Our Complex Conjugate Calculator is a useful tool for performing simple, complex number operations. Continue reading to discover the answer to the question “what is a complex number?” as well as the algebraic and polar forms of complex numbers and how to multiply and divide them. You may also discover information on complex numbers’ characteristics. Most of which are based on the conjugate or absolute value of complex numbers. Also, some practical applications near the end of this article.

Complex numbers

Finding the square root of negative values is easier with complex numbers. When Hero of Alexandria, a Greek mathematician, attempted to discover the square root of a negative integer in the first century, he came across the notion of complex numbers. But all he did was turn the negative into a positive and take the numeric root value. Furthermore, Italian mathematician Gerolamo Cardano established the real identity of a complex number in the 16th century while looking for the negative roots of cubic and quadratic polynomial formulas.

Signal processing, electromagnetic, fluid dynamics, quantum physics, and vibration analysis are just a few of the fields where complex numbers are used. The sum of a real and an imaginary number is a complex number. A complex number is denoted by the letter z and has the form a + ib. Both a and b are genuine numbers in this case. The value ‘a’ is known as the real component and is indicated by Re(z), whereas ‘b’ is known as the imaginary portion and is denoted by Im (z). ib is also known as an imaginary number.

What is conjugation? The conjugate definition

A conjugate is a pair of items that are connected together. Except for one set of qualities that are diametrically opposed, these two items are identical. When you look at these two faces, you’ll notice that they’re identical except for their facial expressions: one is smiling, and the other is frowning. A math conjugate is created by altering the sign of two binomial expressions. The conjugate of x + y, for example, is x – y. x + y is also known as the conjugate of x – y. To put it another way, the two binomials are conjugates. Math conjugates have positive and negative sign instead of a grin and a frown.

“Looking at the same group from a new point of view” is the best way I’ve found to conceive of conjugation. Normal subgroups are subgroups for which the constituents “all seem the same when seen from a different perspective.” Conjugacy classes are collections of items that “all look the same when viewed from a different perspective.” When the group in issue acts on objects, this is the best option because conjugation usually results in a new element with a fairly comparable action. For example, Matn(F) The group of n x n matrices over a field F (and you can conceive of the group operating on the n x n vectors over the field F if you wish) is probably the easiest example to explain that.

How to find the complex conjugate?

Consider the two complex integers z and w and their complex conjugates z and w (with – above them).

  • The product of the complex conjugates of two complex numbers is equal to the product of the complex conjugates, i.e. z w = z w
  • The quotient of the complex conjugates of two complex numbers is equal to the quotient of the complex conjugates of the two complex numbers, that is, ( z / w ) = z / w
  • The total of the complex conjugates of two complex numbers is equal to the sum of the complex conjugates of the two complex numbers, that is, z + w = z + w
  • The difference of two complex numbers’ complex conjugates is equal to the difference of the two complex numbers’ complex conjugates, i.e. z w = z w
  • The sum of a complex number plus its complex conjugate equals twice the real component of the complex number, i.e. z + z = 2 R e ( z ).
  • The difference between a complex number and its complex conjugate is equal to twice the complex number’s imaginary portion, or z z = 2 I m ( z ).
  • The square of the magnitude of a complex number, that is, z, is equal to the product of the complex number and its complex conjugate.
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  • The real portion of a complex number is equal to the real part of its complex conjugate. The imaginary part of a complex number is equal to the negative of the imaginary part of its complex conjugate. Therefore R e ( z ) = R e ( z ) and I m ( z ) = I m ( z ).

Complex conjugate example

x + iy has a complex conjugate of x – iy, and x – iy has a complex conjugate of x + iy. When you multiply a complex number by its complex conjugate, you get a real number with a value equal to the square of the complex number’s magnitude. A polynomial’s complex roots are found in pairs.

If z = 2 – 3i and w = -4 – 7i, find the complex conjugate of the complex number 4z – i2w.

Solution: 4z – i2w = 4(2 – 3i) – i2(-4 – 7i) = 8 – 12i + 8i -14 = -6 – 4i will be simplified first. We’ll modify the sign of I to find the complex conjugate of 4z – i2w = -6 – 4i. As a result, -6 – 4i’s complex conjugate is -6 + 4i.

Multiplying complex conjugates

It’s similar to multiplying binomials when it comes to multiplying complex numbers. The main distinction is that we operate independently with the actual and imagined portions. Multiply to find the product when given a complex number and a real number.

  • Use the distributive property to your advantage.
  • Simplify.

4(2+5i). Distribute the 4. 4(2+5i)=(4⋅2)+(4⋅5i)=8+20i