Namba hii ina maana gani? 1.6180339887.

Sokoro waito

JF-Expert Member
Nov 21, 2014
2,201
2,587
Heko kwenu wadau,

Wakati fulani nilipokuwa nasoma kitabu cha THINK LIKE A CHAMPION kilichoandikwa na Raisi Donald J. Trump nilisoma mahali fulani ambapo aliitaja hii namba 1.6180339887 kama GOLDEN RATIO ambayo imewahi kutumiwa na watu mbalimbali kwenye kazi zao, kwa mfano amemtaja DA VINCI.

Lakini pia ameielezea namba hii kwa ufupi sana huku akisisitiza kuwa ni namba ya maajabu. Trump akaandika pia kuwa yeye hashauri watu kuamini katika maajabu ya namba ili kufanikiwa katika uvumbuzi na kazi zao bali wafanye kazi kwa bidii na akili ili wafanikiwe.

sasa wadau, kama unajua chochote kuhusu namba hiyo karibu ututoe tongotongo kwa ujuzi wako.
 
Heko kwenu wadau,

Wakati fulani nilipokuwa nasoma kitabu cha THINK LIKE A CHAMPION kilichoandikwa na Raisi Donald J. Trump nilisoma mahali fulani ambapo aliitaja hii namba 1.6180339887 kama GOLDEN RATIO ambayo imewahi kutumiwa na watu mbalimbali kwenye kazi zao, kwa mfano amemtaja DA VINCI.

Lakini pia ameielezea namba hii kwa ufupi sana huku akisisitiza kuwa ni namba ya maajabu. Trump akaandika pia kuwa yeye hashauri watu kuamini katika maajabu ya namba ili kufanikiwa katika uvumbuzi na kazi zao bali wafanye kazi kwa bidii na akili ili wafanikiwe.

sasa wadau, kama unajua chochote kuhusu namba hiyo karibu ututoe tongotongo kwa ujuzi wako.
Siti ya mbele kabisa hapa
Nitarudi kuielezea inafanyaje kazi...
Huyu mshenzi ndio alitengeza hiki kitu alikua mwanafunzi wa Johannes Kepler



Line segments in the golden ratio

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi }
94b17cee919ba893572e106101885e9947e2d8c3
.
  • List of numbers
  • Irrational numbers

  • ζ(3)
  • √2
  • √3
  • √5
  • φ
  • ψ
  • ρ
  • δS
  • e
  • π
Binary1.1001111000110111011...Decimal1.6180339887498948482...
Hexadecimal1.9E3779B97F4A7C15F39...Continued fraction{\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}
7fbad6d8c8d284ea5391a39db22c14858d696c1f
Algebraic form{\displaystyle {\frac {1+{\sqrt {5}}}{2}}}
2102ba6ed802cb9a98dc1a0fc1ac99b1a03b4047
Infinite series{\displaystyle {\frac {13}{8}}+\sum _{n=0}^{\infty }{\frac {(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}}}
9402466d8bf07ab67260f520c5360d36020cef34


Two quantities a and b are said to be in the golden ratio φ if

{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi .}
cf7e51356f54a831db021e7e46922c778bbd91c0

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,

{\displaystyle {\frac {a+b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}.}
60bdf40e1de260ce8adbfc22d4301b5c52699e3c

Therefore,

{\displaystyle 1+{\frac {1}{\varphi }}=\varphi .}
73516b60e9391da6113e1df0eab685d732451cc2

Multiplying by φ gives

{\displaystyle \varphi +1=\varphi ^{2}}
7452e786dfd2a1a3e0cdc7cf38b508d0ccb43796

which can be rearranged to

{\displaystyle {\varphi }^{2}-\varphi -1=0.}
909a0da5459476692cb089d9b951d4284387e927

Using the quadratic formula, two solutions are obtained:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }
1c9b7ed65783d1cdb2dfa7abd1361b1c1553bbe6

and

{\displaystyle \varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.6180\,339887\dots }
bcee7ad3a39865b95e89b8aa7fdb15afa913e750

Because φ is the ratio between positive quantities φ is necessarily positive:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }
Cc
Wick snowhite Malcom Lumumba lifecoded
 
Nitarudi kuielezea inafanyaje kazi...
  • List of numbers
  • Irrational numbers

  • ζ(3)
  • √2
  • √3
  • √5
  • φ
  • ψ
  • ρ
  • δS
  • e
  • π
Binary1.1001111000110111011...Decimal1.6180339887498948482...
Hexadecimal1.9E3779B97F4A7C15F39...Continued fraction{\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}
7fbad6d8c8d284ea5391a39db22c14858d696c1f
Algebraic form{\displaystyle {\frac {1+{\sqrt {5}}}{2}}}
2102ba6ed802cb9a98dc1a0fc1ac99b1a03b4047
Infinite series{\displaystyle {\frac {13}{8}}+\sum _{n=0}^{\infty }{\frac {(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}}}
9402466d8bf07ab67260f520c5360d36020cef34


Two quantities a and b are said to be in the golden ratio φ if

{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi .}
cf7e51356f54a831db021e7e46922c778bbd91c0

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,

{\displaystyle {\frac {a+b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}.}
60bdf40e1de260ce8adbfc22d4301b5c52699e3c

Therefore,

{\displaystyle 1+{\frac {1}{\varphi }}=\varphi .}
73516b60e9391da6113e1df0eab685d732451cc2

Multiplying by φ gives

{\displaystyle \varphi +1=\varphi ^{2}}
7452e786dfd2a1a3e0cdc7cf38b508d0ccb43796

which can be rearranged to

{\displaystyle {\varphi }^{2}-\varphi -1=0.}
909a0da5459476692cb089d9b951d4284387e927

Using the quadratic formula, two solutions are obtained:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }
1c9b7ed65783d1cdb2dfa7abd1361b1c1553bbe6

and

{\displaystyle \varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.6180\,339887\dots }
bcee7ad3a39865b95e89b8aa7fdb15afa913e750

Because φ is the ratio between positive quantities φ is necessarily positive:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }
mkuu wahi utupe maujuzi hapa
 
Nitarudi kuielezea inafanyaje kazi...


Line segments in the golden ratio

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi }
94b17cee919ba893572e106101885e9947e2d8c3
.
  • List of numbers
  • Irrational numbers

  • ζ(3)
  • √2
  • √3
  • √5
  • φ
  • ψ
  • ρ
  • δS
  • e
  • π
Binary1.1001111000110111011...Decimal1.6180339887498948482...
Hexadecimal1.9E3779B97F4A7C15F39...Continued fraction{\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}
7fbad6d8c8d284ea5391a39db22c14858d696c1f
Algebraic form{\displaystyle {\frac {1+{\sqrt {5}}}{2}}}
2102ba6ed802cb9a98dc1a0fc1ac99b1a03b4047
Infinite series{\displaystyle {\frac {13}{8}}+\sum _{n=0}^{\infty }{\frac {(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}}}
9402466d8bf07ab67260f520c5360d36020cef34


Two quantities a and b are said to be in the golden ratio φ if

{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi .}
cf7e51356f54a831db021e7e46922c778bbd91c0

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,

{\displaystyle {\frac {a+b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}.}
60bdf40e1de260ce8adbfc22d4301b5c52699e3c

Therefore,

{\displaystyle 1+{\frac {1}{\varphi }}=\varphi .}
73516b60e9391da6113e1df0eab685d732451cc2

Multiplying by φ gives

{\displaystyle \varphi +1=\varphi ^{2}}
7452e786dfd2a1a3e0cdc7cf38b508d0ccb43796

which can be rearranged to

{\displaystyle {\varphi }^{2}-\varphi -1=0.}
909a0da5459476692cb089d9b951d4284387e927

Using the quadratic formula, two solutions are obtained:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }
1c9b7ed65783d1cdb2dfa7abd1361b1c1553bbe6

and

{\displaystyle \varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.6180\,339887\dots }
bcee7ad3a39865b95e89b8aa7fdb15afa913e750

Because φ is the ratio between positive quantities φ is necessarily positive:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }
Nimetoka empty
 
Nitarudi kuielezea inafanyaje kazi...
Huyu mshenzi ndio alitengeza hiki kitu alikua mwanafunzi wa Johannes Kepler



Line segments in the golden ratio

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi }
94b17cee919ba893572e106101885e9947e2d8c3
.
  • List of numbers
  • Irrational numbers

  • ζ(3)
  • √2
  • √3
  • √5
  • φ
  • ψ
  • ρ
  • δS
  • e
  • π
Binary1.1001111000110111011...Decimal1.6180339887498948482...
Hexadecimal1.9E3779B97F4A7C15F39...Continued fraction{\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}
7fbad6d8c8d284ea5391a39db22c14858d696c1f
Algebraic form{\displaystyle {\frac {1+{\sqrt {5}}}{2}}}
2102ba6ed802cb9a98dc1a0fc1ac99b1a03b4047
Infinite series{\displaystyle {\frac {13}{8}}+\sum _{n=0}^{\infty }{\frac {(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}}}
9402466d8bf07ab67260f520c5360d36020cef34


Two quantities a and b are said to be in the golden ratio φ if

{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi .}
cf7e51356f54a831db021e7e46922c778bbd91c0

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,

{\displaystyle {\frac {a+b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}.}
60bdf40e1de260ce8adbfc22d4301b5c52699e3c

Therefore,

{\displaystyle 1+{\frac {1}{\varphi }}=\varphi .}
73516b60e9391da6113e1df0eab685d732451cc2

Multiplying by φ gives

{\displaystyle \varphi +1=\varphi ^{2}}
7452e786dfd2a1a3e0cdc7cf38b508d0ccb43796

which can be rearranged to

{\displaystyle {\varphi }^{2}-\varphi -1=0.}
909a0da5459476692cb089d9b951d4284387e927

Using the quadratic formula, two solutions are obtained:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }
1c9b7ed65783d1cdb2dfa7abd1361b1c1553bbe6

and

{\displaystyle \varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.6180\,339887\dots }
bcee7ad3a39865b95e89b8aa7fdb15afa913e750

Because φ is the ratio between positive quantities φ is necessarily positive:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }
Ngwini hapati hata punje hapo.
 
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