* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Lesson 104: Review of Complex Numbers, Subsets of the Real
Survey
Document related concepts
Abuse of notation wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Infinitesimal wikipedia , lookup
Large numbers wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Location arithmetic wikipedia , lookup
Real number wikipedia , lookup
Approximations of π wikipedia , lookup
Transcript
Lesson 104: Review of Complex Numbers, Subsets of the Real Numbers, Representing Repeating Decimals as Fractions We review complex numbers by remember that this number 4/7 + √2i Is a complex number written in standard form. The real part is 4/7 and is written first. The imaginary part is √2i and is written after the real part. We often use the letters a and b to designate the standard form of a complex number by writing a + bi And say that a and b can be any real numbers. Since zero is a real number, either a or b can be zero. If b is 0, then only a is left, and thus, 4 ¾ -5√2 -19/3 Are all complex numbers whose imaginary parts are zero. If a is zero, then only the imaginary part b remains. Thus, the imaginary numbers -√2i 4√2/3i -3i i Are all complex numbers whose real part equal zero. The set of real numbers has an infinite number of members, and these can be used to form an infinite number of subsets. Normally, however, we restrict our attention to five major subsets of the set of real numbers. The first three are: The counting (natural) numbers {1, 2, 3, …} The whole numbers {0, 1, 2, 3…} The integers {…-3, -2, -1, 0, 1, 2, 3…} These three sets account for the numbers that are designated when the number line is drawn, because we usually designate the location of the integers below a number line. All integers can be written as fractions of other integers. For example, -4, 0, and 13 can be written as fractions as shown here. -4 = 12/-3 0 = 0/2 13 = -39/-3 We say that a number that can be written as a fraction of integers is a rational number, because ratio is another name for fraction. The rest of the set of real numbers I made up of all the positive numbers or arithmetic and their negative counterparts. Some of these numbers can be written as fractions of integers and thus are rational numbers. The rest cannot be written as fractions of integers and are irrational numbers. Irrational numbers cannot be represented exactly with decimal numerals that contain a finite number of digits. The square root of 2 is an irrational number and thus can only be approximated with a decimal numeral. A calculator gives an approximation of the square root of 2 as √2 = 1.4142136 The complete representation of this number would require a numeral with an infinite number of digits, and the digits would occur in a nonrepeating pattern. 1. If the digits in a decimal numeral terminate, the number is a rational number. 2. If the digits in a nonterminating decimal numeral repeat in a pattern, the number is a rational number. Example: Show that 0.00314 is a rational number by writing it as a fraction of integers. Answer: 314/100,000 Example: Show that 0.00000623 is a rational number by writing it as a fraction of integers. Answer: 623/100,000,000 We indicate that digits in a decimal fraction repeat by drawing a bar over the repeating digits. Thus, in the following numerals, the digits under the bars repeat in an endless pattern. _____ 0.01623 = 0.01623232323… _______ 1.0031543 = 1.0031543543543… Each of these numerals represents a rational number, and any rational number can be written as a quotient of integers. To write the first numeral as a quotient of integers, we must get rid of the repeating digits. We can eliminate these repeating digits by subtracting the number from the product of the number and 100. this product has the same repeating digits that the number has. 100N = 1.623 23 23 23 N = 0.016 23 23 23 99N = 1.607 (repeating digits eliminated) The equation with 100N is the same as the equation with N except that each side has been multiplied by 100. we multiplied by 100 because there were two repeating digits. Three repeating digits would require a multiplier of 1000, four repeating digits would require a multiplier of 10,000 etc. We will investigate this procedure in the next three examples. Example: ____ Show that 0.01623 is a rational number by writing it as a fraction of integers. Answer: 1607/99,000 Example: _______ Show that 1.0031543 is a rational number by writing it as a quotient of integers. Answer: 10,021,512/9,990,000 Example: __ Show that 13.012 is a rational number by writing it as a fraction of integers. Answer: 11,711/900 HW: Lesson 104 #1-30