Let’s understand what are complex numbers before learning their properties including the Modulus of a Complex Number which is important to understand to use properly in algebraic operations involving complex numbers.
What are Complex Numbers?
A complex number is created by multiplying two real numbers and one imaginary number. A complex number is denoted by the symbol z, and its formula is a + ib. In this situation, both a and b are real numbers. The value ‘a represents the real component, which is represented by Re(z), while the value ‘b’ represents the imaginary component I’m (z). ib is also referred to as an imaginary number.
The square root of a negative value is calculated using complex numbers. It is used in a variety of scientific fields, including signal processing, electromagnetism, fluid dynamics, quantum physics, and vibration analysis.
Some examples of complex numbers are 2+3i,-2-5i,1/2+i3/2, etc.
Power of i
The iota is the letter ‘I’ that represents the imaginary component of a complex number. We have the number i2 = -1, which is used to compute the value -4 = i24 = +2i.
Now, let’s dive deep into the properties of complex numbers with examples for better understanding.
Properties of a Complex Number
The properties of complex numbers listed below can help you better understand them and perform various arithmetic/algebraic operations on them.
Modulus of a Complex Number
The distance between the origin and the point on the argand plane that represents the complex number z is the modulus of a complex number.
Let P be the point corresponding to the complex number z = x + iy.
Then OP = |z| = √(x2 + y2 ).
Since the modulus of a complex number is the distance, its value is always non-negative. Also, all complex numbers with the same modulus lie on a circle with the origin at the center and the radius r = |z|.
Conjugate a Complex Number
The conjugate of a complex number is created by taking the complex number’s identical real part and changing the imaginary part to its additive inverse. Conjugate complex numbers exist when the sum and product of two complex numbers are both real numbers. A complex number z = a + it’s conjugate is z=a-ib.
The complex number and its conjugate are added to form z+z=(a+ib)+(a-ib)=2a, and their product is z.z=(a+ib)*(a-ib)=a2+b2.
Reciprocal of a Complex Number
When dividing one complex number by another, the complex number reciprocal comes in handy. The product of one complex number and the reciprocal of another complex number equals complex number division. The reciprocal of the complex number z = a + ib is z-1=1/a+ib=a-ib/a2+b2=a/a2+b2=i(-b)/a2+b2. This also shows that z ≠z-1.
Equality of Complex Numbers
The concept of complex number equality is equivalent to that of real number equality. If the real parts, a1=a2, and imaginary parts, b1=b2, of two complex numbers, z1=a1+ib1 and z2=a2+ib2, are equal, they are said to be identical. Furthermore, two complex numbers in the polar form are equal if and only if their magnitudes differ by an integral multiple of 2π.
Ordering of Complex Numbers
Complicated numbers are incapable of being ranked. Complex numbers cannot be sorted like real numbers or other similar number systems. There is no ordering of addition and multiplication compatible complex numbers, therefore complex numbers lack the structure of an ordered field.
Furthermore, in an ordered field, the non-trivial sum of squares is not equal to 0, but in a complex number, it is comparable to 2i + 12 = 0. The magnitude of a complex number, or its distance from the origin, can be measured and represented in a two-dimensional grand plane.