[2] – Operation of Complex Numbers
複數的四則運算

影片內容:
00:05 – addition and subtraction of complex numbers 複數的加減運算
02:06 – multiplication and division of complex numbers 複數的乘除運算
07:31 – illustrative example 1 例子 1
09:04 – complex numbers on calculator 如何使用計數機進行複數運算
10:37 – illustrative example 2 例子 2

所屬主題:
[高中數學] – Complex Numbers 複數

Category: S4-S6 Algebra

4 Comments

  • admin
    admin
    Reply

    回應 千羽樱 喺 Youtube 問嘅幾條問題:

    Q1: quadratic equation (二次方程) 會有兩個 real roots (實根) 或者一個 double real root (重複實根),咁 discriminant (判別式) 係負數嘅話,係咪會有兩個 complex roots (複數根) 或者一個 double complex roots (重複複數根)?

    A1: 如果 discriminant 係負數嘅話,咁條 quadratic equation 係一定有 complex roots,而且一定會有兩個,而唔會有重複複數根。因為睇返條 quadratic formula:
    \large x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
    我地之所以有重複嘅根,係因為 \ \Delta=b^2-4ac=0 ,令到條 formula 嘅 \ \pm \ 部分消失。如果 discriminant 係負數嘅話,根據條 formula,你要加或者減 \ \sqrt{b^2-4ac} ,所以逢親有 complex roots,都一定係打孖上。而且正如片入面最尾個例子所講,如果 \ a+bi \ 係其中一個 root 嘅話, a-bi \ 就一定係另外一個 root。

    Q2: 用 factorization solve quadratic equation 嘅話,計唔計到 complex roots?

    A2: 唔係話唔得嘅,不過都幾高難度。例如以下呢條 equation:
    \large x^2-8x+25=0
    咁你當然可以將條 equation factorize 為:
    \large (x-(4+3i))(x-(4-3i))=0
    去搵佢嘅兩個 complex roots \ 4+3i \ \ 4-3i \ (你將佢 expand 就會得出返上面嘅 equation),但好明顯呢個 factorization 好難憑空諗到,所以我地通常都用 quadratic formula。

    Q3: 要 solve quadratic equation,用 quadratic formula 定 factorization 方便啲?quadratic formula 又係咪可以 solve 到所有嘅 quadratic equation?

    A3: 如果係一啲好簡單嘅 quadratic equation,factorization 係會快啲 (太前題係你熟嘅話),但 quadratic formula 就真係比較萬能,佢係可以 solve 到所有嘅 quadratic equation,所以冇得唔背。不過坦白講,factorization 係一個好基礎好重要嘅 technique,唔剩係用喺 solve quadratic equations,所以冇得唔熟㗎。

    Q4: coordinate plane (坐標平面) 上畫嘅 quadratic polynomial (其實嚴格嚟講應該叫 quadratic function 二次函數) 係唔係都係呈 parabola (拋物線)?

    A4: 係呀,如果 \ x^2 \ 嘅 coefficient 大過 0 嘅話就開口向上,細過 0 就開口向下。

    • Avatar
      Angus

      \large \begin{aligned}
      x^2-8x+25=0\\
      x^2-8x+16+9=0\\
      (x-4)^2-9i^2=0\\
      (x-4+3i)(x-4-3i)=0\\
      \end{aligned}

      (by admin: 記住係 [/latex] 唔係 [\latex],同埋要加返 \begin{aligned} … \end{aligned},再喺等號前加 &,先可以令到算式隔行對齊。)

    • Admin
      Admin

      @Angus
      不過要咁諗都唔易呀,所以都係 quadratic formula 比較方便

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請在此留言,如要輸入數學符號,可使用 [latex] ... [/latex],例如 [latex] a^2+b^2=c^2 [/latex] (數式編輯器推薦:HostMath)。
如欲瀏覽詳細教學,請按此;如欲試用 [latex] ... [/latex] 功能,可到塗鴉牆