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Complex Numbers 1-5

Complex Numbers 1-5. Kayla McGoran, Allie Gloor, Megan White, and Sarah Kidder. Real Numbers. Real Numbers are used for measurement and that can be represented on a continuous number line.

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Complex Numbers 1-5

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  1. Complex Numbers1-5 Kayla McGoran, Allie Gloor, Megan White, and Sarah Kidder

  2. Real Numbers • Real Numbers are used for measurement and that can be represented on a continuous number line. • They consist of zeros and all positive and negative integers, rational numbers, and irrational numbers. • A basic property of real numbers is that their squares are never negative.

  3. Rational and Irrational Numbers • Rational Numbers are ratios of integers. They are used to represent fractional parts of quantities. • Irrational Numbers are numbers that cannot be expressed as a ratio.

  4. Complex Numbers • A complex number is any number of the form a+bi, where a and b are real numbers and iis the imaginary unit, a is the real part, and b is the imaginary part of a+bi.

  5. Complex Numbers • An imaginary unit iwith the following properties: • i=√-1 and i2= -1 • We define the square root of any negative number as: • If a>0, √-a=i√a • When a and b are both positive real numbers, but not when a and b are both negative real numbers. • √a●√b=√ab

  6. Parts of the imaginary unit… a+bi • The real part is a • The imaginary part is b • If b≠0, the number is an imaginary number • Imaginary numbers that a≠0 are called pure numbers. • a+bi and d+ci are equal if and only if a=c and b=d.

  7. Complex Conjugates • a+bi and a-bi are complex conjugates • Their sum is a real number and their product is a non-negative real number.

  8. Imaginary Numbers i= i i2= -1 i3= -i i4= 1 i5= i i6= -1

  9. Imaginary Numbers • To find in, you divide the exponent by 4, and the remainder is which nth power you match it up with. i46= i2 i349= i i567= i3 or -i

  10. Simplify: • √-3•√-6 i√3•i√6 i2√18 -√18 -3√2 2. √-4•√-36• √-144 i√4+i√36-i√144 2i+6i-12i 8i-12i -4i

  11. Simplify: 3. (8+6i)+(3-2i) 11+6i-2i 11+4i 5. (3+5i)(4+2i) 12+6i+20i+10i2 12+26i+10i2 12+26i-10 2+26i 4. (6+i)(6-i) 36-6i+6i-i2 36-i2 36-(-1) 37 6. (6-3i)-(2-7i) 6-3i-2+7i 6-2+4i 4+4i

  12. Simplify: 7. (2+i√4)(2-i√4) 4-2i√4+2i√4-i2√4 4-i2√4 4-2i2 4-(-2) 6 8. (6+4i)2 (6+4i)(6+4i) 36+24i+24i+16i2 36+48i+16i2 36+48i-16 20+48i 9. 3 • 4+5i 4-5i 4+5i 12+15i 16+20i-20i-25i2 12+15i 16-25i2 12+15i 16-(-25) 12+15i 41

  13. Simplify: 7. 7+4i•7+4i 7-4i7-4i 49+28i+28i+16i2 49+28i-28i-16i2 49+56i+16i2 49-16i2 49+56i-16 49-(-16) 33+56i 65 8. 26• i = 26i iii2 26i =-26i -1 • i+i2+i3+i4+i5 i-1-i+1+i i

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