αိα်းα α ်αျα်း (real number line) αေါ်αွα် αေါ်αြαိုα်αော αိα်းα‘ားαုံးαို αိα်းα α ် (real number) αုαေါ်αα်။ α€αို့αိုαျှα် αိα်းα α ်αျα်းαေါ်αွα် ααော်αြαိုα်αော αိα်းαှိαါααား။ αှိαါαα်။ α‘ောα်αါ α₯ααာαိုαေ့αာαြα့်αα်။
α‘αα်αော်αြαါ αီαျှαြα်းαို αြေαှα်းαိုα်αါααား။
αα်αα့်αိα်းα α ်αို ααို αှα ်αα် (α ုံαα်αိα်း) αα်αျှα် α‘αါα်းαိα်းαာ ααα်αြα ်αα်။ α‘αုα်αိα်းαααိုα်αါ။ αို့αြောα့် $x^2+1=0$ αိုαြေαα်α ေαော αိα်းα α ်α‘αြေααှိαါ။ αို့αြောα့် αိα်းα α ်αα်αα်αိုαာ αေ့αာαဲ့α α်α α‘αိုαါαီαျှαြα်းαို αြေαα်α ေαော α‘αြေααှိαု αα်αှα်αြီး αြေαှα်းαြα်းαို αα်αိုα်းαဲ့αြαα်။ αို့αော် αα်္αျာααာαှα်αျားα α‘αိုαါαီαျှαြα်းαျိုးαို αြေαှα်းαိုα်αα် αိα်းα αα ်αို αျဲ့αွα်αဲ့αြαα်။
Imaginary Number
αိα်းα α ်αα်αα်αွα် αြေαှα်းαိုα်းαြα်းααှိαော $\sqrt{-1}$ αို imaginary unit (αα်္αေαα‘ားαြα့် $i$) αုαα်αှα်αဲ့αြαα်။ αို့αြောα့် $i^2= \left( \sqrt{-1}\right)^2=-1$ αု αα်αှα်αိုα်αါαα်။ α‘ောα်αါ α₯ααာαျားαို αα်αα်αေ့αာαြα့်αါ။
α€αေαာαွα် square root of positive number αျား α‘αွα် principal root (positive root) αို αာαူαါαα်။
Complex Number
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a + bi where ais the real part and b is the imaginary part.Complex numbers are usually denoted by the symbol z.
$\quad\quad$ a = Re (z)= Real part of z
$\quad\quad$ b = Im (z)= imaginary part of z
A complex number is a number of the form a + bi where
- a is the real part of the complex number.
- bi is the imaginary part of the complex number.
- If b = 0, then a + bi is a real number.
- If a = 0 and b is not equal to 0, the complex number is called an imaginary number.
- An imaginary number is an even root of a negative number.
| Example 1 Express $\sqrt{-16}$ in the form of $a+bi$. Solution $\sqrt{-16}= \sqrt{16}\sqrt{-1}=4i=0+4i$ |
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Plotting a Complex Number on the Complex Plane
Complex number αျားαα် real part + imaginary part αြα ်αောαြောα့် real number line αွေ αေαာααα်αှα်αိုα်αါ။ αို့αာαွα် real dimension (real axis) αှα့် imaginary dimension (imaginary axis)αို့αိုαုံး၍ αေαာαα်αှα်αိုα်αα်။ α‘αိုαါ real axis αှα့် imaginary axis αှα ်αုαြα့်αွဲ့α α်းαားαော two-dimenssional plane αို complex plane αုαေါ်αα်။ Complex plane αα ်αုαွα် horizontal axis ($x$-axis) αို real axis αု αα်αှα်αြီး vertical axis ($y$-axis) αို imaginary axis αု αα်αှα်αα်။
In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in figure.
| Example 2 Plot the complex number $z=3 + 4i$ on the complex plane. Solution |
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Equality of Complex Numbers
if and only if $a = c$ and $b = d$.
Adding and Subtracting Complex Numbers
Complex number αျား αေါα်းαြα်း αုα်αြα်းαွα်
- real part α‘αျα်းαျα်း αေါα်း/αုα် ααα်။
- imaginary part α‘αျα်းαျα်း αေါα်း/αုα် ααα်။
For example, if $z_1=a + bi$ and $z_2=c + di$, then
| $\begin{array}{l} z_1+z_2=(a + bi)+(c+di)=(a+c)+(b+d)i\\\\ z_1-z_2=(a + bi)-(c+di)=(a-c)+(b-d)i \end{array}$ |
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| Example 3 If $z_1=3 + 4i$ and $z_2=5 - 12i$, find (a) $z_1+z_2$ (b) $z_1-z_2$ Solution $z_1=3 + 4i$ and $z_2=5 - 12i$ (a) $z_1+z_2 = (3 + 4i) + (5 - 12i) = 8 - 8i$ (b) $z_1-z_2 = (3 + 4i) - (5 - 12i) = -2 + 16i$ |
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Multiplying Complex Numbers
Complex Numbers αျားαြှောα်αြα်းαွα် Real Numbers αျား αြှောα်αြα်းαှာαဲ့αို့αα် αြα့်αေααုα်αα္αိ (distributive law) αို αုံးααα်။
Multiplying a Complex Number by a Real Number
If $z =a + bi$ and $k$ is a real number, then
| $\begin{array}{l} kz=k(a + bi)=ka + kb i\\\\ \dfrac{1}{k}z=\dfrac{1}{k}(a + bi)=\dfrac{a}{k}+\dfrac{b}{k}i \end{array}$ |
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| Example 4 If $z_1=3 - 12i$ and $z_2=4 + i$, find (a) $2z_1$ (b) $\dfrac{1}{3}z_1$ (c) $z_1 + 3z_2$ (d) $\dfrac{1}{2}(z_1 - 2z_2)$ Solution $\quad\quad z_{1}=3-12 i$ $\quad\quad z_{2}=4+i$ $\begin{aligned}\text{(a)}\quad 2 z_{1}&=2(3-12 i)\\\\ &=6-24 i \end{aligned}$ $\begin{aligned} \text{(b)}\quad \dfrac{1}{3} z_{1} &=\frac{1}{3}(3-12 i) \\\\ &= 1-4 i \end{aligned}$ $\begin{aligned} \text{(c)}\quad z_1+z_2 &=3-12 i+12+3 i \\\\ &=15-9 i \end{aligned}$ $\begin{aligned} \text{(d)}\quad \dfrac{1}{2}(z_1 - 2z_2) &=\dfrac{1}{2}\left(3-12 i - 2(4+i)\right)\\\\ &=\dfrac{1}{2}\left(3-12 i - 8-2i\right)\\\\ &=\dfrac{1}{2}\left(-5-14 i\right)\\\\ &=-\dfrac{5}{2}-7 i \end{aligned}$ |
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Multiplying a Complex Number by a Complex Number
If $z_1 =a + bi$ and $z_2 =c + di$, then
| $\begin{array}{l} z_1\cdot z_2=(a+b i)(c+d i)=a c+a d i+b c i+b d i^{2}\\\\ z_1\cdot z_2=(a+b i)(c+d i)=a c+a d i+b c i-b d\quad (\because\ i^2=-1)\\\\ z_1\cdot z_2=(a+b i)(c+d i)=(a c-b d)+(a d+b c) i \end{array}$ |
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| Example 5 If $z_1=3 + 2i$ and $z_2=2 -5 i$, find (a) $z_1\cdot z_2$ (b) ${z_1}^2$ Solution $z_1=3 + 2i$ $z_2=2 -5 i$, $\begin{aligned} \text{(a)}\quad z_1\cdot z_2 &= (3 + 2i)(2 -5 i) \\\\ &= 6 - 15i + 4i -10i^2\\\\ &= 6 - 11i +10 \quad(\because\quad i^2=-1)\\\\ &= 16 - 11i \end{aligned}$ $\begin{aligned} \text{(b)}\quad {z_1}^2 &= (3 + 2i)^2 \\\\ &= 9+12i+4i^2\\\\ &= 9 +12i-4 \quad(\because\quad i^2=-1)\\\\ &= 5+12i \end{aligned}$ |
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Conjugate of a Complex Number
Complex Number αα ်αု၏ conjugate αိုαα်αှာ αေးαားαော complex number ၏ imaginary part αို αα့်αျα်αα် αα္ααာαို့ αြောα်းαေးαိုα်αြα်းαα် αြα ်αα်။ $z=a+bi$ ၏ conjugate αှာ $\overline {z}= a-bi$ αြα ်αα်။
The complex conjugate of a complex number which is obtained by changing the sign of the imaginary part. So if $z=a+b i$, its complex conjugate, $\bar{z}$, is defined by
- When a complex number is multiplied by its complex conjugate, the result is a real number.
- When a complex number is added to its complex conjugate, the result is a real number.
| Example 6 Find the complex conjugate of each number (a) $z=3+5i$ (b) $w=-2i$ Solution $\begin{array}{lll} \text{(a)}& \text{conjugate of}\ z & = \overline{z} \\\\ & & = \overline{3+5i} \\\\ & & = 3-5i\\\\ \text{(b)} &\quad\quad\quad\quad\quad w & = -2i\\\\ &\therefore \quad\quad\quad\quad w & = 0-2i\\\\ &\text{conjugate of}\ w & = \overline{w} \\\\ & & = \overline{0-2i} \\\\ & & = 0+2i\\\\ & & = 2i \end{array}$ |
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| Example 7 If $z=2 + i$ , find (a) $\overline{z}$ (b) $z\cdot \overline{z}$ (b) $z + \overline{z}$ Solution $z=2 + i$ $\begin{array}{lll} \text{(a)}& \quad\quad\overline{z} & = \overline{2 + i} \\\\ & & = 2-i\\\\ \text{(b)}& z\cdot \overline{z} & = (2+i)(2-i)\\\\ & & = 4-i^2\\\\ & & = 4-(-1)\\\\ & & = 5\\\\ \text{(c)}&z + \overline{z} &= (2+i)+(2-i)\\\\ & & = 4 \end{array}$ |
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Dividing Complex Numbers
Complex number αျားα ားαα့်α‘αါ α ားαိα်း (αိုα်းαြေ) αို real number αြα ်α‘ောα် αြုαုα်αေးααα်။ αိုα်းαြေαα် complex number αြα ်αေαါα αိုα်းαေαှα့် αိုα်းαြေ αှα ်αုαုံးαို conjugate αြα့် αြှောα်αေးαြα်းα‘ားαြα့် αိုα်းαြေαို real number αြα ်α‘ောα် αြောα်းαေးαိုα်αα်။
| If $z_1=a+bi$ and $z_2=c+di$, then $\begin{aligned} \dfrac{z_1}{z_2} =&\dfrac{a+bi}{c+di}\\\\ =&\dfrac{a+bi}{c+di}\times\dfrac{c-di}{c-di}\\\\ =&\dfrac{ac-adi+bci-bdi^2}{c^2-d^2i^2}\\\\ =&\dfrac{(ac+bd)+ (bc-ad)i}{c^2+d^2} \end{aligned}$ |
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| Example 8 Express $\dfrac{2+3i}{4-2i}$ in standard form (in the form of $a+bi$). Solution $\begin{aligned} \dfrac{2+3i}{4-2i} =&\dfrac{2+3i}{4-2i}\times\dfrac{4+2i}{4+2i}\\\\ =&\dfrac{8+4i+12i+6i^2}{16-4i^2}\\\\ =&\dfrac{8+16i-6}{16+4}\quad (\because\quad i^2=-1)\\\\ =&\dfrac{2+16i}{20}\\\\ =&\dfrac{2}{20}+\dfrac{16i}{20}\\\\ =&\dfrac{1}{10}+\dfrac{4}{5}i \end{aligned}$ |
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Exercise
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