Rationalizing: We can apply this rule to \rationalize" a complex number such as z = 1=(a+ bi). Remember a real part is any number OR letter that … Addition / Subtraction - Combine like terms (i.e. = + ∈ℂ, for some , ∈ℝ In this T & L Plan, some students Rings also were studied in the 1800s. Complex Numbers and the Complex Exponential 1. Questions can be pitched at different levels and can move from basic questioning to ones which are of a higher order nature. Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. The real complex numbers lie on the x–axis, which is then called the real axis, while the imaginary numbers lie on the Paul Garrett: Basic complex analysis (September 5, 2013) Proof: Since complex conjugation is a continuous map from C to itself, respecting addition and multiplication, ez = 1 + z 1! 1 Basic Theorems of Complex Analysis 1.1 The Complex Plane A complex number is a number of the form x + iy, where x and y are real numbers, and i2 = −1. The representation is known as the Argand diagram or complex plane. A complex number is any number that is written in the form a+ biwhere aand bare real numbers. Example: 7 + 2i A complex number written in the form a + bi or a + ib is written in standard form. + = ez Then jeixj2 = eixeix = eixe ix = e0 = 1 for real x. If z= a+biis a complex number, we say Re(z) = ais the real part of the complex number and we say Im(z) = bis the imaginary part of the complex number. Noether (1882{1935) gave general concept of com- Several elds were studied in mathematics for some time including the eld of real numbers the eld of rational number, and the eld of complex numbers, but there was no general de nition for a eld until the late 1800s. Complex numbers are often denoted by z. Basic Concepts of Complex Numbers If a = 0 and b ≠ 0, the complex number is a pure imaginary number. + z2 2! The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has Basic rules of arithmetic. 6.1 Video 21: Polar exponential form of a complex number 41 6.2 Revision Video 22: Intro to complex numbers + basic operations 43 6.3 Revision Video 23: Complex numbers and calculations 44 6.4 Video 24: Powers of complex numbers via polar forms 45 7 Powers of complex numbers 46 7.1 Video 25: Powers of complex numbers 46 + ::: = 1 + z 1 + z2 2! 2. The real numbers … Basic Arithmetic: … (See chapter2for elds.) Complex numbers can be represented as points in the plane, using the cor-respondence x + iy ↔ (x, y). For instance, for any complex numbers α,β,γ, we have • Commutative laws: α+β= β+αand αβ= βα. Complex numbers obey many of the same familiar rules that you already learned for real numbers. Basic rule: if you need to make something real, multiply by its complex conjugate. • Associative laws: (α+β)+γ= γ+(β+γ) and (αβ)γ= α(βγ). Example: 3i If a ≠0 and b ≠ 0, the complex number is a nonreal complex number. Complex numbers are built on the concept of being able to define the square root of negative one. If two complex numbers are equal then the real parts on the left of the ‘=’ will be equal to the real parts on the right of the ‘=’ and the imaginary parts will be equal to the imaginary parts. (Note: and both can be 0.) Complex Number – any number that can be written in the form + , where and are real numbers.

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