Dividing Complex Numbers

The Organic Chemistry Tutor
28 Jan 201805:55
EducationalLearning
32 Likes 10 Comments

TLDRThis transcript offers a detailed explanation of how to simplify complex fractions involving imaginary numbers. It demonstrates the process of eliminating the imaginary component in the denominator by multiplying both the numerator and denominator by the conjugate or by 'i' when appropriate. The examples provided illustrate the steps to convert complex expressions into standard form (a + bi), including rationalizing the denominator to remove radicals and imaginary numbers. The transcript is a practical guide for those looking to master complex arithmetic.

Takeaways
  • 📚 To simplify a fraction with a complex number in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.
  • 🔢 When the denominator is a complex number, multiplying by the conjugate helps to eliminate the imaginary part, resulting in a real number in the denominator.
  • 👉 For example, to simplify 3 divided by 5i, multiply both the numerator and the denominator by i, resulting in 3i over -5.
  • 📉 The expression 3i/-5 can be rewritten in standard form as -3/5i, where 'a' is 0 and 'b' is -3/5.
  • ✅ To convert an expression to standard form (a + bi), ensure that the real part 'a' and the imaginary part 'b' are clearly identified.
  • 🌟 In the case of 4 + 2i divided by 3i, multiply by i to get rid of the imaginary number in the denominator, resulting in 4i + 2(-1) over -3.
  • 🧩 When simplifying, combine like terms and simplify the fractions to their lowest terms, as shown with the example 4 - 2i / 3 + 5i.
  • 📉 For rationalizing the denominator, multiply by the square root of the denominator's conjugate, as demonstrated with the example involving root 5i.
  • 👨‍🏫 The process involves distributing, combining like terms, and simplifying the resulting expressions, as shown in the step-by-step walkthrough.
  • 📝 The final simplified form of a complex fraction should be in the form of a + bi, with 'a' and 'b' being real numbers.
Q & A

  • What is the best way to simplify the fraction 3 divided by 5i?

    -To simplify the fraction 3 divided by 5i, multiply the top and bottom by i, resulting in 3i on the top and 5i squared on the bottom. The simplified fraction is then -3/5i, which can be further simplified to 0 - 3/5i, or -3/5i in standard form.

  • How can you convert the expression 4 + 2i divided by 3i to standard form?

    -First, multiply the top and bottom by the conjugate of the denominator, which in this case is 3i. The top becomes 4i + 2i^2, and the bottom becomes 3i^2. Simplify the terms to get 4i + 4i^2. Then, divide by 3i^2 to get 4/9i^2. The standard form is a + bi, where a is 4/9 and b is 4/9i, so the expression is 4/9i + 4/9i^2.

  • What is the standard form of the expression 8 divided by 3 plus 2i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 - 2i. The top becomes 8 times 3 and 2i times 3, simplifying to 24 and 6i. The bottom becomes 3 times 3 and 2i times 3, simplifying to 9 and 6i. Combine the like terms and simplify, resulting in 24/9 - 16/9i. The standard form is 24/9 - 16/9i.

  • How can you simplify the expression 4 - 2i divided by 3 plus 5i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 - 5i. The top becomes 4 times 3 and 2i times 3, simplifying to 12 and 6i. The bottom becomes 3 times 3 and 5i times 3, simplifying to 9 and 15i. Combine the like terms and simplify, resulting in 12 - 26i and 10i^2. The standard form is 12/34 - 26/34i + 10i^2.

  • What is the standard form of the expression 4 - 2i divided by 3 plus 5i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 + 5i. The top becomes 4 times 3 and 2i times 5i, simplifying to 12i + 10i^2. The bottom becomes 3 times 5i and 2i times 5i, simplifying to 15i^2. Combine the like terms and simplify, resulting in 12i + 10i^2 on the top and 15i^2 on the bottom. The standard form is 12i + 10i^2/(3 + 5i).

  • How can you simplify the fraction 8 divided by 3 plus 2i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 - 2i. The top becomes 8 times 3 and 2i times 3, simplifying to 24 and 6i. The bottom becomes 3 times 3 and 2i times 3, simplifying to 9 and 6i. Combine the like terms and simplify, resulting in 24 - 16i and 6i^2. The standard form is 24/(3 + 2i) - 16/(3 + 2i)i, which simplifies to 24/(3 + 2i) - 16/(3 + 2i)i^2.

  • What is the standard form of the expression 4 - 2i divided by 3 plus 5i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 + 5i. The top becomes 4 times 3 and 2i times 5i, simplifying to 12i + 10i^2. The bottom becomes 3 times 5i and 2i times 5i, simplifying to 15i^2. Combine the like terms and simplify, resulting in 12i + 10i^2 on the top and 15i^2 on the bottom. The standard form is 12i + 10i^2/(3 + 5i).

  • How can you simplify the fraction 4 - 2i divided by 3 plus 5i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 + 5i. The top becomes 4 times 3 and 2i times 5i, simplifying to 12i + 10i^2. The bottom becomes 3 times 5i and 2i times 5i, simplifying to 15i^2. Combine the like terms and simplify, resulting in 12i + 10i^2 on the top and 15i^2 on the bottom. The standard form is 12i + 10i^2/(3 + 5i).

  • What is the standard form of the expression 4 - 2i divided by 3 plus 5i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 + 5i. The top becomes 4 times 3 and 2i times 5i, simplifying to 12i + 10i^2. The bottom becomes 3 times 5i and 2i times 5i, simplifying to 15i^2. Combine the like terms and simplify, resulting in 12i + 10i^2 on the top and 15i^2 on the bottom. The standard form is 12i + 10i^2/(3 + 5i).

  • How can you simplify the fraction 8 divided by 3 plus 2i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 - 2i. The top becomes 8 times 3 and 2i times 3, simplifying to 24 and 6i. The bottom becomes 3 times 3 and 2i times 3, simplifying to 9 and 6i. Combine the like terms and simplify, resulting in 24 - 16i and 6i^2. The standard form is 24/(3 + 2i) - 16/(3 + 2i)i, which simplifies to 24/(3 + 2i) - 16/(3 + 2i)i^2.

  • What is the standard form of the expression 4 - 2i divided by 3 plus 5i?

    -First, multiply the top and bottom by the conjugate of the denominator, which is 3 + 5i. The top becomes 4 times 3 and 2i times 5i, simplifying to 12i + 10i^2. The bottom becomes 3 times 5i and 2i times 5i, simplifying to 15i^2. Combine the like terms and simplify, resulting in 12i + 10i^2 on the top and 15i^2 on the bottom. The standard form is 12i + 10i^2/(3 + 5i).

Outlines
00:00
📚 Simplify Complex Fractions

This paragraph discusses the process of simplifying complex fractions by eliminating the imaginary part. It introduces the concept of multiplying the numerator and denominator by the imaginary unit 'i' to transform the fraction into its standard form, which is a + bi. The example provided includes the fraction 3 divided by 5i, which is simplified to -3/5i. The paragraph then explains how to convert this expression to standard form, showing that if the denominator is squared, the result will be negative. The process of rationalizing the denominator is demonstrated, showing how to multiply the top and bottom by the 5th root of i to simplify the fraction to 0 - 4/5i.

05:01
🔢 Further Simplified Complex Fractions

This paragraph continues the process of simplifying complex fractions by applying the conjugate of the denominator to the top and bottom terms. It provides an example of simplifying the fraction 8 divided by 3 plus 2i, showing how to multiply the top by the conjugate of the denominator (3 - 2i) and the bottom by the original denominator (3 + 2i). The process involves distributing the numbers and combining like terms to simplify the expression. The final result is expressed as 24/13 - 16/13i, which is in standard form. Another example is given, showing how to simplify 4 - 2i divided by 3 plus 5i, leading to the simplified form 0 - 4/5i.

Mindmap
Keywords
💡Simplify
To simplify in mathematics, especially with complex numbers, means to reduce an expression to its simplest or most standard form. In the context of the video, simplifying involves operations such as multiplying by the conjugate or multiplying numerator and denominator by 'i' to eliminate complex denominators. For example, the script describes simplifying '3 divided by 5i' by multiplying by 'i' to get '3i over -5'.
💡Complex Number
A complex number is a number that consists of a real part and an imaginary part, typically written in the form a + bi, where 'i' is the imaginary unit with the property that i^2 = -1. The video's theme revolves around simplifying expressions involving complex numbers, such as '3/5i' or '4 + 2i / 3i'.
💡Imaginary Number
The imaginary number is a concept related to complex numbers, represented by the symbol 'i'. It is defined as the square root of -1. In the video, the term is used when discussing the simplification of fractions with complex numbers in the denominator, such as multiplying by 'i' to eliminate the imaginary unit from the denominator.
💡Standard Form
Standard form for complex numbers is a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. The video explains how to convert complex expressions into this form, such as changing 'negative 3 over 5 times i' to '0 - 3/5 times i'.
💡Conjugate
The conjugate of a complex number is obtained by changing the sign of the imaginary part while keeping the real part the same. For example, the conjugate of '3 + 2i' is '3 - 2i'. In the script, the conjugate is used to simplify expressions by multiplying the numerator and denominator by the conjugate of the complex denominator to eliminate the imaginary part.
💡Rationalize
To rationalize a denominator means to eliminate any radicals or irrational numbers from the denominator of a fraction. In the context of the video, rationalizing involves steps such as multiplying by 'root 5i' to clear the imaginary part from the denominator, as seen in the example '4 root 5 times i / (root 5 * root 5)'.
💡Foil Method
The foil method is a technique used to multiply two binomials. It stands for 'First, Outer, Inner, Last' and helps in organizing the multiplication process. In the video, this method is used when multiplying the numerator and denominator by the conjugate, as shown in the example '8 * (3 - 2i) / (3 + 2i) * (3 - 2i)'.
💡i Squared
In complex number arithmetic, i squared (i^2) equals -1. This property is fundamental to simplifying complex expressions and is used multiple times throughout the video script, such as when simplifying '3i^2' to '-3'.
💡Divide Complex Numbers
Dividing complex numbers involves manipulating the expression to eliminate the complex denominator. The video demonstrates this by multiplying the numerator and denominator by the conjugate of the denominator or by 'i' if the denominator contains 'i'. An example from the script is dividing '4 - 2i' by '3 + 5i'.
💡Like Terms
Like terms are terms in a mathematical expression that have the same variables raised to the same powers. In the context of the video, like terms are combined when simplifying complex expressions, such as combining '12 - 26i' and '10i^2' to get '2 - 26i'.
Highlights

Simplifying 3 divided by 5i by multiplying the numerator and denominator by i.

Resulting in -3i/5 after simplification.

Converting complex numbers to standard form a + bi.

Multiplying numerator and denominator by i to eliminate imaginary unit in the denominator.

Calculating 2i^2 as -2 and 3i^2 as -3 for simplification.

Dividing by negative three to get the simplified form.

Rationalizing the denominator to eliminate the radical and imaginary number.

Multiplying by root 5i to simplify the expression.

Using the conjugate of the denominator to simplify complex fractions.

Foil method to eliminate middle terms and simplify the denominator.

Simplifying complex fractions step by step.

Combining like terms to simplify the numerator.

Reducing fractions by dividing by common factors.

Final simplified form of complex fractions.

Transcripts

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Complex NumbersSimplificationMathematicsAlgebraEducationalImaginary NumbersRationalizationConjugateFractionsTutorial