The complex conjugate is another complex number that is connected with every complex number. A complex conjugate of a complex number is another complex number with the same real portion as the original complex number and the same magnitude but opposite sign as the original complex number. A real number is the product of a complex number and its complex conjugate.

A complex conjugate is the mirror image of a complex number in the Argand plane about the horizontal axis (real axis). In this article, we will look at the notion of a complex conjugate, its properties, the complex root theorem, and various complex conjugate applications.

Meaning and Origin of the Asterisk

The asterisk * is a typographical symbol. It was given its name because it resembles a typical portrayal of a heraldic star. An asterisk * is usually referred to as a star in computer programming. As a wildcard character, the asterisk is frequently used in computer science to represent pointers, repetition, or multiplication.

In ice age cave art, the asterisk was already used as a symbol. There is also a two-thousand-year-old character known as the asterisks, which Aristarchus of Samothrace used to identify duplicated lines in Homeric poetry when editing it. In his Hexapla, Origen also employed the asterisks to signify missing Hebrew lines.

Use of Asterisk in Mathematics

The asterisk has numerous applications in mathematics. The list that follows shows some frequent applications but is not exhaustive.

stand-alone

• A random point in a set. When computing Riemann sums or contracting a simply linked group to the singleton set,{*} for example.

s a unary operator, denoted by the prefix.

• On vector spaces, the Hodge dual operator.

written as a subscript as a unary operator

• f denotes the pushforward (differential) of a smooth map f between two smooth manifolds.

written as a superscript as a unary operator

• A complex number’s complex conjugate (the more common notation is ).
• Hermitian Adjoint
• A matrix’s conjugate transpose, Hermitian transpose, or adjoint matrix.
• A ring’s multiplicative group, especially when the ring is a field. For example, V* is the dual space of a vector space V.
• Combining an indexed collection of objects into a single example, such as combining all cohomology groups Hk(X) into the cohomology ring H* (X).
• z* and t* are important points in statistics for z-distributions and t-distributions, respectively, infix notation, as a binary operator
• An arbitrary binary operator’s notation.
• Two groups’ free products.
• f + g is the convolution of f and g.

In all disciplines of mathematics, the asterisk is used to indicate a relationship between two quantities indicated by the same letter – one with and one without the asterisk.

The Complex Conjugate of a Number

A complex conjugate of a complex number is yet another complex number with the same real portion as the original complex number and the same magnitude as the opposite sign. A complex number has the formula a + ib, where a and b are real numbers, a is the real component, b is the imaginary part, and I is an imaginary number equivalent to the root of a negative one. The complex conjugate of a + ib with real component ‘a’ and imaginary part ‘b’ is a – ib, where ‘a’ is the real part and ‘-b’ is the imaginary part. a – ib is the argand plane reflection of a + ib about the real axis (X-axis). A complex number’s complex conjugate is used to justify the complex number.

A complex number’s complex conjugate, z, is its mirror counterpart with respect to the horizontal axis (or x-axis). z represents the complex conjugate of the complex number z*. The complex conjugate of the complex number reix in polar form is re-ix. To find the conjugate of a complex number, just substitute I with ‘-i’ in the original complex number. x + iy’s complex conjugate is x – iy, while x – iy’s complex conjugate is x + iy. Consider the following examples: 3 + 2i is the complex conjugate of 3 – 2i and 2 + 3i is the complex conjugate of 2 + 3i.

Multiplication of z*

When a complex number is multiplied by its complex conjugate, the product is a real number with a value equal to the complex number’s magnitude squared. We utilize the algebraic identities (x+y)(x-y)=x²-y² and i² = -1 to get the product value. If we multiply the complex number a + ib by its complex conjugate a – ib, we get

(ib + a)(a – ib) = a² – i²b² = a² + b²

Properties of Z*

Let us now look at a few complex conjugate qualities that can help us simplify and simplify our computations. Consider two complex numbers, z and w, as well as their complex conjugates, z* and w*.

• The complex conjugate of the product of the two complex numbers is simply the product of the complex conjugates of the two complex numbers. (zw)* = z*.w*
• The complex conjugate of the quotient of two complex numbers equals the quotient of the complex conjugates of the two complex numbers, i.e. (z/w)* = z*/w*
• The complex conjugate of the sum of two complex numbers is the sum of the complex conjugates of the two complex numbers, i.e. (z+w)* = z* +w*
• The complex conjugate of the difference between two complex numbers is equal to the difference between their complex conjugates, i.e. (z – w)* = z* – w*
• The sum of a complex number and its complex conjugate is equal to two times the complex number’s real portion, z + z* = 2Re (z)
• The difference between such a complex number and its complex conjugate is equal to twice the complex number’s imaginary portion, that is, z – z* = 2Im (z)
• The product of a complex number and its complex conjugate is equal to the complex number’s magnitude squared, that is, z.z* = |z|²
• The real component of a complex number equals the real part of its complex conjugate, and the imaginary part equals the negative of the imaginary part of its complex conjugate, so

Re(z) = Re (z*) and Im(z) = – Im(z*)

Conclusion

A complex number is one that is expressed as (x + I y), where x and y are real numbers and i =√-1  is known as iota (an imaginary unit). It’s also referred to as imaginary numbers or quantities. Depending on the values of x and y, a complex number might be wholly real or purely fictitious. x + iy’s complex conjugate is x – iy, while x – iy’s complex conjugate is x + iy. When a complex number is multiplied by its complex conjugate, the product is a real number with a value equal to the complex number’s magnitude squared. A polynomial’s complex roots come in pairs.