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## Complex Numbers 2

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**Complex Numbers 2**www.mathxtc.com**Complex NumbersWho uses themin real life?**The navigation system in the space shuttle depends on complex numbers!**-2**Can you see a problem here?**-2**Who goes first?**-2**Complex numbers do not have order**What is a complex number?**• It is a tool to solve an equation.**What is a complex number?**• It is a tool to solve an equation. • It has been used to solve equations for the last 200 years or so.**What is a complex number?**• It is a tool to solve an equation. • It has been used to solve equations for the last 200 years or so. • It is defined to be isuch that ;**What is a complex number?**• It is a tool to solve an equation. • It has been used to solve equations for the last 200 years or so. • It is defined to be isuch that ; • Or in other words;**Complex**• i is an imaginary number**Complex**• i is an imaginary number • Or a complex number**Complex**• i is an imaginary number • Or a complex number • Or an unreal number**Complex?**• i is an imaginary number • Or a complex number • Or an unreal number • The terms are inter-changeable unreal complex imaginary**Some observations**• In the beginning there were counting numbers 1 2**Some observations**• In the beginning there were counting numbers • And then we needed integers 1 2**Some observations**• In the beginning there were counting numbers • And then we needed integers 1 2 -1 -3**Some observations**• In the beginning there were counting numbers • And then we needed integers • And rationals 1 0.41 2 -1 -3**Some observations**• In the beginning there were counting numbers • And then we needed integers • And rationals • And irrationals 1 0.41 2 -1 -3**Some observations**• In the beginning there were counting numbers • And then we needed integers • And rationals • And irrationals • And reals 1 0.41 2 -1 0 -3**So where do unreals fit in ?**We have always used them. 6 is not just 6 it is 6 + 0i. Complex numbers incorporate all numbers. 3 + 4i 2i 1 0.41 2 -1 0 -3**A number such as 3i is a purely imaginary number**• A number such as 6 is a purely real number**A number such as 3i is a purely imaginary number**• A number such as 6 is a purely real number • 6 + 3i is a complex number**A number such as 3i is a purely imaginary number**• A number such as 6 is a purely real number • 6 + 3i is a complex number • x + iy is the general form of a complex number**A number such as 3i is a purely imaginary number**• A number such as 6 is a purely real number • 6 + 3i is a complex number • x + iy is the general form of a complex number • If x + iy = 6 – 4i then x = 6 and y = -4**A number such as 3i is a purely imaginary number**• A number such as 6 is a purely real number • 6 + 3i is a complex number • x + iy is the general form of a complex number • If x + iy = 6 – 4i then x = 6 and y = –4 • The ‘real part’ of 6 – 4i is 6**Worked Examples**• Simplify**Worked Examples**• Simplify**Worked Examples**• Simplify • Evaluate**Worked Examples**• Simplify • Evaluate**Worked Examples**3. Simplify**Worked Examples**3. Simplify**Worked Examples**3. Simplify 4. Simplify**Worked Examples**3. Simplify 4. Simplify**Worked Examples**3. Simplify 4. Simplify 5. Simplify**Addition Subtraction Multiplication**3. Simplify 4. Simplify 5. Simplify**Division**6. Simplify**Division**6. Simplify The trick is to make the denominator real:**Division**6. Simplify The trick is to make the denominator real: