document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181354>Question 1181354</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that |z1 + z2 | < or = |z1|+|z2|. Use |z|^2 = mod z times z, Re(z) = (z + mod z)/ 2, and Re(z) < or = |z|. The inequality is known as the Triangle Inequality.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850056 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850056\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the proof of the triangle inequality using the properties you provided:\r<BR>\n" );
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document.write( "1. **Start with the square of the absolute value:**\r<BR>\n" );
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document.write( "   |z₁ + z₂|² = (z₁ + z₂)(z₁ + z₂)*  (where * denotes the complex conjugate)\r<BR>\n" );
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document.write( "2. **Expand the expression:**\r<BR>\n" );
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document.write( "   |z₁ + z₂|² = z₁z₁* + z₁z₂* + z₂z₁* + z₂z₂*\r<BR>\n" );
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document.write( "3. **Use the property |z|² = z * z*:**\r<BR>\n" );
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document.write( "   |z₁ + z₂|² = |z₁|² + z₁z₂* + z₂z₁* + |z₂|²\r<BR>\n" );
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document.write( "4. **Rewrite the middle terms using conjugates:** Notice that z₂z₁* is the conjugate of z₁z₂*.  So we can write:\r<BR>\n" );
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document.write( "   z₂z₁* = (z₁z₂*)*\r<BR>\n" );
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document.write( "5. **Substitute this back into the equation:**\r<BR>\n" );
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document.write( "   |z₁ + z₂|² = |z₁|² + z₁z₂* + (z₁z₂*)* + |z₂|²\r<BR>\n" );
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document.write( "6. **Use the property Re(z) = (z + z*)/2:** The middle terms can be expressed using the real part:\r<BR>\n" );
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document.write( "   z₁z₂* + (z₁z₂*)* = 2 * Re(z₁z₂*)\r<BR>\n" );
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document.write( "7. **Substitute this into the equation:**\r<BR>\n" );
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document.write( "   |z₁ + z₂|² = |z₁|² + 2 * Re(z₁z₂*) + |z₂|²\r<BR>\n" );
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document.write( "8. **Use the property Re(z) ≤ |z|:**\r<BR>\n" );
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document.write( "   |z₁ + z₂|² ≤ |z₁|² + 2|z₁z₂*| + |z₂|²\r<BR>\n" );
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document.write( "9. **Use the property |ab| = |a||b|:**\r<BR>\n" );
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document.write( "   |z₁ + z₂|² ≤ |z₁|² + 2|z₁||z₂*| + |z₂|²\r<BR>\n" );
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document.write( "10. **Since |z| = |z*|, we have:**\r<BR>\n" );
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document.write( "    |z₁ + z₂|² ≤ |z₁|² + 2|z₁||z₂| + |z₂|²\r<BR>\n" );
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document.write( "11. **Recognize the perfect square:** The right side is a perfect square:\r<BR>\n" );
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document.write( "    |z₁ + z₂|² ≤ (|z₁| + |z₂|)²\r<BR>\n" );
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document.write( "12. **Take the square root of both sides:** Since absolute values are non-negative, we can take the square root without changing the inequality:\r<BR>\n" );
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document.write( "    |z₁ + z₂| ≤ |z₁| + |z₂|\r<BR>\n" );
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document.write( "This completes the proof of the triangle inequality.<BR>\n" );
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