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Questions and Answers

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 $z = 3 - 3i$ ؟

  • π/3
  • -π/4 (correct)
  • -π/3
  • π/4
  • 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 $z_1 z_2$ ؟

    <p>3 + 4i</p> Signup and view all the answers

    عند مضاعفة عدد مركب بـ $[0.5, π]$، ماذا يحدث لمقياسه؟

    <p>يقل بمقدار النصف</p> Signup and view all the answers

    ما هو الناتج النهائي لتبسيط التعبير $\frac{1}{i}$؟

    <p>-i</p> Signup and view all the answers

    عند تبسيط التعبير $\frac{3}{1+i}$، ما هو الشكل النهائي؟

    <p>$\frac{3}{2}-\frac{3}{2}i$</p> Signup and view all the answers

    إذا كان ز=4+5i، ما هو الحل لـ $z - (1 - i)$؟

    <p>3 + 4i</p> Signup and view all the answers

    ما هو الناتج النهائي لتبسيط التعبير $\frac{4+7i}{2+5i}$؟

    <p>$\frac{43}{29}-\frac{6}{29}i$</p> Signup and view all the answers

    ما هو الشكل الحل لـ المعادلة $(1 + 2i)z = 2 + 5i$؟

    <p>$\frac{11}{5} + \frac{2}{5}i$</p> Signup and view all the answers

    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.

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