If z1 = a1 + b1j and z2 = a2 + b2j then z1 + z2 = (a1 + a2) + (b1 + b2)j z1 − z2 = (a1 − a2) + (b1 − b2)j

Consequently, in order to add the complex numbers, all that has to be done is to add the real portions and the imaginary parts together.

Example

If z1 = 13 + 5j and z2 = 8 − 2j find a) z1 + z2, b) z2 − z1.

Solution

- a) z1 + z2 = (13 + 5j) + (8 − 2j) = 21 + 3j.
- b) z2 − z1 = (8 − 2j) − (13 + 5j) = −5 − 7j

The operation of multiplying complex numbers:-

When multiplying two complex integers, in addition to using the standard rules of algebra, we also take into account the fact that j2 = 1.

If the two complex numbers are z1 and z2, then the product of those values is expressed as z1z2.

**Example**

If z1 = 5 − 2j and z2 = 2 + 4j find z1z2.

Solution

z1z2 = (5 − 2j)(2 + 4j) = 10 + 20j − 4j − 8j2

When j2 is changed to 1, we get the following:

z1z2 = 10 + 16j − 8(−1) = 18 + 16j

In general, the following is the result that we get:

Given that z1 = a1 + b1j and z2 = a2 + b2j, the conclusion is that

z1z2 = (a1 + b1j)

(a2 + b2j) = a1a2 + a1b2j + b1a2j + b1b2j2 = (a1a2 − b1b2) + j(a1b2 + a2b1)

**Construct**

A number that can be expressed using a finite amount of integer operations such as addition, subtraction, multiplication, division, and square root extraction.

These values relate to line segments that can be created using a straightedge and a compass alone.

If the value of z1 is equal to 12 and the value of z2-3-4i is equal to 5, then the minimal value of z1 minus z2 is

Solution:

Given |z1| = 12 and |z2-3-4i| = 5

|z1-z2| ≥|z1| -|z2| |z2-3-4i| ≥ |z2| -|3+4i| |z2-3-4i| ≥ |z2| -√(32+42) |z2| ≥ 10

Minimum value of |z1-z2| ≥|z1| -|z2|

|z1-z2| ≥ 12 -10 |z1-z2| ≥ 2

**Modulus**

The word measure or method comes from the Latin word modus, from which we get the diminutive “modulus.” It, or more specifically, its plural moduli.

The modulus is the number that determines where the counting begins again after reaching the maximum value.

Mathematically speaking, when we claim that a mod n is congruent to b mod n, we are stating that both a and b have the same residual when they are divided by n.

This is what we mean when we say that a mod n is congruent to b mod n.

**Characteristics of the modulus**

The fundamental building block of modular arithmetic is called the modulus.

The absolute value of a real or complex number, often known as the modulus (a),

In mathematics, a moduli space is a geometric space in which the points represent various algebraic and geometric objects.

The conformal modulus is a measure that can be used to determine how large a curve family is.

The modulus of continuity is a function that evaluates the degree to which a given function maintains its uniform continuity.

In a similar vein, the modulus of a Dirichlet character is a formal product of places within a number field. This concept originates from algebraic number theory.

The modular function in the theory of Haar measure, which is more commonly referred to as the modulus alone.

**Vector z**

The movement from the origin to the tip of the vector is specified by the components of the vector.

The z axis, which is not depicted, extends off the page and in your direction.

Both vectors and scalars are used.

A scalar is a quantity that may be expressed using a single number as its representation without the direction.

**Conclusion**

A vector can be broken down into its components, which provide a split of the vector.

We can compute the components of a vector by first dividing it with reference to each of the axes, and then we may compute the components.

After that, the separate components of a vector can be merged with one another to produce the full representation of the vector.

In general, vectors are represented in either a two-dimensional coordinate plane with an x-axis and a y-axis or in a three-dimensional space with an x-axis, a y-axis, and a z-axis correspondingly.

Vectors are a special kind of mathematical representation that have both magnitude and direction associated with them.