Evaluate
2+i
Real Part
2
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1\times 2+i+2i+i^{2}+\frac{5}{1+2i}
Multiply complex numbers 1+i and 2+i like you multiply binomials.
1\times 2+i+2i-1+\frac{5}{1+2i}
By definition, i^{2} is -1.
2+i+2i-1+\frac{5}{1+2i}
Do the multiplications in 1\times 2+i+2i-1.
2-1+\left(1+2\right)i+\frac{5}{1+2i}
Combine the real and imaginary parts in 2+i+2i-1.
1+3i+\frac{5}{1+2i}
Do the additions in 2-1+\left(1+2\right)i.
1+3i+\frac{5\left(1-2i\right)}{\left(1+2i\right)\left(1-2i\right)}
Multiply both numerator and denominator of \frac{5}{1+2i} by the complex conjugate of the denominator, 1-2i.
1+3i+\frac{5\left(1-2i\right)}{1^{2}-2^{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}.
1+3i+\frac{5\left(1-2i\right)}{5}
By definition, i^{2} is -1. Calculate the denominator.
1+3i+\frac{5\times 1+5\times \left(-2i\right)}{5}
Multiply 5 times 1-2i.
1+3i+\frac{5-10i}{5}
Do the multiplications in 5\times 1+5\times \left(-2i\right).
1+3i+\left(1-2i\right)
Divide 5-10i by 5 to get 1-2i.
1+1+\left(3-2\right)i
Combine the real and imaginary parts.
2+i
Do the additions.
Re(1\times 2+i+2i+i^{2}+\frac{5}{1+2i})
Multiply complex numbers 1+i and 2+i like you multiply binomials.
Re(1\times 2+i+2i-1+\frac{5}{1+2i})
By definition, i^{2} is -1.
Re(2+i+2i-1+\frac{5}{1+2i})
Do the multiplications in 1\times 2+i+2i-1.
Re(2-1+\left(1+2\right)i+\frac{5}{1+2i})
Combine the real and imaginary parts in 2+i+2i-1.
Re(1+3i+\frac{5}{1+2i})
Do the additions in 2-1+\left(1+2\right)i.
Re(1+3i+\frac{5\left(1-2i\right)}{\left(1+2i\right)\left(1-2i\right)})
Multiply both numerator and denominator of \frac{5}{1+2i} by the complex conjugate of the denominator, 1-2i.
Re(1+3i+\frac{5\left(1-2i\right)}{1^{2}-2^{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(1+3i+\frac{5\left(1-2i\right)}{5})
By definition, i^{2} is -1. Calculate the denominator.
Re(1+3i+\frac{5\times 1+5\times \left(-2i\right)}{5})
Multiply 5 times 1-2i.
Re(1+3i+\frac{5-10i}{5})
Do the multiplications in 5\times 1+5\times \left(-2i\right).
Re(1+3i+\left(1-2i\right))
Divide 5-10i by 5 to get 1-2i.
Re(1+1+\left(3-2\right)i)
Combine the real and imaginary parts in 1+3i+\left(1-2i\right).
Re(2+i)
Do the additions in 1+1+\left(3-2\right)i.
2
The real part of 2+i is 2.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\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}