Complex Numbers and Basic Properties (Benjamin C. Kuo) (Kompleks Sayฤฑlar ve Temel Özellikleri) Rectangular Form (Kartezyen Biçim) A complex number is represented in rectangular form. Rectangular Form: ๐ง = ๐ฅ + ๐๐ฆ where, ๐ = √−1 , (๐ 2 = −1, ๐ 3 = −๐, ๐ 4 = +1) (x, y) are real and imaginary coefficients of z respectively ๐ฅ = ๐ ๐{๐ง}, ๐ฆ = ๐ผ๐{๐ง} We can treat (x, y) as a point in the Cartesian coordinate frame. Complex number z representation in Rectangular and Polar forms (Cartesian coordinate frame) Polar Form (Kutupsal Biçim) Rectangular coordinate frame may also be defined by a vector R and an angle ๏ฑ. ๐ฅ = ๐ ๐๐๐ ๐, ๐ฆ = ๐ ๐ ๐๐๐ R = magnitude of z=๏ฝz๏ฝ and ๏ฑ =phase of z ๏ฑ is measured from the x axis. ๐ = |๐ง| = √๐ฅ 2 + ๐ฆ 2 , ๐ฆ ๐ = ๐ก๐๐−1 ๐ฅ Positive phase is in counter clockwise direction. R y ๏ฑ x Polar Form: ๐ง = ๐ ๐๐๐ ๐ + ๐๐ ๐ ๐๐๐ = ๐ (๐๐๐ ๐ + ๐๐ ๐๐๐), or substituting (๐๐๐ ๐ + ๐๐ ๐๐๐) → ๏๐ Polar Form: ๐ง = ๐ ๏๐ Exponential Form (Üstel Biçim) Using Euler Formula: ๐ ๏ฑ๐๐ = ๐๐๐ ๐๏ฑ๐๐ ๐๐๐ Polar Form may also be represented in exponential form as Exponential Form: ๐ง = ๐ ๐ ๐๐ , or substituting ๐ ๐๐ → ๏๐ Polar Form: ๐ง = ๐ ๏๐ Düzenleyen: Prof. Dr. Herman Sedef (YTÜ) V6 Conjugate of the Complex Number (Kompleks Sayฤฑnฤฑn Eลleniฤi) ๐ง ∗ = ๐ฅ − ๐๐ฆ = ๐ ๐๐๐ ๐ − ๐๐ ๐ ๐๐๐ = ๐ ๐ −๐๐ = ๐ ๏ − ๐, ๐ง โ ๐ง ∗ = |๐ง|2 = ๐ 2 = ๐ฅ 2 + ๐ฆ 2 Basic Properties of Complex Numbers (Kompleks Sayฤฑlarฤฑn Temel Özellikleri) CASIO fx-82MS ve fx-82ES Hesap Makinesi ile Koordinat Dönüลümleri Uyarฤฑ: Makine derece modunda olmalฤฑdฤฑr! Bu durumda ekranda D harfi görülür. CASIO fx-82ES’ te RCL E ve RCL F yapmaya gerek yoktur (x, y) veya (r, ๏ฑ) çifti ekranda aynฤฑ anda görülür. Kutupsaldan-Kartezyene Dönüลüm (Pol๏ฎ Rec) Düzenleyen: Prof. Dr. Herman Sedef (YTÜ) Kartezyenden-Kutupsala Dönüลüm (Rec๏ฎPol) V6