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Lösung 3.1:1f

Aus Online Mathematik Brückenkurs 2

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
Zeile 1: Zeile 1:
Let's begin by calculating some powers of ''i'',
Let's begin by calculating some powers of ''i'',
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
i^2 &= i\cdot i = -1\,,\\[5pt]
i^2 &= i\cdot i = -1\,,\\[5pt]
i^3 &= i^2\cdot i = (-1)\cdot i = -i\,,\\[5pt]
i^3 &= i^2\cdot i = (-1)\cdot i = -i\,,\\[5pt]
Zeile 9: Zeile 9:
Now, we observe that because <math>i^4=1</math>, we can try to factorize <math>i^{11}</math> and <math>i^{20}</math> in terms of <math>i^4</math>,
Now, we observe that because <math>i^4=1</math>, we can try to factorize <math>i^{11}</math> and <math>i^{20}</math> in terms of <math>i^4</math>,
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
i^{11} &= i^{4+4+3} = i^4\cdot i^4\cdot i^3 = 1\cdot 1 \cdot (-i) = -i\,,\\[5pt]
i^{11} &= i^{4+4+3} = i^4\cdot i^4\cdot i^3 = 1\cdot 1 \cdot (-i) = -i\,,\\[5pt]
i^{20} &= i^{4+4+4+4+4} = i^4\cdot i^4\cdot i^4\cdot i^4\cdot i^4 = 1\cdot 1 \cdot 1\cdot 1 \cdot 1 = 1\,\textrm{.}
i^{20} &= i^{4+4+4+4+4} = i^4\cdot i^4\cdot i^4\cdot i^4\cdot i^4 = 1\cdot 1 \cdot 1\cdot 1 \cdot 1 = 1\,\textrm{.}
Zeile 16: Zeile 16:
The answer becomes
The answer becomes
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{{Displayed math||<math>i^{20}+i^{11}=1-i\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>i^{20}+i^{11}=1-i\,\textrm{.}</math>}}

Version vom 13:05, 10. Mär. 2009

Let's begin by calculating some powers of i,

i2i3i4=ii=−1=i2i=(−1)i=−i=i2i2=(−1)(−1)=1.

Now, we observe that because i4=1, we can try to factorize i11 and i20 in terms of i4,

i11i20=i4+4+3=i4i4i3=11(−i)=−i=i4+4+4+4+4=i4i4i4i4i4=11111=1.

The answer becomes

i20+i11=1−i.
Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.1:1f“