Podcast
Questions and Answers
ما هو مقياس العدد المركب $z_1 = 1 + i$ ؟
ما هو مقياس العدد المركب $z_1 = 1 + i$ ؟
- $ rac{ oot{2}{2}}$
- $ oot{2}{1}$
- $ oot{2}{3}$
- $ oot{2}{2}$ (correct)
ما هي قيمة الزاوية (الحجة) للعدد المركب $z = 3 - 3i$ ؟
ما هي قيمة الزاوية (الحجة) للعدد المركب $z = 3 - 3i$ ؟
- π/3
- -π/4 (correct)
- -π/3
- π/4
ماذا يحدث عندما يتم ضرب عدد مركب بـ $[1, θ]$ ؟
ماذا يحدث عندما يتم ضرب عدد مركب بـ $[1, θ]$ ؟
- يظل دون تغيير
- يؤثر على الجزء الحقيقي فقط
- يزيد المقياس فقط
- يؤثر على الزاوية فقط (correct)
إذا كانت $z_1 = 1 + i$ و $z_2 = 3 + i$، ما هو ناتج $z_1 z_2$ ؟
إذا كانت $z_1 = 1 + i$ و $z_2 = 3 + i$، ما هو ناتج $z_1 z_2$ ؟
عند مضاعفة عدد مركب بـ $[0.5, π]$، ماذا يحدث لمقياسه؟
عند مضاعفة عدد مركب بـ $[0.5, π]$، ماذا يحدث لمقياسه؟
ما هو الناتج النهائي لتبسيط التعبير $\frac{1}{i}$؟
ما هو الناتج النهائي لتبسيط التعبير $\frac{1}{i}$؟
عند تبسيط التعبير $\frac{3}{1+i}$، ما هو الشكل النهائي؟
عند تبسيط التعبير $\frac{3}{1+i}$، ما هو الشكل النهائي؟
إذا كان ز=4+5i، ما هو الحل لـ $z - (1 - i)$؟
إذا كان ز=4+5i، ما هو الحل لـ $z - (1 - i)$؟
ما هو الناتج النهائي لتبسيط التعبير $\frac{4+7i}{2+5i}$؟
ما هو الناتج النهائي لتبسيط التعبير $\frac{4+7i}{2+5i}$؟
ما هو الشكل الحل لـ المعادلة $(1 + 2i)z = 2 + 5i$؟
ما هو الشكل الحل لـ المعادلة $(1 + 2i)z = 2 + 5i$؟
Flashcards
Modulus of a complex number
Modulus of a complex number
The modulus (or absolute value) of a complex number z = a + bi is the distance from the origin to the point (a, b) in the complex plane. It is calculated as |z| = √(a² + b²).
Argument of a complex number
Argument of a complex number
The argument of a complex number z = a + bi is the angle θ between the positive real axis and the line connecting the origin to the point (a, b) in the complex plane. It is often represented as arg(z) or θ and is measured in radians.
Multiplying complex numbers in polar form
Multiplying complex numbers in polar form
When multiplying complex numbers in polar form (like [r, θ]), the moduli multiply and the arguments add. [r₁, θ₁] * [r₂, θ₂] = [r₁r₂, θ₁ + θ₂].
Complex number in polar form
Complex number in polar form
Signup and view all the flashcards
Argand Diagram
Argand Diagram
Signup and view all the flashcards
Complex Conjugate
Complex Conjugate
Signup and view all the flashcards
Simplifying Complex Fractions
Simplifying Complex Fractions
Signup and view all the flashcards
Solving Equations with Complex Numbers
Solving Equations with Complex Numbers
Signup and view all the flashcards
Complex Number Form (a + bi)
Complex Number Form (a + bi)
Signup and view all the flashcards
Solving for z (Example)
Solving for z (Example)
Signup and view all the flashcards
Study Notes
Chapter 1: Number Systems and Inequalities
- Mathematics deals with various objects, including numbers, points, lines, planes, triangles, circles, angles, equations, functions, etc.
- Sets are collections of objects with a common property, which can be finite or infinite.
- An element "a" belongs to a set "M" (written as a ∈ M), read as "a is an element of M."
- If an object doesn't belong to a set "M" (written as a ∉ M), read as "a is not an element of M."
- Sets can be described by listing their elements or by defining a rule.
Subsets, Power Sets, Equality of Sets
- A set A is a subset of a set B (written as A ⊆ B) if every element of A is also an element of B.
- The power set of a set A, denoted by P(A) is the set of all subsets of A.
- Two sets A and B are equal (written as A = B) if A ⊆ B and B ⊆ A.
Finite and Infinite Sets
- A set M is finite if its elements can be numbered from 1 to n such that each element appears only once. Otherwise, it is infinite.
- Examples of infinite sets include natural numbers (N), integers (Z), rational numbers (Q), and real numbers (R).
Set Operations
- Intersection (A ∩ B) of sets A and B is the set of elements that belong to both A and B.
- Union (A ∪ B) of sets A and B is the set of elements that belong to A or B or both.
- Set difference (A \ B) of sets A and B is the set of elements that belong to A but not to B.
- Disjoint sets are sets with no common elements (their intersection is empty).
Additional Lemmas and Theorems
- Various lemmas and theorems, such as those related to the laws of commutativity, associativity, and De Morgan's rules, govern set operations and relationships among sets.
- Different set properties are highlighted, such as the empty set being a subset of every set.
Chapter 1.2: Numbers
- Whole numbers, fractions, and zero, together with their positive and negative counterparts, are called rational numbers.
- Rational numbers can be expressed as a ratio p/q, where p and q are integers and q ≠ 0.
- Numbers with non-terminating, non-repeating decimal representations are called irrational numbers.
- Examples of irrational numbers include √2, √3, and π.
- The collection of all rational and irrational numbers makes up the set of real numbers (R).
Absolute Value
- The absolute value of a real number x, written as |x|, is the non-negative value of x regardless of its sign.
- |x| = x if x ≥ 0, and |x| = −x if x < 0.
- There are properties associated with the absolute value operation.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
