Solution use the distributive property to write this as. Simplify the powers of i, specifically remember that i 2 1. Note that real numbers are complex a real number is simply a. Use the commutative, assoc iative, and distributive properties to add and subtract complex numbers. To divide complex numbers, you must multiply by the conjugate. To find the conjugate of a complex number all you have to do is change the sign between the two terms in the denominator. A complex number is any expression that is a sum of a pure imaginary number and a real number. Use complex conjugates to write the quotient of two complex numbers in standard form. Download englishus transcript pdf i assume from high school you know how to add and multiply complex numbers using the relation i squared equals negative one.
This means that if two complex numbers are equal, their real and imaginary parts must be equal. This is more or less the simplest equation with no solution in r. Complex numbers complex numbers pearson schools and fe. We can plot complex numbers on the complex plane, where the xaxis is the real part. I say almost because after we multiply the complex numbers, we have a little bit of simplifying work. Multiplying complex numbers is almost as easy as multiplying two binomials together. The product of complex conjugates is always a real number.
In spite of this it turns out to be very useful to assume that there is a number ifor which one has 1 i2. A frequently used property of the complex conjugate is the following formula 2 ww. Complex numbers are a combination of a real number with an imaginary one. Hence, they would supply you only with the theory part i covered. Use the relation i 2 1 to multiply two imaginary numbers to get a real number. You can click on the pic to view it in another tab to get an enlarged view of the same. Imaginary numbers a number whose square is less than zero negative. Full teaching notes for a2 complex numbers tes resources. Simplify the powers of i, specifically remember that i 2. Im a little less certain that you remember how to divide them.
A complex number zis an ordered pair of real numbers a,b. Try 1 5th roots of 32 ve of them, 2 4th roots of 81. Finding nth roots of other real numbers is just as easy. Classroom size graphic organizer and postit notes labeled with the various numbers in the system. Furthermore, each real number is in the set of complex numbers, so that the real numbers are a subset of the complex numbers see figure 1. Algebraic, geometric, cartesian, polar, vector representation of the complex numbers.
Adding, multiplying, dividing, subtracting in rectangular form finding the modulus and argument of a complex number converting between rectangular and polar form finding the square root of a complex number loci of complex numbers ive also included a. If complex numbers and are represented by the vectors and respectively, then is represented by example 10. Distribute or foil in both the numerator and denominator to remove the parenthesis step 3. The multiplication of conjugates always results in a real number. I hope you read last night by way of preparation for that, but since thats something were going to have to do a lot of a differential equations, so remember that the. However, there is still one basic procedure that is missing from the algebra of complex numbers. Here is a way to understand some of the basic properties of c using our knowledge of linear algebra. To divide complex numbers one should multiply numerator and denomi nator by the. Stolp 101800 95 complex numbers imaginary j 1 the imaginary number rectangular form a a b. Given that the complex numbers and are represented in an argand diagram by the points a and b respectively, find the length of ab. Read all the important points and chapterwise notes on iitjee. These are my teaching notes for the cie a2 pure complex numbers unit. Finding nthroots of complex numbers is just a little more di cult. It is presented solely for those who might be interested.
Nov 23, 2015 these are my teaching notes for the cie a2 pure complex numbers unit. To see this, consider the problem of finding the square root of a complex number. Notes for day 4 andrew geng hssp spring 2008 1 taylor series for a function fx, its taylor series can be thought of as a polynomial possibly of in. Derive the equation of a parabola given the focus and directrix 10. Also, these notes were only guiding my theory behind the chapter. The relationship between exponential and trigonometric functions. To divide two complex numbers one always uses the following trick. For the love of physics walter lewin may 16, 2011 duration. Next we investigate the values of the exponential function with complex arguments.
See more ideas about complex numbers, algebra and maths algebra. To compute in for n 4, we divide n by 4 and write it in the form n. The following notation is used for the real and imaginary parts of a complex number z. I am enclosing a scanned version of my notes on complex numbers. Use the imaginary unit i to write complex numbers, and add, subtract, and multiply. Subscript r means the real part of complex number, and the subscript i means the. Its all about complex conjugates and multiplication. In this unit we are going to look at how to divide a complex number by another complex number.
Students use the concept of conjugate to divide complex numbers. Mathematics extension 2 complex numbers dux college. The mathematical jargon for this is that c, like r, is a eld. This is the second day of a twoday lesson on complex number division and applying this knowledge to further questions about linearity. Complex numbers and imaginary numbers the set of all numbers in the form a bi, with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The first section is a more mathematical definition of complex numbers and is not really required for understanding the remainder of the document. It can be constructed by choosing a point and stipulating that f. An introduction to complex numbers homepages of uvafnwi. Now, i guess in the time remaining, im not going to talk about in the notes, i, r, at all, but i would like to talk a little bit about the extraction of the complex roots, since you have a problem about that and because its another beautiful application of this polar way of writing complex numbers.
Mathematical institute, oxford, ox1 2lb, november 2003 abstract cartesian and polar form of a complex number. But let us imagine that there is some number ithat satis. Use pythagorean theorem to determine the absolute value of this point. In other words, elements of c are pairs of real numbers. Note that the set of real numbers rrrr is a proper subset of the set of complex numbers cccc, since any real number a. Complex numbers absolute value of a complex number. This will leaf to the wellknown euler formula for complex numbers. The real and imaginary parts of a complex number are given by re3.
Finally, any quadratic equation with real coefficients, or even any polynomial with real coefficients, has solutions that can be represented as complex numbers. The complex plane the real number line below exhibits a linear ordering of the real numbers. Note that this rule says that to multiply two complex numbers you multiply moduli and add arguments, whereas to divide two complex numbers you divide. It uni es the mathematical number system and explains many mathematical phenomena. Algebra revision notes on complex numbers for iit jee.
Distribute or foil in both the numerator and denominator to remove the parenthesis. In fact, for any complex number z, its conjugate is given by z rez imz. Complex solutions of a quadratic equation solve a x2 4 4 o and b 312 2x 5 o example 4 42i writing a quotient of complex numbers in standard form a. Full teaching notes for a2 complex numbers teaching. First, find the complex conjugate of the denominator, multiply the numerator and denominator by that conjugate and simplify. In spite of this it turns out to be very useful to assume that there is a number ifor which one has. Notes on complex numbers university of british columbia, vancouver yuexian li march 17, 2015 1.
They will gain an understanding of the definition of each type of number. Graphically the absolute value of complex number is the distance from the origin to the complex point in the complex plane. Division of complex numbers relies on two important principles. When d 0, roots of the quadratic equation are real and equal.
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