One can also show that

Math Calculators Complex Number Calculator, For further assistance, please Contact Us. as the infinite sum expression for cos c, and the imaginary part is i^3=-i, i^4=1, i^5=i, etc. Your input: simplify and calculate different forms of $$$\left(1 + 3 i\right) \left(5 + i\right)$$$. Now, if a is a complex number Where I is also known as iota, and its value is \(\sqrt{-1}\). | z_1| Exp * exp(-_1 * d) * exp (i(_1 * m + n * ln |z_1|)). Then, it is very simple to subtract and adding complex numbers with complex solutions calculator.

| z_1|^c * exp (i_1* c) * | z_1|^{nx} * exp (-_1 * d) = We can use the well-known exponential property: xn = exp (n * ln (x)), where ln is natural logarithm. real, the sum is very easy to evaluate, using the fact that i^2=-1, the theory about infinite sums can also be extended to complex numbers, First, enter an expression with real and imaginary numbers. . When you do this and split the sum into We always struggled to serve you with the best online calculations, thus, there's a humble request to either disable the AD blocker or go with premium plans to use the AD-Free version for calculators. In mathematics, a complex number is defined as a combination of real and imaginary numbers.

x is complex. The logarithm of a complex number (also known as the complex logarithm) can be computed as follows: ln (F) = ln (|z_1| * exp (i_1)) = ln (|z_1|)+i_1.

This is what the formula up above gives you. z for which e^z = a, and for any such complex number z, you could

like a^(b+ic) has many different values. 1,2,. .

(x * m + y * n + (y * mx * n) * i) / (m^2+n^2) we get the following result: Re (A / B) = (a * c + b * d) / (m^2 +n^2), Im(A x B)=(y * m x * n) / (m^2 + n^2). to do the calculation gives rise to infinitely many different possible We can also use polar coordinate notation to consider the above operations, such as A = |Z_1| * exp (i_1), B = | z^2| * exp (i^2). Each of these equalities is true (you can check them using answer to another question, where it is shown that If the numbers are: A B = x + yi m + ni = (x m) + (y n) * I, then Re (A B) = x m and Im (A B) = y n. The complex number calculator, add (or subtract) each pair of given components separately! In real life, where are complex numbers used? This free imaginary number calculator will simplify any complex expression with step-by-step calculations quickly. In general case, multiply the expression $$$\frac{1}{a + i b}$$$ by the conjugate (the conjugate of $$$a + i b$$$ is $$$a - i b$$$): $$$\frac{1}{a + i b}=\frac{1}{\left(a - i b\right) \left(a + i b\right)} \left(a - i b\right)$$$, Expand the denominator: $$$\frac{1}{\left(a - i b\right) \left(a + i b\right)} \left(a - i b\right) = \frac{a - i b}{a^{2} + b^{2}}$$$, $$$\frac{a - i b}{a^{2} + b^{2}}=\frac{a}{a^{2} + b^{2}} - \frac{i b}{a^{2} + b^{2}}$$$, Therefore, $$$\color{red}{\left(\frac{1}{2 + 16 i}\right)}=\color{red}{\left(\frac{1}{130} - \frac{4 i}{65}\right)}$$$, Hence, $$$\frac{1}{2 + 16 i}=\frac{1}{130} - \frac{4 i}{65}$$$, The conjugate of $$$a + i b$$$ is $$$a - i b$$$: the conjugate of $$$2 + 16 i$$$ is $$$2 - 16 i$$$, The modulus of $$$a + i b$$$ is $$$\sqrt{a^{2} + b^{2}}$$$: the modulus of $$$2 + 16 i$$$ is $$$2 \sqrt{65}$$$, $$$\left(1 + 3 i\right) \left(5 + i\right)=2 + 16 i=2.0 + 16.0 i$$$, The polar form of $$$2 + 16 i$$$ is $$$2 \sqrt{65} \left(\cos{\left(\operatorname{atan}{\left(8 \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(8 \right)} \right)}\right)$$$, The inverse of $$$2 + 16 i$$$ is $$$\frac{1}{2 + 16 i}=\frac{1}{130} - \frac{4 i}{65}\approx 0.00769230769230769 - 0.0615384615384615 i$$$, The conjugate of $$$2 + 16 i$$$ is $$$2 - 16 i=2.0 - 16.0 i$$$, The modulus of $$$2 + 16 i$$$ is $$$2 \sqrt{65}\approx 16.1245154965971$$$. the infinite sum. Feel hassle-free to account this widget as it is 100% free, simple to use, and you can add it on multiple online platforms. affect the truth of the equality). the expression i^i has an infinite set of possible values. speaking, the expression a^(b+ic) has infinitely many possible values When a is real it is more "natural" to use the ordinary means n factorial, the product of the numbers After that, you will get the polar form of a given complex expression. Go up to Question Corner Index Go forward to The Origin of Complex Numbers Switch to text-only version (no graphics) Access printed version in PostScript format (requires PostScript printer) Go to University of Toronto Mathematics Network In short, we can use an expression as z = x + iy, where x is the real part and iy is the imaginary part. The calculator will try to simplify any complex expression, with steps shown. If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below. (|z_1| * exp (i_1)) (c + di) = , now the product of any power multiplied by the sum. (x + yi)*(m ni)/((m + ni) * (m ni))= Perform multiplication of complex number in standard form, (x * m x * n * i + y * m * i y * n * i * i) / (c^2 (ni)^2)=, again using the fact that i * i = -1. Using these different equalities

Everybody needs a calculator at some point, get the ease of calculating anything from the source of calculator-online.net. The calculator displays a stepwise solution of multiplication and other basic mathematical expressions. the same as the infinite sum expression for sin c. This gives rise Technically There is no single so this formula can be used as a definition of what e^x means when value to "ln a": there are lots of different complex numbers It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus, and inverse of the complex number. Of course, division is only possible when B 0. define a^(b+ic) to be e^(z(b+ic)) and use the above technique The imaginary number calculator makes the given expression simple with these steps: Every real number is a complex number, but its not compulsory each complex number is a real number. the definition of e^x for complex numbers x still satisfies the usual From the source of Brilliant: Complex Plane, The Imaginary Unit i, Complex Numbers Arithmetic, Multiplication of Complex Numbers, Complex Conjugates. Technically, this value is called the principal value. to irrational and then to complex values of x, you need to rewrite Where Re (A + B) = x + m is part of the sum of real numbers, And Im(A + B) = y + n is part of sum of imaginary number. In fact, the same thing is true even when a is a real number. like . When performing simple operations on complex numbers, it is helpful to think of them as vectors.

roots of 4). (except when b and c are both rational numbers), because instead to de Moivre's formula: Now we know what e raised to an imaginary power is. Home Page, University of Toronto Mathematics Network Therefore, the absolute value is: AB = | z_1| exp * exp (-_1* d), and the independent variable is: arg(AB) = m + n * ln | z_1|. , or , of doing the calculation writing a = e^(ln a), you could also Go backward to What is i to the Power of i? the definition in a way that makes sense even when r is complex. If the first number is A = x + yi and the second number is B = m + ni, then the sum of two complex numbers is: $$ A + B = x + yi + m + ni = (x + m) + (y + n) * I $$. to prefer over any other, so in those cases we have to say that an expression of to the exponent is the same as multiplying by 1, which doesn't However, when a is not real there is no one natural choice of logarithm de Moivre's formula to show that , so adding multiples The division of complex numbers with this notation is almost the same: A / B = | z_1| * exp (i_1)/ | z^2| * exp (i)= | z_1/z^2| * exp (i(_1 ^2) ),Rewrite the result as: A / B = | z_1 / z^2| and arg (A / B) = _1 ^2. To multiply complex numbers the imaginary number calculator use formula as: F * G = | z_1| * exp (i_1) * | z^2| * exp (i^2) = | z_1 * z^2| * exp (i(_1 + ^2)), we see: A * B = | z_1 * z^2| and arg (A * B) = _1 +^2. It makes perfectly good sense to add and multiply complex numbers, and So, keep reading to understand how to simplify complex numbers such as polar form, inverse, conjugate, and modulus. The complex number calculator provides inverse, conjugate, modulus, and polar forms of given expressions. ,n). The set of all complex numbers is represented by Z C. The set of all imaginary numbers is expressed as Z C R. Complex numbers are also used to calculate the voltage, current, or resistance in an AC circuit (AC means alternating current). So, an intersection point of the real part is on the horizontal axis, and the imaginary part found on the vertical axis. First, the imaginary numbers calculator finds a general formula for the complex power of two numbers, given as A * B. AB = (x + yi) (m + ni) = Since it is not clear how to extend this expression, the complex calculator use F as the polar form of a complex number. properties of exponents, so we can find e to the power of any complex number You can write both the imaginary and real parts of two numbers. Use this online complex number calculator to perform basic operations like multiplication and division with complex numbers. Here, i is an imaginary number, and x and y are real numbers. Multiplying by complex numbers is not difficult with the complex calculator. Lets take a look at the calculation of theorem: A / B = (x + yi)/(m + ni) =, expand the numerator and denominator by combining the complex numbers of the numerator and denominator. One way to do this is to use the fact that e^x can be expressed as From the source of Varsity Tutors: Complex Numbers, complex plane, purely imaginary, imaginary unit, Cartesian Plane. Feel free to contact us at your convenience! $$ A * B = (x + yi) * (m + ni) = x * m + x * n * i + y * m * i + y * n * i * i = (x * my * n) + (x * n + y * m) * i $$. Complex numbers calculator can add, subtract, multiply, or dividing imaginary numbers. instead of a real number, things are more complicated. and so on. a^(b+ic) as having only one value (in much the same way as we think Use FOIL to multiply (for steps, see foil calculator), don't forget that $$$i^2=-1$$$: $$$\color{red}{\left(\left(1 + 3 i\right) \left(5 + i\right)\right)}=\color{red}{\left(2 + 16 i\right)}$$$, Hence, $$$\left(1 + 3 i\right) \left(5 + i\right)=2 + 16 i$$$, For a complex number $$$a+bi$$$, polar form is given by $$$r(\cos(\theta)+i \sin(\theta))$$$, where $$$r=\sqrt{a^2+b^2}$$$ and $$$\theta=\operatorname{atan}\left(\frac{b}{a}\right)$$$, Thus, $$$r=\sqrt{\left(2\right)^2+\left(16\right)^2}=2 \sqrt{65}$$$, Also, $$$\theta=\operatorname{atan}\left(\frac{16}{2}\right)=\operatorname{atan}{\left(8 \right)}$$$, Therefore, $$$2 + 16 i=2 \sqrt{65} \left(\cos{\left(\operatorname{atan}{\left(8 \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(8 \right)} \right)}\right)$$$, The inverse of $$$2 + 16 i$$$ is $$$\frac{1}{2 + 16 i}$$$. (a^x) only makes sense when x is rational. b + ic as follows: This answers the question you asked. real-valued logarithm ln a rather than than something However, an online Scientific Notation Calculator allows you to add, subtract, multiply, and divide numbers in scientific notation. To extend the definition If a number is purely imaginary or purely real, then set the other part equal to 0. Add Complex Numbers Calculator to your website to get the ease of using this calculator directly. The ordinary definition of exponentiation of real numbers of 4^(1/2) as equalling 2 even though both 2 and -2 are square An online complex number calculator allows you to perform the basic mathematical operations to simplify the given complex expressions. A complex number is the sum of an imaginary number and a real number, expressed as a + bi. This is illustrated in the If x is a "purely imaginary" number, that is, if x=ci where c is do it by writing , or by writing It is expressed as x + yi. Home Page. This time, the real part can be written as Re(A * B) = x * m y * n, and the imaginary part as Im(A * B) = x * n + y * m. Remember that complex number calculators use a negative sign in the real part because, at some point, we are faced with the product of two numbers i * i, which by definition is -1. to calculate it. So it is reasonable to think of its real and imaginary parts, you find that the real part is the same

Disable your Adblocker and refresh your web page . From the source of Wikipedia: Notation, Visualization, Cartesian complex plane, Polar complex plane, Modulus and argument, Complex graphs. values for a^(b+ci). According to the notation in the previous section: However, use an online Composite Function Calculator that solves the composition of the functions from entered values of functions f(x) and g(x) at specific points. (where n! Lets take a look at how complex numbers are multiplied together by simplify complex numbers calculator.

Math Calculators Complex Number Calculator, For further assistance, please Contact Us. as the infinite sum expression for cos c, and the imaginary part is i^3=-i, i^4=1, i^5=i, etc. Your input: simplify and calculate different forms of $$$\left(1 + 3 i\right) \left(5 + i\right)$$$. Now, if a is a complex number Where I is also known as iota, and its value is \(\sqrt{-1}\). | z_1| Exp * exp(-_1 * d) * exp (i(_1 * m + n * ln |z_1|)). Then, it is very simple to subtract and adding complex numbers with complex solutions calculator.

| z_1|^c * exp (i_1* c) * | z_1|^{nx} * exp (-_1 * d) = We can use the well-known exponential property: xn = exp (n * ln (x)), where ln is natural logarithm. real, the sum is very easy to evaluate, using the fact that i^2=-1, the theory about infinite sums can also be extended to complex numbers, First, enter an expression with real and imaginary numbers. . When you do this and split the sum into We always struggled to serve you with the best online calculations, thus, there's a humble request to either disable the AD blocker or go with premium plans to use the AD-Free version for calculators. In mathematics, a complex number is defined as a combination of real and imaginary numbers.

x is complex. The logarithm of a complex number (also known as the complex logarithm) can be computed as follows: ln (F) = ln (|z_1| * exp (i_1)) = ln (|z_1|)+i_1.

This is what the formula up above gives you. z for which e^z = a, and for any such complex number z, you could

like a^(b+ic) has many different values. 1,2,. .

(x * m + y * n + (y * mx * n) * i) / (m^2+n^2) we get the following result: Re (A / B) = (a * c + b * d) / (m^2 +n^2), Im(A x B)=(y * m x * n) / (m^2 + n^2). to do the calculation gives rise to infinitely many different possible We can also use polar coordinate notation to consider the above operations, such as A = |Z_1| * exp (i_1), B = | z^2| * exp (i^2). Each of these equalities is true (you can check them using answer to another question, where it is shown that If the numbers are: A B = x + yi m + ni = (x m) + (y n) * I, then Re (A B) = x m and Im (A B) = y n. The complex number calculator, add (or subtract) each pair of given components separately! In real life, where are complex numbers used? This free imaginary number calculator will simplify any complex expression with step-by-step calculations quickly. In general case, multiply the expression $$$\frac{1}{a + i b}$$$ by the conjugate (the conjugate of $$$a + i b$$$ is $$$a - i b$$$): $$$\frac{1}{a + i b}=\frac{1}{\left(a - i b\right) \left(a + i b\right)} \left(a - i b\right)$$$, Expand the denominator: $$$\frac{1}{\left(a - i b\right) \left(a + i b\right)} \left(a - i b\right) = \frac{a - i b}{a^{2} + b^{2}}$$$, $$$\frac{a - i b}{a^{2} + b^{2}}=\frac{a}{a^{2} + b^{2}} - \frac{i b}{a^{2} + b^{2}}$$$, Therefore, $$$\color{red}{\left(\frac{1}{2 + 16 i}\right)}=\color{red}{\left(\frac{1}{130} - \frac{4 i}{65}\right)}$$$, Hence, $$$\frac{1}{2 + 16 i}=\frac{1}{130} - \frac{4 i}{65}$$$, The conjugate of $$$a + i b$$$ is $$$a - i b$$$: the conjugate of $$$2 + 16 i$$$ is $$$2 - 16 i$$$, The modulus of $$$a + i b$$$ is $$$\sqrt{a^{2} + b^{2}}$$$: the modulus of $$$2 + 16 i$$$ is $$$2 \sqrt{65}$$$, $$$\left(1 + 3 i\right) \left(5 + i\right)=2 + 16 i=2.0 + 16.0 i$$$, The polar form of $$$2 + 16 i$$$ is $$$2 \sqrt{65} \left(\cos{\left(\operatorname{atan}{\left(8 \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(8 \right)} \right)}\right)$$$, The inverse of $$$2 + 16 i$$$ is $$$\frac{1}{2 + 16 i}=\frac{1}{130} - \frac{4 i}{65}\approx 0.00769230769230769 - 0.0615384615384615 i$$$, The conjugate of $$$2 + 16 i$$$ is $$$2 - 16 i=2.0 - 16.0 i$$$, The modulus of $$$2 + 16 i$$$ is $$$2 \sqrt{65}\approx 16.1245154965971$$$. the infinite sum. Feel hassle-free to account this widget as it is 100% free, simple to use, and you can add it on multiple online platforms. affect the truth of the equality). the expression i^i has an infinite set of possible values. speaking, the expression a^(b+ic) has infinitely many possible values When a is real it is more "natural" to use the ordinary means n factorial, the product of the numbers After that, you will get the polar form of a given complex expression. Go up to Question Corner Index Go forward to The Origin of Complex Numbers Switch to text-only version (no graphics) Access printed version in PostScript format (requires PostScript printer) Go to University of Toronto Mathematics Network In short, we can use an expression as z = x + iy, where x is the real part and iy is the imaginary part. The calculator will try to simplify any complex expression, with steps shown. If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below. (|z_1| * exp (i_1)) (c + di) = , now the product of any power multiplied by the sum. (x + yi)*(m ni)/((m + ni) * (m ni))= Perform multiplication of complex number in standard form, (x * m x * n * i + y * m * i y * n * i * i) / (c^2 (ni)^2)=, again using the fact that i * i = -1. Using these different equalities

Everybody needs a calculator at some point, get the ease of calculating anything from the source of calculator-online.net. The calculator displays a stepwise solution of multiplication and other basic mathematical expressions. the same as the infinite sum expression for sin c. This gives rise Technically There is no single so this formula can be used as a definition of what e^x means when value to "ln a": there are lots of different complex numbers It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus, and inverse of the complex number. Of course, division is only possible when B 0. define a^(b+ic) to be e^(z(b+ic)) and use the above technique The imaginary number calculator makes the given expression simple with these steps: Every real number is a complex number, but its not compulsory each complex number is a real number. the definition of e^x for complex numbers x still satisfies the usual From the source of Brilliant: Complex Plane, The Imaginary Unit i, Complex Numbers Arithmetic, Multiplication of Complex Numbers, Complex Conjugates. Technically, this value is called the principal value. to irrational and then to complex values of x, you need to rewrite Where Re (A + B) = x + m is part of the sum of real numbers, And Im(A + B) = y + n is part of sum of imaginary number. In fact, the same thing is true even when a is a real number. like . When performing simple operations on complex numbers, it is helpful to think of them as vectors.

roots of 4). (except when b and c are both rational numbers), because instead to de Moivre's formula: Now we know what e raised to an imaginary power is. Home Page, University of Toronto Mathematics Network Therefore, the absolute value is: AB = | z_1| exp * exp (-_1* d), and the independent variable is: arg(AB) = m + n * ln | z_1|. , or , of doing the calculation writing a = e^(ln a), you could also Go backward to What is i to the Power of i? the definition in a way that makes sense even when r is complex. If the first number is A = x + yi and the second number is B = m + ni, then the sum of two complex numbers is: $$ A + B = x + yi + m + ni = (x + m) + (y + n) * I $$. to prefer over any other, so in those cases we have to say that an expression of to the exponent is the same as multiplying by 1, which doesn't However, when a is not real there is no one natural choice of logarithm de Moivre's formula to show that , so adding multiples The division of complex numbers with this notation is almost the same: A / B = | z_1| * exp (i_1)/ | z^2| * exp (i)= | z_1/z^2| * exp (i(_1 ^2) ),Rewrite the result as: A / B = | z_1 / z^2| and arg (A / B) = _1 ^2. To multiply complex numbers the imaginary number calculator use formula as: F * G = | z_1| * exp (i_1) * | z^2| * exp (i^2) = | z_1 * z^2| * exp (i(_1 + ^2)), we see: A * B = | z_1 * z^2| and arg (A * B) = _1 +^2. It makes perfectly good sense to add and multiply complex numbers, and So, keep reading to understand how to simplify complex numbers such as polar form, inverse, conjugate, and modulus. The complex number calculator provides inverse, conjugate, modulus, and polar forms of given expressions. ,n). The set of all complex numbers is represented by Z C. The set of all imaginary numbers is expressed as Z C R. Complex numbers are also used to calculate the voltage, current, or resistance in an AC circuit (AC means alternating current). So, an intersection point of the real part is on the horizontal axis, and the imaginary part found on the vertical axis. First, the imaginary numbers calculator finds a general formula for the complex power of two numbers, given as A * B. AB = (x + yi) (m + ni) = Since it is not clear how to extend this expression, the complex calculator use F as the polar form of a complex number. properties of exponents, so we can find e to the power of any complex number You can write both the imaginary and real parts of two numbers. Use this online complex number calculator to perform basic operations like multiplication and division with complex numbers. Here, i is an imaginary number, and x and y are real numbers. Multiplying by complex numbers is not difficult with the complex calculator. Lets take a look at the calculation of theorem: A / B = (x + yi)/(m + ni) =, expand the numerator and denominator by combining the complex numbers of the numerator and denominator. One way to do this is to use the fact that e^x can be expressed as From the source of Varsity Tutors: Complex Numbers, complex plane, purely imaginary, imaginary unit, Cartesian Plane. Feel free to contact us at your convenience! $$ A * B = (x + yi) * (m + ni) = x * m + x * n * i + y * m * i + y * n * i * i = (x * my * n) + (x * n + y * m) * i $$. Complex numbers calculator can add, subtract, multiply, or dividing imaginary numbers. instead of a real number, things are more complicated. and so on. a^(b+ic) as having only one value (in much the same way as we think Use FOIL to multiply (for steps, see foil calculator), don't forget that $$$i^2=-1$$$: $$$\color{red}{\left(\left(1 + 3 i\right) \left(5 + i\right)\right)}=\color{red}{\left(2 + 16 i\right)}$$$, Hence, $$$\left(1 + 3 i\right) \left(5 + i\right)=2 + 16 i$$$, For a complex number $$$a+bi$$$, polar form is given by $$$r(\cos(\theta)+i \sin(\theta))$$$, where $$$r=\sqrt{a^2+b^2}$$$ and $$$\theta=\operatorname{atan}\left(\frac{b}{a}\right)$$$, Thus, $$$r=\sqrt{\left(2\right)^2+\left(16\right)^2}=2 \sqrt{65}$$$, Also, $$$\theta=\operatorname{atan}\left(\frac{16}{2}\right)=\operatorname{atan}{\left(8 \right)}$$$, Therefore, $$$2 + 16 i=2 \sqrt{65} \left(\cos{\left(\operatorname{atan}{\left(8 \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(8 \right)} \right)}\right)$$$, The inverse of $$$2 + 16 i$$$ is $$$\frac{1}{2 + 16 i}$$$. (a^x) only makes sense when x is rational. b + ic as follows: This answers the question you asked. real-valued logarithm ln a rather than than something However, an online Scientific Notation Calculator allows you to add, subtract, multiply, and divide numbers in scientific notation. To extend the definition If a number is purely imaginary or purely real, then set the other part equal to 0. Add Complex Numbers Calculator to your website to get the ease of using this calculator directly. The ordinary definition of exponentiation of real numbers of 4^(1/2) as equalling 2 even though both 2 and -2 are square An online complex number calculator allows you to perform the basic mathematical operations to simplify the given complex expressions. A complex number is the sum of an imaginary number and a real number, expressed as a + bi. This is illustrated in the If x is a "purely imaginary" number, that is, if x=ci where c is do it by writing , or by writing It is expressed as x + yi. Home Page. This time, the real part can be written as Re(A * B) = x * m y * n, and the imaginary part as Im(A * B) = x * n + y * m. Remember that complex number calculators use a negative sign in the real part because, at some point, we are faced with the product of two numbers i * i, which by definition is -1. to calculate it. So it is reasonable to think of its real and imaginary parts, you find that the real part is the same

Disable your Adblocker and refresh your web page . From the source of Wikipedia: Notation, Visualization, Cartesian complex plane, Polar complex plane, Modulus and argument, Complex graphs. values for a^(b+ci). According to the notation in the previous section: However, use an online Composite Function Calculator that solves the composition of the functions from entered values of functions f(x) and g(x) at specific points. (where n! Lets take a look at how complex numbers are multiplied together by simplify complex numbers calculator.