  Assignment 28

 "With the help of God, and with His precious assistance, I say that algebra is a scientific art" -- (Omar Khayyam, 1048-1131) Fibonacci (Leonardo of Pisa, 1170-1250): While he is famous for his sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... Fibonacci also made a major contribution to mathematics by advocating for the acceptance of the decimal system in the western world. Born in Italy, he studied under Islamic teachers during a period of his life when he lived in North Africa. He saw the advantage of the decimal system over the cumbersome systems used in Europe. He wrote many mathematical manuscripts that made use of the decimal system. Interestingly, he signed his works "Blockhead." Fibonacci numbers can be seen in the scales of pinkness, the knobbles of pineapples, the leaves of plants, and many other elements of nature. If you carefully examine piano keys, you will see Fibonacci numbers. Where did Herkimer expect to find landscaping companies in the Middle East? Answer:In Soddy Arabia. Herky's friends: CAROL LING...she loves to sing Christmas songs. SID DOWN...this guy never liked to stand around for long periods of time. ASSIGNMENT #28 Reading: Section 5.4, pages 272-277. Written: Pages 277-279/18-69 (multiples of 3). Mathematical word analysis:CONJUGATE: From the union of the Latin prefix com (together) and the root juge (yoke). Basically it means to bind together in a pair. Mathematically, it is often used to reference things that are opposite in some way. (See this use of the word in the written material below.)

There is an extension of the real number system that has practical use in the real world. Unfortunately, history has attached terms such as "imaginary" and "complex," to this extension, so we just live with the terminology. Use of the word complex in this number system does not imply "difficult," and imaginary does not imply "something that is not reality in the real world."

In the real number system, the equation x2 + 1 = 0 ==> x2 = -1 has no solutions. That is, you can't square a real number and produce an negative number. However, if we define a number i = ÷(-1), and further say that i2 = -1, interesting things can happen. Now, i is called an imaginary number . (OK, it's imaginary only in the sense that it is not a real number.)

This allows us to create numbers such as 8 + 3i. These (unfortunately) are called complex numbers . A complex number is the sum of a real number and an imaginary number. Referencing 8 + 3i, the 8 is the real part of the number, and the 3i is the imaginary part.

As the text indicates, we can create a 2-dimensional system with the real numbers on the horizontal axis and imaginary numbers on the vertical axis. This is called the complex number plane . Any complex number can be represented in the complex number plane.

Complex numbers "behave" like real numbers under addition and multiplication. For instance

(3 + 6i) + (4 - 2i) = 9 + 4i, and

(3 + 6i)(4 - 2i) = 12 - 6i + 24i - 12(i2) = 12 + 18i -12(-1) = 24 + 18i.

The conjugate of the complex number a + bi is a - bi. The product of a complex number and its conjugate is a real number. Note that (a + bi)(a - bi) = a2 + b2. This is useful in writing complex number quotients in standard from. (See example in the problems presented below.)

The absolute value of a complex number a + bi is written |a + bi| , and this is defined to be ÷(a2 + b2). Hence

|3 + 4i| = ÷(32 + 42) = ÷25 = 5.

Now, you are simply being introduced to a new idea. Accept the challenge! We won't spend a lot of time working with complex numbers, but you should certainly be aware of their existence. (Yes, they do exist despite the word imaginary .)

MATH IS POWER. COMPLEX NUMBERS ARE EXTREMELY USEFUL IN OUR MATHEMATICALLY-DESIGNED UNIVERSE.

 Problem: Simplify the expression (2 + 5i) + (7 - 2i) - 12i Solution (with communication): (2 + 5i) + (7 - 2i) - 12i = (9 + 3i) - 12i = 9 - 9i. Problem: Perform the indicated multiplication. (8 - 2i)(3 + 4i) Solution (with communication): (8 - 2i)(3 + 4i) = 24 + 32i - 6i - 8i2 = 24 + 26i - 8(-1) = 32 + 26i. Problem: Write (2 + 5i)/(3 + i) in standard from. Solution (with communication): Multiply numerator and denominator of the given quotient by 3 - i, which is the conjugate of 3 + i. (2 + 5i)/(3 + i) = [(2 + 5i)/(3 + i)]/[(3 - i)/(3 - i)] = (6 - 2i + 15i -5i2)/(9 - i2) = (11 + 13i)/10 = 1.1 + 1.3i. Problem: Find the absolute value of the complex number 6 - 8i. Solution (with communication): | 6 - 8i | = ÷[62 + (-8)2] = ÷100 = 10.