Evaluate
\frac{3}{5}-\frac{4}{5}i=0.6-0.8i
Real Part
\frac{3}{5} = 0.6
Quiz
Complex Number
5 problems similar to:
\frac { ( 4 + 3 i ) ( 1 - 2 i ) } { ( 4 - 3 i ) ( 1 + 2 i ) }
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\frac{4\times 1+4\times \left(-2i\right)+3i\times 1+3\left(-2\right)i^{2}}{\left(4-3i\right)\left(1+2i\right)}
Multiply complex numbers 4+3i and 1-2i like you multiply binomials.
\frac{4\times 1+4\times \left(-2i\right)+3i\times 1+3\left(-2\right)\left(-1\right)}{\left(4-3i\right)\left(1+2i\right)}
By definition, i^{2} is -1.
\frac{4-8i+3i+6}{\left(4-3i\right)\left(1+2i\right)}
Do the multiplications in 4\times 1+4\times \left(-2i\right)+3i\times 1+3\left(-2\right)\left(-1\right).
\frac{4+6+\left(-8+3\right)i}{\left(4-3i\right)\left(1+2i\right)}
Combine the real and imaginary parts in 4-8i+3i+6.
\frac{10-5i}{\left(4-3i\right)\left(1+2i\right)}
Do the additions in 4+6+\left(-8+3\right)i.
\frac{10-5i}{4\times 1+4\times \left(2i\right)-3i-3\times 2i^{2}}
Multiply complex numbers 4-3i and 1+2i like you multiply binomials.
\frac{10-5i}{4\times 1+4\times \left(2i\right)-3i-3\times 2\left(-1\right)}
By definition, i^{2} is -1.
\frac{10-5i}{4+8i-3i+6}
Do the multiplications in 4\times 1+4\times \left(2i\right)-3i-3\times 2\left(-1\right).
\frac{10-5i}{4+6+\left(8-3\right)i}
Combine the real and imaginary parts in 4+8i-3i+6.
\frac{10-5i}{10+5i}
Do the additions in 4+6+\left(8-3\right)i.
\frac{\left(10-5i\right)\left(10-5i\right)}{\left(10+5i\right)\left(10-5i\right)}
Multiply both numerator and denominator by the complex conjugate of the denominator, 10-5i.
\frac{\left(10-5i\right)\left(10-5i\right)}{10^{2}-5^{2}i^{2}}
Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{\left(10-5i\right)\left(10-5i\right)}{125}
By definition, i^{2} is -1. Calculate the denominator.
\frac{10\times 10+10\times \left(-5i\right)-5i\times 10-5\left(-5\right)i^{2}}{125}
Multiply complex numbers 10-5i and 10-5i like you multiply binomials.
\frac{10\times 10+10\times \left(-5i\right)-5i\times 10-5\left(-5\right)\left(-1\right)}{125}
By definition, i^{2} is -1.
\frac{100-50i-50i-25}{125}
Do the multiplications in 10\times 10+10\times \left(-5i\right)-5i\times 10-5\left(-5\right)\left(-1\right).
\frac{100-25+\left(-50-50\right)i}{125}
Combine the real and imaginary parts in 100-50i-50i-25.
\frac{75-100i}{125}
Do the additions in 100-25+\left(-50-50\right)i.
\frac{3}{5}-\frac{4}{5}i
Divide 75-100i by 125 to get \frac{3}{5}-\frac{4}{5}i.
Re(\frac{4\times 1+4\times \left(-2i\right)+3i\times 1+3\left(-2\right)i^{2}}{\left(4-3i\right)\left(1+2i\right)})
Multiply complex numbers 4+3i and 1-2i like you multiply binomials.
Re(\frac{4\times 1+4\times \left(-2i\right)+3i\times 1+3\left(-2\right)\left(-1\right)}{\left(4-3i\right)\left(1+2i\right)})
By definition, i^{2} is -1.
Re(\frac{4-8i+3i+6}{\left(4-3i\right)\left(1+2i\right)})
Do the multiplications in 4\times 1+4\times \left(-2i\right)+3i\times 1+3\left(-2\right)\left(-1\right).
Re(\frac{4+6+\left(-8+3\right)i}{\left(4-3i\right)\left(1+2i\right)})
Combine the real and imaginary parts in 4-8i+3i+6.
Re(\frac{10-5i}{\left(4-3i\right)\left(1+2i\right)})
Do the additions in 4+6+\left(-8+3\right)i.
Re(\frac{10-5i}{4\times 1+4\times \left(2i\right)-3i-3\times 2i^{2}})
Multiply complex numbers 4-3i and 1+2i like you multiply binomials.
Re(\frac{10-5i}{4\times 1+4\times \left(2i\right)-3i-3\times 2\left(-1\right)})
By definition, i^{2} is -1.
Re(\frac{10-5i}{4+8i-3i+6})
Do the multiplications in 4\times 1+4\times \left(2i\right)-3i-3\times 2\left(-1\right).
Re(\frac{10-5i}{4+6+\left(8-3\right)i})
Combine the real and imaginary parts in 4+8i-3i+6.
Re(\frac{10-5i}{10+5i})
Do the additions in 4+6+\left(8-3\right)i.
Re(\frac{\left(10-5i\right)\left(10-5i\right)}{\left(10+5i\right)\left(10-5i\right)})
Multiply both numerator and denominator of \frac{10-5i}{10+5i} by the complex conjugate of the denominator, 10-5i.
Re(\frac{\left(10-5i\right)\left(10-5i\right)}{10^{2}-5^{2}i^{2}})
Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
Re(\frac{\left(10-5i\right)\left(10-5i\right)}{125})
By definition, i^{2} is -1. Calculate the denominator.
Re(\frac{10\times 10+10\times \left(-5i\right)-5i\times 10-5\left(-5\right)i^{2}}{125})
Multiply complex numbers 10-5i and 10-5i like you multiply binomials.
Re(\frac{10\times 10+10\times \left(-5i\right)-5i\times 10-5\left(-5\right)\left(-1\right)}{125})
By definition, i^{2} is -1.
Re(\frac{100-50i-50i-25}{125})
Do the multiplications in 10\times 10+10\times \left(-5i\right)-5i\times 10-5\left(-5\right)\left(-1\right).
Re(\frac{100-25+\left(-50-50\right)i}{125})
Combine the real and imaginary parts in 100-50i-50i-25.
Re(\frac{75-100i}{125})
Do the additions in 100-25+\left(-50-50\right)i.
Re(\frac{3}{5}-\frac{4}{5}i)
Divide 75-100i by 125 to get \frac{3}{5}-\frac{4}{5}i.
\frac{3}{5}
The real part of \frac{3}{5}-\frac{4}{5}i is \frac{3}{5}.
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{ x } ^ { 2 } - 4 x - 5 = 0
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\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}