Brief Discussion On Subtraction Of Complex Numbers

We use the formulas (a + ib) – (c + id) = (a – c) + i (b – d) for removing complex numbers and (a + ib) + (c + id) = (a + c) + i(b + d) for adding complex numbers.

When subtracting complex numbers, the minus sign must first be distributed into the 2nd complex number. Then rearrange the terms so that similar terms are near each other.

We’ll talk about a common mathematics operation called subtraction of 2 complex numbers.

What is the best way to subtract Complex Numbers?

If z1 = p + iq and z2 = r + is any 2 complex numbers,

Then z1 – z2 = z1 + (-z2)

                       = (p + iq) + (-r – is) 

                       = (p – r) + i is the subtraction of z2 from z1 (q – s)

The following are the steps for subtracting complex numbers:

First, disperse the negatives.

Step 2: Combine the real and imaginary parts of the complex number in a single group.

Step 3: Combine and simplify similar terms.

If z1 = 6 + 4i and z2 = -7 + 5i,

then z1 – z2 = (6 + 4i) – (-7 + 5i)

                       = (6 + 4i) + (7 – 5i) 

= (6 + 4i) + (7 – 5i), [Distributing the negative sign] 

                       = (6 + 7) + (4 – 5) i [The real and imaginary parts of a complex number are combined.]

                       = 13 – i [Combining and simplifying similar phrases]

   and z2 – z1 = (-7 + 5i) – (6 + 4i) 

  = (-7 + 5i) + (-6 – 4i), [Distributing the negative value]

  = (-7 – 6) + (5 – 4) i,

Complex Number Subtraction

Two difficult numbers by combining the real and imaginary components of both complex numbers & applying the subtraction operation independently on each of them, z1 = a + ib and z2 = c + id can be subtracted. z1 – z2 = (a + ib) – (c + id) = (a – c) + I (b – d) is the formula for subtracting complex numbers.

When z = z1 – z2, z = (a – c) + I (b – d)

Subtracting Complex Numbers: Properties

• Complex numbers generated by adding or subtracting complex numbers have the same closure feature as complex numbers.

• The associative attribute only applies to the addition of complex numbers. That is, (z1 + z2) + z3 = z1 + (z2 + z3) for any three complex numbers z1, z2, & z3.

• The commutative property is true when two complex numbers are added. We have z1 + z2 = z2 + z1 for any two complex numbers z1 and z2.

• The additive identity of complex numbers is 0 in this case, because z + 0 = 0 + z.

• Additive inverse: The additive inverse of a complex number z is -z, because z + (-z) = 0.

Subtraction and Negation

Negation has a great geometric interpretation as well. Because the negative of x + yi = –x – yi, the negation of a complex number will be opposite 0 and at the same distance. For example, z = 2 + I is 2 units to the right and one unit up, while -z = –2 – I is 2 units to the left and one unit down.

Negation can also be understood as a plane C transformation. Every point z is directed to its negation –z when the plane is rotated 180 degrees around 0. Negation produces a 180° rotation.

The geometric rule for subtraction can be deduced from addition and negation. To identify the location of z – w, first negate w by locating the point opposite 0 and then apply the parallelogram rule.

Subtraction of w can be interpreted as a modification of C: the plane is moved all along vector from 0 to –w. Another way to put it is that the plane is translated all along w0 vector.

Conclusion

Mathematical operations on complex numbers include subtracting complex numbers. Let us review the definition of complex numbers before diving into the details of adding complex numbers. A complex number is made up of two numbers: a real and an imagined one. It is commonly symbolised by z and has the form a + ib. 

When subtracting complex numbers, the real and imaginary components of the number are subtracted separately. In the same way, we remove the real and imaginary components of complex numbers separately when subtracting them.

Subtracting two binomials is the same as adding complex numbers. We simply need to combine similar phrases. Although all real number are complex numbers, not all complex numbers must be real. Commutative law does not apply when subtracting complex numbers. To subtract complex numbers in polar form, we must first convert them to rectangle form before performing the operation. The final result is then converted to polar form.