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Introduction

Introduction Until now, you have been told that you cannot take the square of –1 because there is no number that when squared will result in a negative number. In other words,

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Introduction

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  1. Introduction Until now, you have been told that you cannot take the square of –1 because there is no number that when squared will result in a negative number. In other words, the square root of –1, or , is not a real number. French mathematician René Descartes suggested the imaginary unit ibe defined so that i2= –1. The imaginary unit enables us to solve problems that we would not otherwise be able to solve. Problems involving electricity often use the imaginary unit. 4.3.1: Defining Complex Numbers, i, and i2

  2. Key Concepts All rational and irrational numbers are real numbers. The imaginary unit iis used to represent the non-real value, . An imaginary number is any number of the form bi, where b is a real number, i= , and b ≠ 0. Real numbers and imaginary numbers can be combined to create a complex number system. A complex number contains two parts: a real part and an imaginary part. 4.3.1: Defining Complex Numbers, i, and i2

  3. Key Concepts, continued All complex numbers are of the form a + bi, where a and b are real numbers and iis the imaginary unit. In the general form of a complex number, a is the real part of the complex number, and bi is the imaginary part of the complex number. Note that if a = 0, the complex number a + bi is wholly imaginary and contains no real part: 0 + bi = bi. If b = 0, the complex number a + bi is wholly real and contains no imaginary part: a + (0)i= a. Expressions containing imaginary numbers can also be simplified. 4.3.1: Defining Complex Numbers, i, and i2

  4. Key Concepts, continued To simplify infor n > 4, divide n by 4, and use the properties of exponents to rewrite the exponent. The exponent can be rewritten using the quotient: n ÷ 4 = m remainder r, or , where r is the remainder when dividing n by 4, and m is a whole number. Then n = 4m + r, and r will be 0, 1, 2, or 3. Use the properties of exponents to rewrite in. in= i4m + r = i4m• ir 4.3.1: Defining Complex Numbers, i, and i2

  5. Key Concepts, continued i4= 1, so ito a multiple of 4 will also be 1: i4m= (i4)m= (1)m= 1. The expression irwill determine the value of in. Use i0, i1, i2, and i3to find in. If r = 0, then: If r = 1, then: in= i4m + r in= i4m + r in= i4m• ir in= i4m• ir in= 1 • i0in= 1 • i1 in= 1 • 1 = 1 in= i, or 4.3.1: Defining Complex Numbers, i, and i2

  6. Key Concepts, continued If r = 2, then: If r = 3, then: in= i4m + r in= i4m + r in= i4m• ir in= i4m• ir in= 1 • i2in= 1 • i3 in= 1 • –1 = –1 in= i• –1 = –i, or Only the value of the remainder when n is divided by 4 is needed to simplify in. i0= 1, i1= i, i2= –1, and i3= –i 4.3.1: Defining Complex Numbers, i, and i2

  7. Key Concepts, continued Properties of exponents, along with replacing iwith its equivalent value of , can be used to simplify the expression in. Start with n = 0. 4.3.1: Defining Complex Numbers, i, and i2

  8. Key Concepts, continued When simplifying i3, use the property i3= i2• i1, and the determined values of i2and i. When simplifying i4, use the property i4= i2• i2, and the determined value of i2. 4.3.1: Defining Complex Numbers, i, and i2

  9. Common Errors/Misconceptions incorrectly identifying the real and imaginary parts of a complex number assuming that a real number isn’t part of the complex number system incorrectly dividing a power of iby 4 incorrectly simplifying the expressions i2, i3, and i4, including stating i2= 1 4.3.1: Defining Complex Numbers, i, and i2

  10. Guided Practice Example 3 Rewrite the radical using the imaginary unit i. 4.3.1: Defining Complex Numbers, i, and i2

  11. Guided Practice: Example 3, continued Rewrite the value under the radical as the product of –1 and a positive value. 4.3.1: Defining Complex Numbers, i, and i2

  12. Guided Practice: Example 3, continued Rewrite the radical as i. 4.3.1: Defining Complex Numbers, i, and i2

  13. Guided Practice: Example 3, continued If possible, rewrite the positive value as the product of a square number and another whole number. 32 = 16 • 2, and 16 is a square number. 4.3.1: Defining Complex Numbers, i, and i2

  14. Guided Practice: Example 3, continued Simplify the radical by finding the square root of the square number. The simplified expression will be in the form whole number • i • radical. ✔ 4.3.1: Defining Complex Numbers, i, and i2

  15. Guided Practice: Example 3, continued 4.3.1: Defining Complex Numbers, i, and i2

  16. Guided Practice Example 4 Simplify i57. 4.3.1: Defining Complex Numbers, i, and i2

  17. Guided Practice: Example 4, continued Find the remainder of the power of i when divided by 4. 14 • 4 = 56; therefore, 57 ÷ 4 = 14 remainder 1. The remainder is 1. 4.3.1: Defining Complex Numbers, i, and i2

  18. Guided Practice: Example 4, continued Use the remainder to simplify the power of i. i57= 1 • i1= i ✔ 4.3.1: Defining Complex Numbers, i, and i2

  19. Guided Practice: Example 4, continued 4.3.1: Defining Complex Numbers, i, and i2

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