Comments on: Complex Numbers for Rotating, Translating, and Scaling the Plane http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/ software development and consulting Sun, 20 May 2012 18:07:58 +0000 hourly 1 http://wordpress.org/?v=3.3.2 By: pat http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-2665 pat Tue, 06 Mar 2012 17:36:46 +0000 http://nklein.com/?p=511#comment-2665 I agree with you if we are talking about arithmetic for real numbers and complex numbers. We are not though. We are talking about something altogether different. This is not a statement about arithmetic. It is a statement about geometry formalized with notation that makes it look like arithmetic. If it helps, think of them just as 4-dimensional vectors: $$(a,b,c,d)$$ that add just like vectors add: $$(a,b,c,d) + (w,x,y,z) = (a+w,b+x,c+y,z+d)$$. But, unlike vectors, they can also be multiplied to create new vectors. The multiplication is such that: $$(s,0,0,0) \cdot (a,b,c,d) = (sa,sb,sc,sd) = (a,b,c,d) \cdot (s,0,0,0)$$ $$(0,s,0,0) \cdot (a,b,c,d) = (-sb,sa,-sd,sc)$$ $$(a,b,c,d) \cdot (0,s,0,0) = (-sb,sa,sc,-sd)$$ $$(0,0,s,0) \cdot (a,b,c,d) = (-sc,sd,sa,-sb)$$ $$(a,b,c,d) \cdot (0,0,s,0) = (-sc,-sd,sa,sb)$$ $$(0,0,0,s) \cdot (a,b,c,d) = (-sd,-sc,sb,sa)$$ $$(a,b,c,d) \cdot (0,0,0,s) = (-sd,sc,-sb,sa)$$ And, any other product can be determined if we agree that this multiplication distributes over addition just like we're used to. So, $$(s,t,u,v) \cdot (a,b,c,d) = \left[(s,0,0,0) + (0,t,0,0) + (0,0,u,0) + (0,0,0,v)\right] \cdot (a,b,c,d)$$ I agree with you if we are talking about arithmetic for real numbers and complex numbers. We are not though. We are talking about something altogether different.

This is not a statement about arithmetic. It is a statement about geometry formalized with notation that makes it look like arithmetic. If it helps, think of them just as 4-dimensional vectors: (a,b,c,d) that add just like vectors add: (a,b,c,d) + (w,x,y,z) = (a+w,b+x,c+y,z+d). But, unlike vectors, they can also be multiplied to create new vectors. The multiplication is such that:

(s,0,0,0) \cdot (a,b,c,d) = (sa,sb,sc,sd) = (a,b,c,d) \cdot (s,0,0,0)

(0,s,0,0) \cdot (a,b,c,d) = (-sb,sa,-sd,sc)

(a,b,c,d) \cdot (0,s,0,0) = (-sb,sa,sc,-sd)

(0,0,s,0) \cdot (a,b,c,d) = (-sc,sd,sa,-sb)

(a,b,c,d) \cdot (0,0,s,0) = (-sc,-sd,sa,sb)

(0,0,0,s) \cdot (a,b,c,d) = (-sd,-sc,sb,sa)

(a,b,c,d) \cdot (0,0,0,s) = (-sd,sc,-sb,sa)

And, any other product can be determined if we agree that this multiplication distributes over addition just like we’re used to. So,

(s,t,u,v) \cdot (a,b,c,d) = \left[(s,0,0,0) + (0,t,0,0) + (0,0,u,0) + (0,0,0,v)\right] \cdot (a,b,c,d)

]]> By: Peter L. Griffiths http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-2664 Peter L. Griffiths Tue, 06 Mar 2012 17:02:27 +0000 http://nklein.com/?p=511#comment-2664 There is no law of arithmetic which makes ij equal to anything but +1. There is no law of arithmetic which makes ij equal to anything but +1.

]]> By: pat http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-2638 pat Sun, 04 Mar 2012 18:18:05 +0000 http://nklein.com/?p=511#comment-2638 It is not possible for a number to have three square roots in the Complex numbers. But, the quaternions are not in tge complex numbers. In fact, there are infinitely many ways to map subsets of the quaternions down to the complex numbers. Note, to that ij = -ji in the quaternions whereas all complex numbers commute under multiplication. It is not possible for a number to have three square roots in the Complex numbers. But, the quaternions are not in tge complex numbers. In fact, there are infinitely many ways to map subsets of the quaternions down to the complex numbers. Note, to that ij = -ji in the quaternions whereas all complex numbers commute under multiplication.

]]> By: Peter L.Griffiths http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-2618 Peter L.Griffiths Fri, 02 Mar 2012 18:02:14 +0000 http://nklein.com/?p=511#comment-2618 I refer to the last sentence of my comments of 21 May 2011. 'In this way all n nth integer roots of complex numbers can be obtained'. This indicates that it is not possible for there to be more than n nth roots of a complex number. This has enormous implications for Hamilton's quaternion equation i^2=j^2=k^2=ijk=-1 where -1 seems to have three square roots i,j, and k, where k seems to be wrong. I refer to the last sentence of my comments of 21 May 2011. ‘In this way all n nth integer roots of complex numbers can be obtained’. This indicates that it is not possible for there to be more than n nth roots of a complex number. This has enormous implications for Hamilton’s quaternion equation i^2=j^2=k^2=ijk=-1 where -1 seems to have three square roots i,j, and k, where k seems to be wrong.

]]> By: Peter L.Griffiths http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-2244 Peter L.Griffiths Sat, 20 Aug 2011 17:55:58 +0000 http://nklein.com/?p=511#comment-2244 Further to my comment of 21 May 2011, the complex number a+ib can be converted into an imaginary number by making a equal 0, and then substituting the Cotes format cos90+isin90 equalling 0+i. It will be recognised that cos180+isin180 equals -1, and cos360+isin360 equals i to the power of 4 which is +1. The same principle also applies for division cos45+isin45 equals the square root of i. In this way the roots and powers of i, -i, +1 and -1 can be calculated all based on the Cotes format. Further to my comment of 21 May 2011, the complex number a+ib can be converted into an imaginary number by making a equal 0, and then substituting the Cotes format cos90+isin90 equalling 0+i. It will be recognised that cos180+isin180 equals -1, and cos360+isin360 equals i to the power of 4 which is +1. The same principle also applies for division
cos45+isin45 equals the square root of i. In this way the roots and powers of i, -i, +1 and -1 can be calculated all based on the Cotes format.

]]> By: Peter L. Griffiths http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-2117 Peter L. Griffiths Sat, 21 May 2011 16:28:26 +0000 http://nklein.com/?p=511#comment-2117 To obtain the roots of the complex number a+ib you take the arcotangent below 90 degrees of the complex ratio a/b and then divide by the root required. The cotangent of the result will give one of the roots of the complex ratio. For the other roots you add 360 degrees and then 720 degrees on to the arcotangent. In this way all n nth integer roots of complex numbers can be obtained. To obtain the roots of the complex number a+ib you take the arcotangent below 90 degrees of the complex ratio a/b and then divide by the root required. The cotangent of the result will give one of the roots of the complex ratio. For the other roots you add 360 degrees and then 720 degrees on to the arcotangent. In this way all n nth integer roots of complex numbers can be obtained.

]]> By: Clifford Algebras for Rotating, Scaling, and Translating Space :: nklein software http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-146 Clifford Algebras for Rotating, Scaling, and Translating Space :: nklein software Mon, 06 Jul 2009 19:52:06 +0000 http://nklein.com/?p=511#comment-146 [...] using complex numbers for rotating, translating, and scaling the plane [...] [...] using complex numbers for rotating, translating, and scaling the plane [...]

]]> By: Clifford Algebras for Rotating, Scaling, and Translating the Plane :: nklein software http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-98 Clifford Algebras for Rotating, Scaling, and Translating the Plane :: nklein software Thu, 11 Jun 2009 18:15:36 +0000 http://nklein.com/?p=511#comment-98 [...] numbers can be used to represent coordinates in the plane and how, once you’ve done that, complex arithmetic leads naturally to rotations, scalings, and translations of the plane. Today, we’re going to do the same with the Clifford algebra [...] [...] numbers can be used to represent coordinates in the plane and how, once you’ve done that, complex arithmetic leads naturally to rotations, scalings, and translations of the plane. Today, we’re going to do the same with the Clifford algebra [...]

]]> By: pat http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-95 pat Sun, 07 Jun 2009 15:45:38 +0000 http://nklein.com/?p=511#comment-95 I haven't seen that site. I will check it out. Thanks... I haven’t seen that site. I will check it out.

Thanks…

]]> By: rory http://nklein.com/2009/06/complex-numbers-for-rotating-translating-and-scaling-the-plane/comment-page-1/#comment-94 rory Sun, 07 Jun 2009 13:04:03 +0000 http://nklein.com/?p=511#comment-94 This was a fun read-- have you checked out the website khanacademy.org? The guy on there is amazing at explaining basic math stuff. This was a fun read– have you checked out the website khanacademy.org? The guy on there is amazing at explaining basic math stuff.

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