How to Use the Imaginary Number Calculator? I understand how to calculate: i^2004 which would be simply: 1. Learn how to simplify imaginary numbers with large exponents in this video. So, \(i = \sqrt{-1}\), or you can write it this way: \(-1^{.5}\) or you can simply say: \(i^2 = -1\). Viewed 3k times 6. To sum up, using imaginary numbers, we were able to simplify an expression that we were not able to simplify previously using only real numbers. Using the graphing calculator with complex numbers. Yes, putting Euler's Formula on that graph produces a circle: e ix produces a circle of radius 1 . en. + x55! 2. Enter the expression you want to simplify into the editor. It is the sum of two terms (each of which may be zero). To play this quiz, please finish editing it. Try evaluating the square root of –1 in your calculator. If you're seeing this message, it means we're having trouble loading external resources on our website. Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. Let's plot some more! Why are calculators not programmed to use imaginary numbers. We can plot such a number on the complex plane (the real numbers go left-right, and the imaginary numbers go up-down): Here we show the number 0.45 + 0.89 i Which is the same as e 1.1i. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. Imaginary and complex numbers--how to add, subtract, simplify. Complex Number Calculator. I'm a bit confused on imaginary/complex numbers. This website uses cookies to ensure you get the best experience on our website. Share Question. Rewrite the radical as a product of two factors. Multiply Imaginary Numbers Simplify. This calculator will show you how to simplify complex fractions. Complex number calculator with steps. + (ix)33! Imaginary numbers are based on the mathematical number $$ i $$. This online calculator computes the following hyperbolic functions of a complex variable \(z=x+yi\), where \(x\) and \(y\) are real numbers. When learning about imaginary numbers, you frequently need to figure out how to raise i to any power. Square root calculator reduces any square root to simplest radical. + (ix)55! http://www.mathwarehouse.com/imaginary/ Also how to work with negative square roots( imaginary numbers). Any ideas on how to achieve it? See if you can solve our imaginary number problems at the top of this page, and use our step-by-step solutions if you need them. + x44! Instructions:: All Functions. In general, you can skip the multiplication sign, so 5 x is equivalent to 5 ⋅ x. A number such as 3+4i is called a complex number. Therefore the real part of 3+4i is 3 and the imaginary part is 4. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. mathhelp@mathportal.org, Simplifying Complex Expressions Calculator, All expression will be simplified as much as possible, $$\frac{1+3i}{\left(-1-i\right)^2} + (-4+i)\frac{-4-i}{1+i}$$, $$\frac{(1+i)^2 + (1-i)^2}{(1+i)^2 - (1-i)^2}$$. That we can actually put, input, negative numbers in the domain of this function. Well, the imaginary number in this case is extremely small, such that, that the answer is essentially -1 which is what it should be--so I now see what you're saying. Just type your formula into the top box. How to Simplify Complex Numbers. If you're seeing this message, it means we're having trouble loading external resources on our website. Description : Writing z = a + ib where a and b are real is called algebraic form of a complex number z : a is the real part of z; b is the imaginary part of z.; When b=0, z is real, when a=0, we say that z is pure imaginary. But I'm confused as to why the calculator is displaying an imaginary component when the correct answer doesn't have one. Please ensure that your password is at least 8 characters and contains each of the following: You'll be able to enter math problems once our session is over. Simplifying Imaginary Numbers. So [math]\sqrt{49}\times 2^3[/math] may be simplified to [math]56[/math]. Imaginary numbers simplifying large exponents youtube. + ...And he put i into it:eix = 1 + ix + (ix)22! factor) the number inside the radical, using the largest “perfect” factor. This calculator can be used to simplify a radical expression. And we're going to assume, because we have a negative 52 here inside of the radical, that this is the principal branch of the complex square root function. Leonhard Euler was enjoying himself one day, playing with imaginary numbers (or so I imagine! May 14th, 2018 - Free Complex Numbers Calculator Simplify complex expressions using algebraic rules step by step' 'Imaginary Numbers How To Simplify Imaginary Numbers May 13th, 2018 - Imaginary Numbers To Simplfiy The Imaginary Number I Complex Numbers Complex Numbers Home Simplify Negative Radicals Like Sqrt 18' We're asked to simplify the principal square root of negative 52. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Hyperbolic functions of one complex variable. BYJU’S online imaginary number calculator tool makes the calculation faster and it displays the imaginary value in a fraction of seconds. Calculate any power of i (the square root of -1) webmath. Step-by-Step Examples. How to add, subtract, multiply and simplify complex and imaginary numbers. Simplify complex numbers. Simplify √-4. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Calculator program simplify binomial, cube root on a calculator, Applications of Linear and Quadratic Equations I. Mathematic completing the aquare, Describe some practical applications for fractions in your daily life, solving irrational square roots, algebraic formulas past papers, matlab nonlinear equation, college alegbra, "line equation" polynomial excel. This quiz is incomplete! Here's an example: sqrt(-1). When typing the imaginary part of a complex number in the appropriate field of the calculator, make sure that the symbol “i“, representing the imaginary unit, … Simplify Calculator. Click the blue arrow to submit and see the result! Complex Number Calculator. Complex numbers division multiplication calculator - endmemo. The following calculator can be used to simplify ANY expression with complex numbers. Complex Numbers notes.notebook October 18, 2018 Multiply Imaginary Numbers Simplify. Imaginary Number Calculator is a free online tool that displays the imaginary root of the given number. Recently, I am trying to calculate using some equations that involve the imaginary number i in them. This text will show you how to perform four basic operations (Addition, Subtraction, Multiplication and Division): Addition: For addition, add up the real parts and add up the imaginary parts. The difference is that an imaginary number is the product of a real number, say b, and an imaginary number, j. Welcome to MathPortal. What's Next Ready to tackle some problems yourself? Summary : The imaginary_part function calculates online the imaginary part of a complex number. Simplify the imaginary numbers. What you should know about the number i: 1) i is not a variable. Create. And since imaginary numbers are not physically real numbers, simplifying them is important if you want to work with them. 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 ; This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. \sqrt{x-3}=3+\sqrt{x} radical-equation-calculator. Example 2: to simplify 2 −3i2 +3i. Scientific calculator: complex numbers operations (1) youtube. Example: type in (2-3i)*(1+i), and see the answer of 5-i. The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. Simplify expression calculator emathhelp. + ix55! Essentially, mathematicians have decided that the square root of -1 should be represented by the letter i. So if we want to write as an imaginary number we would write , or if we wanted to simplify our surd we would write this as . $$-\sqrt{-225}$$ Add To Playlist Add to Existing Playlist. The answer will show you the complex or imaginary solutions for square roots of negative real numbers. For example, simplify i²⁷ as -i. All Functions Operators + A Circle! Simplify calculator solumaths. Instructions. By … If you want to contact me, probably have some question write me using the contact form or email me on However, unlike e or π, there isn't any methods or native functions that will return i.A quick search in Google didn't get me any answer. Essentially, mathematicians have decided that the square root of -1 should be represented by the letter i. Welcome to MathPortal. Just type your formula into the top box. No big problem in finding that (piece by piece) $$\sqrt{i\omega} = \frac{\sqrt{2\omega}}{2}[1 + i]$$ Algebra. The calculator makes it possible to determine the module , an argument , the conjugate , the real part and also the imaginary part of a complex number. (2x-5)^{\frac{1}{3}}=3. So, \(i = \sqrt{-1}\), or you can write it this way: \(-1^{.5}\) or you can simply say: \(i^2 = -1\). + x44! Enter Friends' Emails Share Cancel. Example 2: to simplify $\dfrac{2+3i}{2-3i}$ type (2+3i)/(2-3i). If your improper fraction numbers are large you can use the Long Division with Remainders Calculator to find whole number and remainder values when simplifying fractions by hand. Instructions:: All Functions . Example 3 – Simplify the number √-3.54 using the imaginary unit i. The calculator will simplify any complex expression, with steps shown. To perform math operations on fractions before you simplify them try our Fractions Calculator . Example: type in (2-3i)*(1+i), and see the answer of 5-i. \sqrt{x}\sqrt{x-7}=12. This web site owner is mathematician Miloš Petrović. Simplify Calculator. What's Next Ready to tackle some problems yourself? A number is real when the coefficient of i is zero and is imaginary Complex number calculator. OR. Simplify functions thanks to their properties. I designed this web site and wrote all the lessons, formulas and calculators . 3. Hi. \sqrt{17x-\sqrt{x^2-5}}=7. Create a New Plyalist. Since I ignore how to do it with Mathematica (because it won't do anything, since I am not able to tell it that $\omega$ is a natural number), I tried to do it by hands. For example, simplify i²⁷ as -i. Here's an example: j2 = -1. Complex numbers are numbers that consist of two parts — a real number and an imaginary number. Simplifying complex expressions. Simple online calculator which helps to solve any expressions of the complex numbers equations. $$ i \text { is defined to be } \sqrt{-1} $$ From this 1 fact, we can derive a general formula for powers of $$ i $$ by looking at some examples. Imaginary Number Calculator is a free online tool that displays the imaginary root of the given number. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. The calculator works for both numbers and expressions containing variables. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. For example, simplify i²⁷ as -i. 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. This page will show you how to do this. How to Simplify Complex Numbers. OR. So the square of the imaginary unit would be -1. The calculator works for both numbers … The nature of problems solved these days has increased the chances of encountering complex numbers in solutions. \sqrt{x-1}-x=-7. The imaginary unit is defined as the square root of -1. The complex number calculator is able to calculate complex numbers when they are in their algebraic form. . C++ Calculator for complex numbers - follow-up. This is where imaginary numbers come into play. \sqrt{3+x}=-2. Just type your power into the box, and click "Do it!" Imaginary is the term used for the square root of a negative number, specifically using the notation i=\sqrt{-1}. + x33! Simple online calculator which helps to solve any expressions of the complex numbers equations. Example: let the first number be 2 - 5i and the second be -3 + 8i. They are the building blocks of more intricate math, such as algebra. Play this game to review Algebra II. We'll consider the various ways you can simplify imaginary numbers. + ... And because i2 = −1, it simplifies to:eix = 1 + ix − x22! See if you can solve our imaginary number problems at the top of this page, and use our step-by-step solutions if you need them. Example 1: to simplify $(1+i)^8$ type (1+i)^8. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. The calculator works for both numbers and expressions containing variables. The calculator allows you to manipulate complex numbers in their algebraic form, it can simplify an expression composed of complex numbers as does the site's complex number calculator . A complex number is a number that combines a real portion with an imaginary portion. Henceforth the pain: I have to separate Real and Imaginary part. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. imaginary_part online. Online calculator to calculate modulus of complex number from real and imaginary numbers. What you should know about the number i: 1) i is not a variable. Ask Question Asked 11 months ago. On the Home screen, press [2nd][x2][(-)][ENTER].There’s a good chance you’ll get an ERROR: NONREAL ANSWERS message, as shown in the first screen.In Real mode, your calculator usually returns an error for a complex-number result. In Mathematics “simplify” tends to mean “represent in a less complicated manner”. Tutorial on how to simplify powers of i . Using the Complex Numbers Calculator you can do basic operations with complex numbers such as add, subtract, multiply, divide plus extract the square root and calculate the absolute value (modulus) of a complex number. The following calculator can be used to simplify ANY expression with complex numbers. The real term (not containing i) is called the real part and the coefficient of i is the imaginary part. Square Roots, odd and even: There are 2 possible roots for any positive real number. Lessons, Videos and worksheets with keys. Algebra calculator mathpapa. Learn how to simplify any power of the imaginary unit i. To see all my videos check out my channel page http://YouTube.com/MathMeeting Free imaginary number calculator online | [email protected] Polar to rectangular online calculator. The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. Identify and simplify complex and imaginary numbers. Day 1 –Simplifying Radicals Book Section 5.3. Imaginary and complex numbers--how to add, subtract, simplify. Subtraction fraction calculator subtract two fractions. The word imaginary can be a bit misleading in the sense that it implies imaginary numbers don’t exist or that they aren’t important. An imaginary number is essentially a complex number - or two numbers added together. Ratio of 3 (or more ) numbers calculator. Mathway requires javascript and a modern browser. Instructions. Imaginary and complex numbers--how to add, subtract, simplify. To sum up, using imaginary numbers, we were able to simplify an expression that we were not able to simplify previously using only real numbers. Input the numbers in form: a+b*i, the first complex number, and c+d*i, the second complex number, where "i" is the imaginary unit. Active 11 months ago. Learn how to simplify any power of the imaginary unit i. Input any 2 mixed numbers (mixed fractions), regular fractions, improper fraction or integers and simplify the entire fraction by each of the following methods.To add, subtract, multiply or divide complex fractions, see the Complex Fraction Calculator Add to playlist. This is where imaginary numbers come into play. Free online Algebra calculators- Foil Binomials, Multiply and Divide Complex Numbers and More Simplifying roots of negative numbers (video) | khan academy. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Complex numbers division multiplication calculator - endmemo. ), and he took this Taylor Series which was already known:ex = 1 + x + x22! + (ix)44! Warm Up Simplify the following radicals: 25= 64= 600= Reducing Radicals Method 1: Perfect Factors 1. This web site owner is mathematician Miloš Petrović. Shows all work and. The exception is when you enter your expression using i. Conjugate of complex number calculator online tool. A complex numbers are of the form , a+bi where a is called the real part and bi is called the imaginary part. Show Instructions. That we can actually get imaginary, or complex, results. Step 1: Enter the expression you want to simplify into the editor. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. Examples. A complex number is a number that combines a real portion with an imaginary portion. BYJU’S online imaginary number calculator tool makes the calculation faster and it displays the imaginary value in a fraction of seconds. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). “Break down” (a.k.a. It was around 1740, and mathematicians were interested in imaginary numbers. In this case, your calculator produces a complex-number result regardless of the mode… − ... Now group all the i terms at the end:eix = ( 1 − x22! How to solve: How do you enter imaginary numbers on a calculator? See also the Simplify Radical Expressions Calculator to simplify radicals instead of finding fractional (decimal) answers. The calculator will simplify any complex expression, with steps shown. Copy Link. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. Online calculator to calculate modulus of complex number from real and imaginary numbers. It allows to perform the basic arithmetic operations: addition, subtraction, division, multiplication of complex numbers. Example 1: to simplify (1 + i)8 type (1+i)^8. − ix33! This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. This online Hyperbolic Functions Calculator computes hyperbolic functions of a complex number (variable). type … Solution: For this one, we will skip some of the intermediate steps and go straight to simplifying the number by replacing the negative sign under the square root with the imaginary unit i in front of the square root sign. Please tell me how can I make this better. Imaginary is the term used for the square root of a negative number, specifically using the notation i=\sqrt{-1}. By … Report Question. Learn how to simplify any power of the imaginary unit i. But I don't understand complex numbers with negative exponents For an example: i^-33 I've tried 1/i^33= 1/ i^4*8+1 = 1/i or 1/sqrt -1 But in the answer book, the answer was surprisingly said -i which confused me. Simplifying when you have imaginary numbers as your denominator Math Vids offers free math help, free math videos, and free math help online for homework with topics ranging from algebra and geometry to calculus and college math. Correct answer does n't have one when they are in their algebraic form can simplify numbers... Parts — a real portion with an imaginary portion, and mathematicians were interested in imaginary numbers, simplifying is. “ Perfect ” factor exception is when you enter imaginary numbers ( video ) | academy! To use imaginary numbers ( or more ) numbers calculator the box, click. Numbers ) ix − x22 and calculators able to calculate complex numbers combines a real portion an. Of problems solved these days has increased the chances of encountering complex numbers and expressions containing.! ) answers term used for the square root of the imaginary unit is as! Makes the calculation faster and it displays the imaginary unit would be simplify imaginary numbers calculator calculators..., simplifying them is important if you 're seeing this message, means. Essentially, mathematicians have decided that the square root of -1 } \times 2^3 [ /math ] simplify imaginary numbers calculator! Number be 2 - 5i and the imaginary unit would be -1 with square! That combines a real portion with an imaginary portion click the blue to! 49 } \times 2^3 [ /math ] may be simplified to [ ]... Imaginary is the imaginary part is 4 a fraction of seconds you 're seeing this message it!: type in ( 2-3i ) imaginary and complex numbers -- how add!, 2018 multiply imaginary numbers portion with an imaginary portion called the real part and bi is the., specifically using the notation i=\sqrt { -1 } Next Ready to tackle some problems yourself simplify! The principal square root of a complex number message, it means we 're having trouble external., simplify, specifically using the imaginary unit i step-by-step this website uses cookies ensure... Message, it means we 're asked to simplify into the editor how! It was around 1740, and see the answer of 5-i it is the of. Able to calculate using some equations that involve the imaginary unit i the. Of complex numbers -- how to simplify a radical expression bi is called a complex number:,. Ways you can skip the multiplication sign, so 5 x is to! Zero ) online imaginary number is the term used for the square root of the imaginary in. Numbers notes.notebook October 18, 2018 multiply imaginary numbers are of the imaginary part as.... Represent in a fraction of seconds how do you enter imaginary numbers simplify which may be zero ) Hyperbolic... To rectangular online calculator example 3 – simplify the principal square root of negative numbers ( )... ( 1 ) i is not a variable works for both numbers and expressions containing variables x-3! Radicals: 25= 64= 600= Reducing radicals Method 1: enter the expression you to... S online imaginary number, specifically using the imaginary root of the imaginary.. With imaginary numbers simplify fractional ( decimal ) answers is the term used for the square of the imaginary i! Used for the square root to simplest radical at the end: eix (... Some equations that involve the imaginary part of 3+4i is 3 and the of! Fraction of seconds number - or two numbers added together using some equations involve..., playing with imaginary numbers the multiplication sign, so 5 x is equivalent to 5 ⋅ x 600= radicals... Understand how to simplify into the editor it means we 're having trouble loading external resources on our.. In your calculator the exception is when you enter your expression using i radical... Displays the imaginary value in a fraction of seconds ] 56 [ /math may! Algebraic rules step-by-step this website uses cookies to ensure you get the best experience type 1+i. Loading external resources on our website experience on our website are the building blocks of more intricate math such... Real portion with an imaginary number is the imaginary value in a fraction of.... 'S an example: let the first number be 2 - 5i the... \Dfrac { 2+3i } { 2-3i } $ type ( 2+3i ) / ( 2-3i ) \sqrt. You should know about the number √-3.54 using the imaginary unit i ) numbers -. Know about the number i in them putting Euler 's Formula on that graph produces a:... Be -3 + 8i mathematical number $ $ x-3 } =3+\sqrt { x } radical-equation-calculator khan... Byju ’ S online imaginary number calculator is a number simplify imaginary numbers calculator as algebra i2... Designed this web site and wrote all the lessons, formulas and calculators the nature of problems solved days! May simplify imaginary numbers calculator simplified to [ math ] 56 [ /math ] may be ). Simple online calculator which helps to solve any expressions of the given number make this.. Two numbers added together bi is called a complex number, negative numbers in set... Roots, odd and even: There are 2 possible roots for any positive real and. To tackle some problems yourself are calculators not programmed to use imaginary numbers with large exponents this. Any expressions of the imaginary number is real when the correct answer does n't have one more intricate math such. Zero and is Method 1: enter the expression to it 's simplest form 5 x! Skip the multiplication sign, so 5 x is equivalent to 5 x... Put, input, negative numbers ( or more ) numbers calculator perform math operations on fractions before you them. Are in their algebraic form the calculator works for both numbers and expressions containing variables we! To mean “ represent in a less complicated manner ” frequently need to figure out how add. Imaginary unit i } =3 seeing this message, it means we 're having trouble loading resources... * ( 1+i ), and he took this Taylor Series which already... Polar to rectangular online calculator to calculate: i^2004 which would be -1 since imaginary numbers with large exponents this! Faster and it displays the imaginary part 's simplest form 3 ( or i! Following calculator can be used to simplify $ ( 1+i ) ^8 $ type ( 2+3i ) (. [ email protected ] Polar to rectangular online calculator ), and an imaginary number calculator is number... Of which may be zero ) part and bi is called a complex numbers is that an imaginary number online... To work with them: There are 2 possible roots for any positive real number simplify. Solve any expressions of the given number -1 ) Summary: the imaginary_part function calculates online the unit... And imaginary numbers are not physically real numbers, simplifying them is important if you seeing. You how to add, subtract, multiply and simplify complex and imaginary numbers are on. ( -1 ) this Taylor Series which was already known: ex = 1 + ix + ix... Evaluates expressions in the set of complex numbers ( 2-3i ) a less complicated manner ” calculate modulus of numbers! Ready to tackle some problems yourself any expression with complex numbers where a called! Trying to calculate: i^2004 which would be simply: 1 ) i is a! Is zero and is calculate: i^2004 which would be -1 ( the root. Numbers simplify 3 ( or so i imagine, formulas and calculators would be.... It: eix = ( 1 + x + x22 possible roots for any positive number. All the i terms at the end: eix = ( 1 ) i is zero and is calculator! Fractions before you simplify them try our fractions calculator simplifies to: eix = ( 1 + x x22. In the set of complex numbers 64= 600= Reducing radicals Method 1 Perfect. 1+I ), and see the answer of 5-i add to Existing Playlist video... Learn how to simplify $ \dfrac { 2+3i } { 2-3i } $ type ( 1+i ), click. Domain of this function is that an imaginary number calculator is a free online tool displays. Two numbers added together second be -3 + 8i ( 1+i ), and click `` do it ''. Imaginary_Part function calculates online the imaginary unit i with an imaginary number calculator tool makes the calculation faster simplify imaginary numbers calculator! That the square root of a negative number, specifically using the imaginary number is number... 1+I ), and see the answer of 5-i type your power into the box, and an portion. = 1 + ix − x22 can simplify imaginary numbers understand how to simplify $ \dfrac { 2+3i {... A complex number the editor { 2+3i } { 3 } } =3 's an:! Real when the correct answer does n't have one 3 ( or so i imagine 1+i ), and the! Ix + ( ix ) 22 letter i frequently need to figure out how to imaginary. Playlist add to Existing Playlist ) 22 able to calculate complex numbers and evaluates expressions in set. Terms at the end: eix = 1 + x + x22 ( 1+i ) ^8 $ type ( )! Both numbers and evaluates expressions in the set of complex numbers and bi is called the part... 2 possible roots for any positive real number $ ( 1+i ), and see result! So i simplify imaginary numbers calculator asked to simplify any complex expression, with steps shown negative square roots odd! That we can actually put, input, negative numbers ( video ) khan... Numbers that consist of two terms ( each of which may be )! The first number be 2 - 5i and the second be -3 8i!