be represented as 0+0i.

Graphical representation

In the graph below, check the representation of complex numbers along the axes. Here we can see, the x-axis represents real part and y represents the imaginary part.

Complex numbers graph

Absolute Value

The absolute value of a real number is the number itself. The absolute value of x is represented by modulus, i.e. |x|. Hence, the modulus of any value always gives a positive value, such that;

Now, in case of complex numbers, finding the modulus has a different method.

Suppose, z = x+iy is a complex number. Then, mod of z, will be:

|z| = √(x 2 +y 2 )

This expression is obtained when we apply the Pythagorean theorem in a complex plane. Hence, mod of complex number, z is extended from 0 to z and mod of real numbers x and y is extended from 0 to x and 0 to y respectively. Now these values form a right triangle, where 0 is the vertex of the acute angle. Now, applying Pythagoras theorem ,

|z| 2 = |x| 2 +|y| 2

|z| 2 = x 2 + y 2

Algebraic Operations on Complex Numbers

There can be four types of  algebraic operation on complex numbers  which are mentioned below. Visit the linked article to know more about these algebraic operations along with solved examples. The four operations on the complex numbers include:

Subtraction

Multiplication, roots of complex numbers.

When we solve a quadratic equation in the form of ax 2  +bx+c = 0, the roots of the equations can be determined in three forms;

  • Two Distinct Real Roots
  • Similar Root
  • No Real roots (Complex Roots)

Complex Number Formulas

While performing the arithmetic operations of complex numbers such as addition and subtraction, combine similar terms. It means that combine the real number with the real number and imaginary number with the imaginary number.

(a + ib) + (c + id) = (a + c) + i(b + d)

(a + ib) – (c + id) = (a – c) + i(b – d)

When two complex numbers are multiplied by each other, the multiplication process should be similar to the multiplication of two binomials. It means that the FOIL method (Distributive multiplication process) is used.

(a + ib). (c + id) = (ac – bd) + i(ad + bc)

The division of two complex numbers can be performed by multiplying the numerator and denominator by its conjugate value of the denominator, and then applying the FOIL Method.

(a + ib) / (c + id) = (ac+bd)/ (c 2 + d 2 ) + i(bc – ad) / (c 2 + d 2 )

Power of Iota (i)

Depending upon the power of “i”, it can take the following values;

i 4k+1 = i.i 4k+2  = -1 i 4k+3 =  -i.i 4k = 1

Where k can have an integral value (positive or negative).

Similarly, we can find for the negative power of i , which are as follows;

i -1 = 1 / i

Multiplying and dividing the above term with i, we have;

i -1 = 1 / i  ×  i/i  × i -1   = i / i 2 = i / -1 = -i / -1 = -i

Note: √-1 × √-1 = √(-1  × -1) = √1 = 1 contradicts to the fact that i 2  = -1.

Therefore, for an imaginary number, √a × √b is not equal to √ab.

Let us see some of the identities.

  • (z 1  + z 2 ) 2 = (z 1 ) 2  + (z 2 ) 2  + 2 z 1  × z 2
  • (z 1 – z 2 ) 2 = (z 1 ) 2  + (z 2 ) 2  – 2 z 1  × z 2
  • (z 1 ) 2 – (z 2 ) 2 =  (z 1  + z 2 )(z 1 – z 2 )
  • (z 1  + z 2 ) 3  = (z 1 ) 3   + 3(z 1 ) 2  z 2   +3(z 2 ) 2  z 1   + (z 2 ) 3
  • (z 1  – z 2 ) 3  = (z 1 ) 3   – 3(z 1 ) 2  z 2   +3(z 2 ) 2  z 1   – (z 2 ) 3

The properties of complex numbers are listed below:

  • The addition of two conjugate complex numbers will result in a real number
  • The multiplication of two conjugate complex number will also result in a real number
  • If x and y are the real numbers and x+yi =0, then x =0 and y =0
  • If p, q, r, and s are the real numbers and p+qi = r+si, then p = r, and q=s
  • The complex number obeys the commutative law of addition and multiplication.

          z 1 +z 2   = z 2 +z 1

          z 1 . z 2   = z 2 . z 1

  • The complex number obeys the associative law of addition and multiplication.

         (z 1 +z 2 ) +z 3 = z 1 + (z 2 +z 3 )

          (z 1 .z 2 ).z 3 = z 1 .(z 2 .z 3 )

  • The complex number obeys the distributive law

           z 1 .(z 2 +z 3 ) = z 1 .z 2 + z 1 .z 3

  • If the sum of two complex number is real, and also the product of two complex number is also real, then these complex numbers are conjugate to each other.
  • For any two complex numbers, say z 1 and z 2 , then |z 1 +z 2 | ≤ |z 1 |+|z 2 |
  • The result of the multiplication of two complex numbers and its conjugate value should result in a complex number and it should be a positive value.

Modulus and Conjugate

Let z = a+ib be a complex number.

The Modulus of z is represented by |z|.

Mathematically,

Argand Plane and Polar Form

Similar to the XY plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.

We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis. Check out the detailed  argand plane and polar representation of complex numbers  in this article and understand this concept in a detailed way along with solved examples.

Solved Problems

Example 1:  Simplify

a) 16i + 10i(3-i)

b) (7i)(5i)

c) 11i + 13i – 2i

Solution:  

= 16i + 10i(3) + 10i (-i)

= 16i +30i – 10 i 2

= 46 i – 10 (-1)

b) (7i)(5i) = 35  i 2  = 35 (-1) = -35

c) 11i + 13i – 2i = 22i

Example 2:  Express the following in a+ib form:

(5+√3i)/(1-√3i).

Given: (5+√3i)/(1-√3i)

Complex number example

Practice Questions

Based on the explanation given above in this article, try to solve the following questions.

  • Solve (4i-7)+(1-i)
  • Solve i.(i-1)
  • (1+i)/(1-i) = 
  • Find z 2 , if z = 6i

Frequently Asked Questions on Complex Numbers

What is meant by complex numbers.

The complex number is the combination of a real number and imaginary number. An example of a complex number is 4+3i. Here 4 is a real number and 3i is an imaginary number.

How to divide the complex numbers?

To divide the complex number, multiply the numerator and the denominator by its conjugate. The conjugate of the complex number can be found by changing the sign between the two terms in the denominator value. Then apply the FOIL method to simplify the expression.

Mention the arithmetic rules for complex numbers.

The arithmetic rules of complex numbers are: Addition Rule: (a+bi) + (c+di) = (a+c)+ (b+d)i Subtraction Rule: (a+bi) – (c+di) = (a-c)+ (b-d)i Multiplication Rule: (a+bi) . (c+di) = (ac-bd)+(ad+bc)i

Write down the additive identity and inverse of complex numbers.

The additive identity of complex numbers is written as (x+yi) + (0+0i) = x+yi. Hence, the additive identity is 0+0i. The additive inverse of complex numbers is written as (x+yi)+ (-x-yi) = (0+0i). Hence, the additive inverse is -x-yi.

Write down the multiplicative identity and inverse of the complex number.

The multiplicative identity of complex numbers is defined as (x+yi). (1+0i) = x+yi. Hence, the multiplicative identity is 1+0i. The multiplicative identity of complex numbers is defined as (x+yi). (1/x+yi) = 1+0i. Hence, the multiplicative identity is 1/x+yi.

Learn more about the Identities, conjugate of the complex number, and other complex numbers related concepts at BYJU’S. Also, get additional study materials for various maths topics along with practice questions, examples, and tips to be able to learn maths more effectively.

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Course: Algebra 2   >   Unit 2

Multiplying complex numbers.

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Heavy Machinery Meets AI

  • Vijay Govindarajan
  • Venkat Venkatraman

solving problems using complex number

Until recently most incumbent industrial companies didn’t use highly advanced software in their products. But now the sector’s leaders have begun applying generative AI and machine learning to all kinds of data—including text, 3D images, video, and sound—to create complex, innovative designs and solve customer problems with unprecedented speed.

Success involves much more than installing computers in products, however. It requires fusion strategies, which join what manufacturers do best—creating physical products—with what digital firms do best: mining giant data sets for critical insights. There are four kinds of fusion strategies: Fusion products, like smart glass, are designed from scratch to collect and leverage information on product use in real time. Fusion services, like Rolls-Royce’s service for increasing the fuel efficiency of aircraft, deliver immediate customized recommendations from AI. Fusion systems, like Honeywell’s for building management, integrate machines from multiple suppliers in ways that enhance them all. And fusion solutions, such as Deere’s for increasing yields for farmers, combine products, services, and systems with partner companies’ innovations in ways that greatly improve customers’ performance.

Combining digital and analog machines will upend industrial companies.

Idea in Brief

The problem.

Until recently most incumbent industrial companies didn’t use the most advanced software in their products. But competitors that can extract complex designs, insights, and trends using generative AI have emerged to challenge them.

The Solution

Industrial companies must develop strategies that fuse what they do best—creating physical products—with what digital companies do best: using data and AI to parse enormous, interconnected data sets and develop innovative insights.

The Changes Required

Companies will have to reimagine analog products and services as digitally enabled offerings, learn to create new value from data generated by the combination of physical and digital assets, and partner with other companies to create ecosystems with an unwavering focus on helping customers solve problems.

For more than 187 years, Deere & Company has simplified farmwork. From the advent of the first self-scouring plow, in 1837, to the launch of its first fully self-driving tractor, in 2022, the company has built advanced industrial technology. The See & Spray is an excellent contemporary example. The automated weed killer features a self-propelled, 120-foot carbon-fiber boom lined with 36 cameras capable of scanning 2,100 square feet per second. Powered by 10 onboard vision-processing units handling almost four gigabytes of data per second, the system uses AI and deep learning to distinguish crops from weeds. Once a weed is identified, a command is sent to spray and kill it. The machine moves through a field at 12 miles per hour without stopping. Manual labor would be more expensive, more time-consuming, and less reliable than the See & Spray. By fusing computer hardware and software with industrial machinery, it has helped farmers decrease their use of herbicide by more than two-thirds and exponentially increase productivity.

  • Vijay Govindarajan is the Coxe Distinguished Professor at Dartmouth College’s Tuck School of Business, an executive fellow at Harvard Business School, and faculty partner at the Silicon Valley incubator Mach 49. He is a New York Times and Wall Street Journal bestselling author. His latest book is Fusion Strategy: How Real-Time Data and AI Will Power the Industrial Future . His Harvard Business Review articles “ Engineering Reverse Innovations ” and “ Stop the Innovation Wars ” won McKinsey Awards for best article published in HBR. His HBR articles “ How GE Is Disrupting Itself ” and “ The CEO’s Role in Business Model Reinvention ” are HBR all-time top-50 bestsellers. Follow him on LinkedIn . vgovindarajan
  • Venkat Venkatraman is the David J. McGrath Professor at Boston University’s Questrom School of Business, where he is a member of both the information systems and strategy and innovation departments. His current research focuses on how companies develop winning digital strategies. His latest book is Fusion Strategy: How Real-Time Data and AI Will Power the Industrial Future.  Follow him on LinkedIn . NVenkatraman

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Google’s new gemini ai is having some problems.

Who would've thought AI was hard to get right?

Google just renamed its AI chatbot Bard to "Gemini," the same name as the underlying large language model. As part of this change, there are changes like a more polished UI, image generation, and a paid version that (just like the paid version of ChatGPT) gives you access to a smarter, more capable model. Google is also building Gemini as a potential replacement for Google Assistant, but it's not quite ready for that yet.

Some people trying the updated Gemini chatbot are taking to Reddit and other social media platforms to complain about issues that didn't seem to affect the earlier Bard chatbot or Google Assistant. Some issues are rather silly and not harmful, but still weird. For example, a handful of people have complained that Gemini sometimes switches languages to Russian or Thai mid-sentence to go back (or not) to English. Others have complained that Gemini's restrictions seem a bit excessive, as it outright refuses to comply with a number of requests that aren't harmful and can be done by other chatbots (and even by the previous version of Bard). These restrictions seem to get a little more annoying once you take into account that they can disrupt your workflow. One user complained that Gemini is "unable" to proofread text, which is a basic functionality in other chatbots.

Many issues are related to the fact that you can use Gemini this as your assistant app in place of the old reliable Google Assistant. Trying to do the same things you did previously with Assistant, however, can yield mix results. One person complained that Gemini couldn't figure out Assistant's old routines that have been in place for years.

AI might be the future, but for the time being, it will still need a lot of work before it can actually replace the tech we use on a daily basis. It's certainly weird for an "improvement" from Google to actually be a regression from something that was decent before. Then again, the initial version of Bard also had some considerable problems and things to improve, so hopefully, Google can fix stuff up from here.

Source: Reddit ( 1 , 2 , 3 , 4 , 5 , 6 )

IMAGES

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  3. Solutions of Problems Based on Complex Numbers || Solving Complex

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COMMENTS

  1. Complex number

    The complex numbers arise when we try to solve equations such as . Contents 1 Derivation 2 Formal Definition 3 Parts 4 Operations 4.1 Examples 5 Alternate Forms 6 Topics 7 Problems 7.1 Introductory 7.2 Intermediate 7.3 Olympiad 8 See also Derivation

  2. Algebra

    Section 1.7 : Complex Numbers Perform the indicated operation and write your answer in standard form. (4−5i)(12+11i) ( 4 − 5 i) ( 12 + 11 i) Solution (−3 −i)−(6 −7i) ( − 3 − i) − ( 6 − 7 i) Solution (1+4i)−(−16+9i) ( 1 + 4 i) − ( − 16 + 9 i) Solution 8i(10+2i) 8 i ( 10 + 2 i) Solution (−3 −9i)(1+10i) ( − 3 − 9 i) ( 1 + 10 i) Solution

  3. Complex numbers

    Math Algebra (all content) Unit 16: Complex numbers About this unit This topic covers: - Adding, subtracting, multiplying, & dividing complex numbers - Complex plane - Absolute value & angle of complex numbers - Polar coordinates of complex numbers What are the imaginary numbers? Learn Intro to the imaginary numbers Intro to the imaginary numbers

  4. Intro to complex numbers (article)

    Math > Algebra 2 > Complex numbers > Complex numbers introduction Intro to complex numbers Google Classroom Learn what complex numbers are, and about their real and imaginary parts. In the real number system, there is no solution to the equation x 2 = − 1 .

  5. Algebra

    The standard form of a complex number is a + bi where a and b are real numbers and they can be anything, positive, negative, zero, integers, fractions, decimals, it doesn't matter. When in the standard form a is called the real part of the complex number and b is called the imaginary part of the complex number.

  6. Complex numbers

    Quiz Unit test About this unit Welcome to the world of imaginary and complex numbers. We'll learn what imaginary and complex numbers are, how to perform arithmetic operations with them, represent them graphically on the complex plane, and apply these concepts to solve quadratic equations in new ways. The imaginary unit i Learn

  7. Solving equations using complex numbers.

    Problem 1 Use imaginary numbers to solve this practice equation: X 2 = - 13 Problem 2 Use imaginary numbers to solve this practice equation: X 2 = - 19 Problem 3 Use imaginary numbers to solve this practice equation: X 2 = - 36 How to solve equations using complex numbers

  8. Complex Numbers

    A Complex Number is a combination of a Real Number and an Imaginary Number Real Numbers are numbers like: 1 12.38 −0.8625 3/4 √2 1998 Nearly any number you can think of is a Real Number! Imaginary Numbers when squared give a negative result. Normally this doesn't happen, because: when we square a positive number we get a positive result, and

  9. 9.6: Introduction to Complex Numbers and Complex Solutions

    For example, the real number 5 is also a complex number because it can be written as \(5+0i\) with a real part of 5 and an imaginary part of 0. Hence the set of real numbers, denoted R, is a subset of the set of complex numbers, denoted C. Adding and subtracting complex numbers is similar to adding and subtracting like terms.

  10. Complex Numbers, Defined, with examples and practice problems

    Problem 1 Identify the coordinates of all complex numbers represented in the graph on the right. Problem 2 In what quadrant, is the complex number 2 − i 2 − i ? Problem 3 In what quadrant, is the complex number 2i − 1 2 i − 1 ? Problem 4 In what quadrant, is the complex number −i − 1 − i − 1 ? Definition and examples of complex numbers.

  11. Complex Numbers Calculator

    To multiply two complex numbers z1 = a + bi and z2 = c + di, use the formula: z1 * z2 = (ac - bd) + (ad + bc)i. What is a complex number? A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1.

  12. Practice Complex Numbers

    Concept Quizzes. Complex Numbers Warmup. Complex Numbers Arithmetic. Complex Conjugates - Arithmetic. Complex Numbers - Argand Plane. Complex Numbers - Factoring Polynomials. Complex Numbers Problem Solving.

  13. Complex Numbers: Problems with Solutions

    Complex Numbers Complex Numbers: Problems with Solutions Theory Addition and subtraction of complex numbers: Let (a + bi) and (c + di) be two complex numbers, then: (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) - (c + di) = (a - c) + (b - d)i Reals are added with reals and imaginary with imaginary. Complex numbers multiplication:

  14. 5.1: The Complex Number System

    There is another concept related to complex number that is based on the following bit of algebra. (a + bi)(a − bi) = a2 − (bi)2 = a2 − b2i2 = a2 + b2. The complex number a − bi is called the complex conjugate of a + bi. If we let z = a + bi, we denote the complex conjugate of z as ˉz. So ˉz = ¯ a + bi = a − bi.

  15. Complex Numbers

    Examples and questions with detailed solutions on using De Moivre's theorem to find powers and roots of complex numbers. . Complex Numbers - Basic Operations. A tutorial on how to find the conjugate of a complex number and add, subtract, multiply, divide complex numbers supported by online calculators. Complex Numbers Problems with Solutions ...

  16. Complex Numbers and the Quadratic Formula

    Purplemath. For the time being, you will probably be using complex numbers only in the context of solving quadratic equations for their zeroes. (There are many other practical uses for complexes, but you'll have to wait for more interesting classes like "Engineering 201" to get to the good stuff.) So let's refresh on the topic.

  17. Complex numbers: FAQ (article)

    An irrational number is a real number that cannot be represented as the ratio of two integers. For example, pi (approximately 3.14), is not equal to x/y for any two integers x and y, which is why it is irrational. An imaginary number is any real multiple of the imaginary unit 'i' (equal to the square root of -1). Learn for free about math, art ...

  18. Complex Numbers Problems with Solutions and Answers

    a) Show that the complex number 2i is a root of the equation. z 4 + z 3 + 2 z 2 + 4 z - 8 = 0. b) Find all the roots root of this equation. P (z) = z 4 + a z 3 + b z 2 + c z + d is a polynomial where a, b, c and d are real numbers. Find a, b, c and d if two zeros of polynomial P are the following complex numbers: 2 - i and 1 - i.

  19. Problems on Complex Numbers

    We will learn step-by-step how to solve different types of problems on complex numbers using the formulas. 1. Express \((\frac{1 + i}{1 - i})^{3}\) in the form A + iB where A and B are real numbers. ... 11 and 12 Grade Math From Problems on Complex Numbers to HOME PAGE. New! Comments Have your say about what you just read! Leave me a comment in ...

  20. Complex Numbers Solver

    With its user-friendly interface and accurate results, the MathCrave Fractions Calculator takes the hassle out of dealing with fractions and ensures you get the correct answers every time. Say goodbye to manual calculations and hello to a faster, more efficient way of working with fractions. Solve complex numbers problems with MathCrave complex ...

  21. Complex Number Calculator

    Algebra Complex Number Calculator Step 1: Enter the equation for which you want to find all complex solutions. The Complex Number Calculator solves complex equations and gives real and imaginary solutions. Step 2: Click the blue arrow to submit.

  22. Complex Numbers (Definition, Formulas, Examples)

    Combination of both the real number and imaginary number is a complex number. Examples of complex numbers: 1 + j. -13 - 3i. 0.89 + 1.2 i. √5 + √2i. An imaginary number is usually represented by 'i' or 'j', which is equal to √-1. Therefore, the square of the imaginary number gives a negative value.

  23. Multiplying complex numbers (article)

    Solution If your instinct tells you to distribute the − 4 , your instinct would be right! Let's do that! − 4 ( 13 + 5 i) = − 4 ( 13) + ( − 4) ( 5 i) = − 52 − 20 i And that's it! We used the distributive property to multiply a real number by a complex number. Let's try something a little more complicated.

  24. Heavy Machinery Meets AI

    The Problem. Until recently most incumbent industrial companies didn't use the most advanced software in their products. But competitors that can extract complex designs, insights, and trends ...

  25. How China Broke One Man's Dreams

    Gao Zhibin is among the thousands of migrants disillusioned with their home country who have risked the perilous crossing into the United States.

  26. Google's New Gemini AI Is Having Some Problems

    Google just renamed its AI chatbot Bard to "Gemini," the same name as the underlying large language model. As part of this change, there are changes like a more polished UI, image generation, and a paid version that (just like the paid version of ChatGPT) gives you access to a smarter, more capable model.