Number System and Complex Numbers

This set of flashcards covers key concepts of the number system, including natural numbers, whole numbers, integers, rational and irrational numbers, as well as operations involving complex numbers.

Natural Numbers

Positive whole numbers used in counting, represented as N = {1, 2, 3, 4, 5,…}.

Whole Numbers

The set of natural numbers and zero, represented as W = {0, 1, 2, 3,…}.

Integers

The set of whole numbers and negative whole numbers, represented as T = {…, -2, -1, 0, 1, 2,…}.

Rational Numbers

Numbers that can be expressed as a fraction of integers where the denominator is not zero.

Irrational Numbers

Numbers that cannot be expressed as a ratio of integers, such as √2, π, and e.

Real Numbers

Decimal expressions whose digits may or may not terminate or repeat. They include both rational and irrational numbers.

Imaginary Numbers

Numbers that involve the square root of negative one, represented as i, where i = √-1.

Complex Numbers

Numbers formed by the combination of real and imaginary numbers, expressed in the form Z = a + bi.

Sum Operation for Complex Numbers

Given Z1 = a1 + b1i and Z2 = a2 + b2i, the sum is Z1 + Z2 = (a1 + a2) + (b1 + b2)i.

Difference Operation for Complex Numbers

Given Z1 = a1 + b1i and Z2 = a2 + b2i, the difference is Z1 - Z2 = (a1 - a2) + (b1 - b2)i.

Product Operation for Complex Numbers

Given Z1 = a1 + b1i and Z2 = a2 + b2i, the product is Z1 × Z2 = (a1a2 - b1b2) + (a1b2 + b1a2)i.

Quotient Operation for Complex Numbers

The division of complex numbers involves complex conjugates and follows the formula (a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c² + d²).