PM [B03] Complex Coordinate
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The document discusses the concept of imaginary numbers and their significance in mathematics and engineering, highlighting the creation of the Argand plane by Jean-Robert Argand. It explains the real and imaginary axes in the Argand coordinate system, illustrating how complex numbers are represented as a combination of real and imaginary components. The document also categorizes complex coordinate systems into totally real, totally imaginary, and complex, emphasizing the unique nature of complex numbers compared to standard Cartesian vectors.
![© ABCC Australia 2015 www.new-physics.com
THE COMPLEX COORDINATE SYSTEM
PM [B03]
An imaginary number is unreal
and does not belong to this world.
But It has been proven so useful in
science and engineering that
mathematicians and scientists
embrace the concept like a real
valueless jewel.](https://image.slidesharecdn.com/pmb03complexcoordinate-151009051941-lva1-app6891/85/PM-B03-Complex-Coordinate-1-320.jpg)
![© ABCC Australia 2015 www.new-physics.com
PLANE POLAR COORDINATES
To be continued on [PM B04]](https://image.slidesharecdn.com/pmb03complexcoordinate-151009051941-lva1-app6891/85/PM-B03-Complex-Coordinate-8-320.jpg)
PM [B03] Complex Coordinate
- 1. © ABCC Australia 2015 www.new-physics.com THE COMPLEX COORDINATE SYSTEM PM [B03] An imaginary number is unreal and does not belong to this world. But It has been proven so useful in science and engineering that mathematicians and scientists embrace the concept like a real valueless jewel.
- 2. © ABCC Australia 2015 www.new-physics.com 𝑖3 𝑖0 𝑖1 𝑖2 𝑖-2 𝑖-3 𝑖-4 𝑖5 𝑖4 𝑖6 The Eternal Cycle It seems that the cyclicality of the 𝑖-multiplication can go on forever in both directions from the negative realm to the positive realm.
- 3. © ABCC Australia 2015 www.new-physics.com The Essential Quadrant But the basic elements are only four and this group creates a rigid configuration with its components orthogonal to each other. i1 i0 i3 i2 i4
- 4. © ABCC Australia 2015 www.new-physics.com The Argand plane 𝑖0 = 1 unit for the +x axis 𝑖1 = 𝑖 unit for the +y-axis 𝑖2 = 𝑖 × 𝑖 = −1 unit for the –x axis 𝑖3 = i × 𝑖 × 𝑖 = −𝑖 unit for the –y axis 𝑖4 = 1 back to the +x-axis 1 = 𝑖0 𝑖 = 𝑖1 −1 = 𝑖2 −𝑖 = 𝑖3 1 = 𝑖4 This cyclic feature of the imaginary number enabled Jean-Robert Argand (1768-1822) to set up a coordinate system after his name – the Argand plane: The Argand plane
- 5. © ABCC Australia 2015 www.new-physics.com The Real & Imaginary Axes In mathematics an Argand coordinate system is made up of a standard number line lying horizontally as the 𝑥 axis. The numbers increases positively in magnitude to the right, and negatively increase in magnitude to the left. This axis is called the ‘real axis’. At 0 on this 𝑥-axis, we can erect an imaginary line as the 𝑦-axis. Now ‘positive’ imaginary numbers grow in magnitude upwards, and "negative" imaginary numbers dwindle in magnitude downwards. This vertical axis is called the ‘imaginary axis’ and is denoted by 𝒊 or simply 𝐼𝑚. Real axis Imaginaryaxis 𝑖 -𝑖 r-r
- 6. © ABCC Australia 2015 www.new-physics.com The Complex Number On the Argand plane, any vector can be written as: 𝑂𝐴 = 𝑎 + 𝑖𝑏 By tradition, 𝑎 is called the real part; b with the 𝑖 is called the imaginary part. 𝑎 + 𝑖𝑏 Is called a complex number. Real axis Imaginary axis 𝑖 -𝑖 r-r 𝜃 O A 𝑎 𝑖𝑏
- 7. © ABCC Australia 2015 www.new-physics.com A spooky world So a complex coordinate systems can be broken up into three parts: 1. The totally real. 2. The totally imaginary. 3. The complex. The most importance difference between a complex number and an ordinary Cartesian vector is that except where the thin horizontal line is, all the space are occupied by unreal or semi-unreal beings. The purely imaginary The purely real The complex The complex The complexThe complex
- 8. © ABCC Australia 2015 www.new-physics.com PLANE POLAR COORDINATES To be continued on [PM B04]