Download JH WEEKLIES ISSUE #13 2011

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.
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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