What are the Arithmetic rules for Complex Numbers?

In the Number system, the real numbers can be called the sum of the rational and the irrational numbers. All arithmetic operations may be done on these numbers in general, and they can also be represented on a number line. Real numbers are divided into several groups, including natural and whole numbers, integers, rational and irrational numbers, and so on. A real number is one that can be discovered in the actual world. Numbers may be found all over the place. In addition to other things, natural numbers are utilized to count things, rational numbers are utilized to address fractions, irrational numbers are utilized to calculate the square root of numbers, and integers are used for the measurement of temperatures. Example: 2, 58, -98, 0, 0.5, etc.

Complex Numbers

A complex number is a component of a number system that includes real numbers and a particular element labeled I sometimes known as the imaginary unit, and which obeys the equation i2 = 1. Furthermore, every complex number may be written as a + bi, where a and b are both real values & and I being an imaginary number termed "iota." For example: 12 + 52j, -134 - 13i, √9 + √7i, etc.

Properties of the complex numbers

• Let z = a + ib be a complex number. Then the Modulus of z can be represented by |z|.
• The conjugate of “z” is = a - ib.
• The complex number obeys the distributive law i.e, z₁ × (z₂ + z₃) = z₁ × z₂ + z₁ × z₃
• The complex number obeys the commutative law of addition and multiplication i.e, 

1. z₁ + z₂ = z₂ + z₁
2. z₁ × z₂ = z₂ × z₁

• If two conjugate complex numbers are multiplied, the result will be a real number.

What are the Arithmetic rules for Complex Numbers?

Answer:

Following are the arithmetic rules for complex numbers,

1. Addition: For addition of complex numbers, one can write (a + ib) + (c + id) = (a + c) + i(b + d).
2. Subtraction: For subtraction of complex numbers, one can write (a + ib) - (c + id) = (a - c) + i(b - d).
3. Multiplication: For multiplication of complex numbers,we can write (a + ib). (c + id) = (ac - bd) + i(ad + bc).
4. Division: For division of complex numbers, write (a + ib) / (c + id) = (ac + bd)/ (c² + d²) + i(bc - ad) / (c² + d²)
5. Additive identity: For the additive identity, write, (a + bi) + (0 + 0i) = a + bi
6. Additive Inverse: Also for the additive inverse, (a + bi) + (-a - bi) = (0 + 0i) = 0

Sample Problems

Question 1: Simplify  the value of: 20i + 5i(6 - i)

Solution:

Given, 20i + 5i(6 - i)

= 20i + 30i - 5i²

= 50i - 5 × (-1)

= 50i +5

Question 2: Find the modulus of -6 + 2i.

Solution:

Let z = -6 + 2i.

Then the modulus of z = |z| = \sqrt{6^2 + 2^2}

= |z| = \sqrt{40}

Hence the modulus of -6 + 2i is \sqrt{40}.

Question 3: Evaluate (2 + 3i)(4 - 6i)²  and write the end result in the form of (a + bi).

Solution:

Evaluating the second part,

(4 - 6i)² = 4² - 48i + 36i² = -20 - 48i

Further evaluating,

= (2 + 3i)(-20 - 48i)

= -40 - 96i - 60i + 144 

= 104 - 156i

Question 4: Which of the following is rational in nature?

i, i³, i², i⁵, i²

Solution:

i = √{-1}

i² = -1 

i³ = i × i² = i × -1 = -i 

i⁴ = i² × i² = -1 × -1 = 1 

i⁵ = i × i⁴ = i × 1 = i 

Hence i, i⁵, i³  are not rational, and i², i⁴ is rational in nature.