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JH WEEKLIES ISSUE #13 2011-2012 Mathematics—Numbers Types of Numbers In mathematics, though you may think that a number is just a number, there are actually several classifications for numbers. A number can be Real, Imaginary, Integer, Rational, Irrational, Complex, Whole, or several at the same time. Here, we’ll cover some of the more common types of numbers that arise in quizbowl. Whole Numbers The Whole numbers are positive counting numbers; they are used to indicate the presence of whole objects or concepts, and can be used as ordinal numbers (i.e. 1st, 2nd, 3rd, etc.). Whole numbers include the numbers 0, 1, 2, 3… There is a closely-related set of numbers, the Natural numbers, that exclude 0—therefore consisting of the numbers 1, 2, 3… The set of all Whole numbers is symbolized by a capital, bold-face “ .” The set of all Natural numbers is symbolized by a capital, bold-face “ℕ.” Both of the closely-related sets can be thought of as subsets of the Integers. Integer Numbers The Integer numbers are whole counting numbers that include the Natural numbers (0, 1, 2…) as well as the negative numbers (-1, -2, -3…). With the inclusion of the negative whole numbers, this set is broader than the Natural numbers, and therefore has more applications. The set of all Integer numbers is symbolized by a capital, bold-face “ℤ,” and can be thought of as a subset of the Reals. Questions Galore 319 S. Naperville Road Wheaton, IL 60187 Phone: (630) 580-5735 E-Mail: [email protected] Fax: (630) 580-5765 Rational Numbers When one Integer is divided by another—a ratio is formed—the resulting fractional number is a Rational number. A Rational number is one that can be formed from a ratio of Integers (i.e. 5/2 = 2.5), making terminating decimals (i.e. 0.45 = 9/20) or repeating decimals (i.e. 0.333… = ⅓) also Rational. The set of Rational numbers, symbolized by a capital, bold-face “ℚ,” is a countable subset of the Real numbers. Irrational Numbers The Irrational numbers are Real numbers that have a non-repeating decimal of infinite length (non-repeating, non-terminating decimals). Classic irrational numbers include √2 and π. Real Numbers A Real number is one that may be an Integer, a decimal, a Rational number, an Irrational number–effectively all numbers that do not contain Imaginary elements (see Imaginary Numbers, below). Examples of Real numbers include (though are not limited to): Integers: …-3, -2, -1, 0, 1, 2, 3… Rationals: o Fractions: 7/2 o Terminating Decimals: 5.2 o Repeating decimals: 1/7 = 0.142857142857… Irrationals: √2 ≈ 1.414213562… and π ≈ 3.14159265358… Real numbers are the most common in mathematical analysis. The set of all Real numbers is symbolized by a capital, bold-face “ℝ.” They are often contrasted with the Imaginary numbers. Imaginary Numbers The Imaginary numbers are numbers that are a multiple of the imaginary unit i, which is defined as the square root of -1. Thus, it follows that the square of the imaginary unit is -1, the cube is -i, i4 = 1, and cycles amongst these values. The Imaginary numbers are not normally used as is, and are typically incorporated into Complex numbers with Real components. Complex Numbers A complex number is composed of two components: a Real component and an Imaginary component. Complex numbers typically take the form a + bi, where “i" is the imaginary unit and “a” and “b” are constants. Complex numbers are defined in the imaginary plane, which is an imaginary axis plotted against the x-axis. The set of all Complex numbers is symbolized by a capital, bold-face “ℂ.” Questions Galore 319 S. Naperville Road Wheaton, IL 60187 Phone: (630) 580-5735 E-Mail: [email protected] Fax: (630) 580-5765 Exercise I: Given the following numbers, check in the chart which descriptors can be applied to them. Answers are in the back of the packet. Whole Integer Rational Irrational Real Imaginary Complex 0.24 4 π 3i 2+3i Properties of Real Numbers All types of numbers have a set of properties that can be affixed to operations performed with them. Here, we’ll look at some of the basic properties associated with the Real numbers. Reflexive Property Given any Real number A: A=A Though this property may seem obvious, it is an important factor of set theory. Transitive Property Given Real numbers A, B, and C: If A = B, and B = C, then A = C Example: (2x + 7) = (3x + 4) and (3x + 4) = 13, then (2x + 7) = 13 Properties of Addition Commutative Property The commutative property of addition states that, for all Real numbers A and B: A+B=B+A Basically, this means that you can add Real numbers in any order and still achieve the same result. Example: 2 + 5.5 + 3 = 2 + 3 + 5.5 = 3 + 2 + 5.5 = 10.5 Associative Property The associative property of addition states that, for all Real numbers A, B, C: (A + B) + C = A + (B + C) = A + B + C No matter how you may group Real numbers, as long as the addition operations are applied to them and they retain their sign, the result is the same. Identity Property The identity property of addition states that, for all Real numbers A: A+0=A Basically, adding nothing (0) to A still equals A. Zero is the additive identity. Questions Galore 319 S. Naperville Road Wheaton, IL 60187 Phone: (630) 580-5735 E-Mail: [email protected] Fax: (630) 580-5765 Additive Inverse The additive inverse property states that for all Real A, there is a Real (-A) so: A + (-A) = 0 Simply, this means that adding a Real number to an opposite version of that same Real number equals zero. Example: 5 + (-5) = 5 – 5 = 0 Properties of Multiplication Commutative Property The commutative property of multiplication states that, for all Real numbers A and B: AxB=BxA Example: 5 x 2 = 2 x 5 = 10 Associative Property The associative property of multiplication states that, for all Real numbers A, B, and C: (A x B) x C = A x (B x C) = A x B x C Grouping number will result in the same product provided all numbers retain their signs. Identity Property The identity property of multiplication states that, for all Real numbers A: Ax1=A Multiplying any Real number by 1 will result in the original number. One is the multiplicative identity. Multiplicative Inverse The multiplicative inverse states that for any non-zero Real number A, there exists a Real number 1/A such that: A x (1/A) = 1 Multiplying any non-zero number by its reciprocal will result in 1. Examples: 3 x (1/3) = 1 and (4/5) x (5/4) = 1 Distributive Property For all Real numbers A, B, and C: A x (B + C) = A x B + A x C Example: Example: 3 x (5 + 7) = 3 x 5 + 3 x 7 = 15 + 21 = 36 12ab + 9a2 = 3a(4b + 3a) IMPORTANT NOTE: the Commutative and Associative laws, as a general rule, do NOT function for subtraction and division. Questions Galore 319 S. Naperville Road Wheaton, IL 60187 Phone: (630) 580-5735 E-Mail: [email protected] Fax: (630) 580-5765 Exercise II: 1. 2. 3. Identify the properties utilized in each of the following equations. 3+7=7+3 4. 8 x (1/8) = 1 4x1=4 5. 2 x (8 x 5) = 5 x (8 x 2) 2(3 + 7) = 2 x 3 + 2 x 7 Answers at back of packet Composition of Numbers Let’s now take a look at the composition of Whole numbers. There are two types of whole number composition: Composite numbers and Prime numbers. Composite Numbers A Composite number is any Whole number that has divisors (or factors) other than 1 and itself. That is to say that there is more than a single pair of numbers that multiply to any Composite number. Example: 72 = 1 x 72 = 2 x 36 = 3 x 24 = 4 x 18 = 6 x 12 = 8 x 9 The number 72 has 12 factors and is therefore composite. Prime Numbers A Prime number is a Whole number that has only two divisors (or factors)—1 and itself. Euclid proved in approximately 300 B.C. that there are infinitely-many Prime numbers. As of August 2011, the largest known Prime has nearly 13 million digits (243112609 - 1). It is greatly beneficial to have at least the first 15 prime numbers memorized: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 Please note that the number 1 is not considered a Prime number, and violates the fundamental theorem of arithmetic which is defined on that principle. The number 2 is the lowest Prime number and is the only Prime number that is even. Exercise III: 1. 2. 3. 4. Identify whether the following numbers are Prime or Composite. 53 5. 91 78 6. 14 161 7. 132 71 8. 59 Answers in back of packet Questions Galore 319 S. Naperville Road Wheaton, IL 60187 Phone: (630) 580-5735 E-Mail: [email protected] Fax: (630) 580-5765 Answers to Exercises Exercise I Answers: 0.24 4 π 3i 2+3i Whole Integer X X Rational X X Irrational Real X X X X Imaginary Complex X X Exercise II Answers: 1. Commutative property of addition 2. Identity property of multiplication 3. Distributive property 4. Multiplicative inverse 5. Associative property of multiplication Exercise III Answers: 1. Prime 2. Composite (1 x 78, 2 x 39, 3 x 26, 6 x 13) 3. Composite (1 x 161, 7 x 23) 4. Prime 5. Composite (1 x 91, 7 x 13) 6. Composite (1 x 14, 2 x 7) 7. Composite (1 x 132, 2 x 66, 3 x 44, 4 x 33, 6 x 22, 11 x 12) 8. Prime Questions Galore 319 S. Naperville Road Wheaton, IL 60187 Phone: (630) 580-5735 E-Mail: [email protected] Fax: (630) 580-5765