Module 5: Numbers and Patterns Flashcards
Numbers
universally understood symbols that are often used to convey mathematical ideas
Positional Numeral System
The value of a numeral depends on its position
Earliest numerical system
Tally marks
Why did early civilizations opt to adapt a positional notation in their numerical systems?
A positional notation makes it easier to write or represent larger numbers.
Why were older positional notation systems that lacked a symbol for 0 difficult to use?
Lacking a 0 symbol makes it difficult to distinguish certain numbers from each other (example 63 and 603) and this leads to inconsisten and unreliable notation.
Where did the digits we commonly use today originate from?
North African Maghreb region of the Arab Empire (Hindu-Arabic numeral system)
Characteristics of the Egyptian numeral system (arnd. 3000 BC)
Base 10 hieroglyphics, additive, non-positional
Characteristics of the Babylonian numeral system (arnd. 3500 B.C.)
Sumerian numerals, cuneiform, sexagesimal system (Base 60) and decimal subsytem (Base 10), positional, left to right
What does acrophinic mean in the context of the early Greek numeral system?
the symbol for the numeral is the 1st letter of its name
Characteristics of the early Greek numeral system (arnd 1000 B.C.)
acrophonic, base 10, positional, additive
Characteristics of the Greek numeral system developed around 400 B.C.
alphabetic or ionic, contains 27 letters that stood for numbers, positional
Characterisitics of the early Roman numeral system (arnd. 900 B.C.)
Base 5, positional, additive and subtractive, left to right
Characteristics of the early Mayan numeral system (arnd 2000 B.C)
vigesimal system (base 20), quinary subsystem (base 5), positional, top to bottom
Characteristics of the early Chinese numeral system (1st form)
base 10, non positional, left to right
Characteristics of the early Chinese numeral system (2nd form)
base 10, positional, left to right, used counting boards
Inventor of the Abacus
Akkadians
Characteristics of the Baybayin numeral system (arnd. 2000 B.C)
Base 10
Who was the Indian mathematician that βinventedβ 0 around 600 AD?
Brahmagupta
Characteristics of the Hindu-Arabic numeral system (arnd 6th to 7th century AD)
base 10, positional, left to right
Characteristics of the duodecimal numeral system
base 12 (dozenal), positional, left to right
Characteristics of the octal numeral system
base 8, positional, left to right
Characteristics of the binary numeral system
base 2, positional, left to right
Who promoted Hindu-Arabic numerals in Europe?
Leonardo of Pisa A.K.A Fibonacci
What was the title of Fibonacciβs book wherein he promoted Hindu-Arabic numerals in Europe?
Liber Abaci
What is the Arabic term for 0 according to Fibonacciβs Liber Abaci
zephirum
What are some factors of banned numbers?
political and religious controversies, symbolic representations, ideological threats and differences, etc.
The set of natural numbers is denoted by what symbol
N
The set of natural numbers, together with the number zero, forms the set of whole numbers which is denoted by what symbol
W
The set of natural numbers N, is also referred to as the set of __________, which is denoted by the symbol ____
positive integers; π+
The set of rational numbers is denoted by what symbol
Q
The set of irrational numbers is denoted by what symbol
πβ²
The set of real numbers which is the union of the set of rational and irrational numbers is denoted by what symbol
R
What are the four axioms that satisfy the real number system
closure, commutativity, associativity, distributivity of multiplication over addition, existence of the identity elements for addition and multiplication, and existence of inverse elements under addition and multiplication
Closure axiom
For any pair of real numbers, a and b, their sum (a+b) and product (ab) are also real numbers
Commutativity axiom
For any pair of real numbers, their sum and product give the same result even if the elements combined are swapped
Associativity Axiom
a + ( b + c ) = ( a + b ) + c and a β ( b β c ) = ( a β b ) β c
Distributivity of Multiplication over Addition axiom
a β ( b + c ) = ( a β b ) + (a β c) and ( b + c ) β a = ( b β a ) + (c β a)
Existence of the Identity Elements for Addition and Multiplication axiom
a+0 = 0+a = a and a(1) = 1(a) = a
Existence of Inverse Elements under Addition and Multiplication axiom
a + (-a) = (-a + a) = 0 and a(a^-1) = a^-1(a) = a, provided that a is not equal to 0
True or False: PEMDAS is an axiom, therefore operations must be done left to right.
FALSE
What is an imaginary number?
a real number multiple of i, where i = ββ1
What is a complex numebr?
a combination of a real number and an imaginary number of the form z= a +bi where a,b are real numbers and i = ββ1
What is a prime number?
a natural number whose factors are one and itself
What is a composite number?
a natural number that has other factors in addition to one and itself
TRUE OR FALSE: One (1) is a prime number.
FALSE; it it neither prime nor composite
How is a modular set of n elements (subset of whole numbers) represented?
ππ = {0,1,2,3,β¦ , π β 1}
What is the formula for the additive inverse in modular sets?
a + b β‘ 0 (mod n)
What is the formula for the multiplicative inverse in modular sets?
ab β‘ 1(mod n)
What is cryptography?
The study of encoding and decoding messages
What is the numebr pattern shown: 1,3,5,7,9,11,β¦
positive odd numbers
What is the number pattern shown: 2,7,12,17,22,β¦
arithmetic progression
What number pattern is shown: 1, 10, 100, 1000, 10000, β¦
powers of 10
What number pattern is shown: 2, 6, 18, 54, 162
geometric progression
What number pattern is shown: 1,4,9,16,25
perfect squares
What number pattern is shown: 1,8,27,64,125,β¦
perfect cubes
What number pattern is shown: 1,3,6,19,15,21
triangular numbers
What are patterns used for?
retention and prediction
Who invented Pascalβs Triangle?
Blaise Pascal
Fibonacci series
1,1,2,3,5,8,13,21,32,55,89,144,233,β¦
What is the golden ratio?
a+b/a = a/b = 1.618
an intristic property of a mathmatical object which causes it to remain invariant under certain classes of transformations
Symmetry
What are the different kinds of symmetry?
reflectional, rotational, translational, glide
What is the angle of rotation
the smallest angle that would preserve the figure when the figure is rotated
What is the order of rotation?
the number of positions that preserves the figure when it is repeatedly rotated until it reaches one revolution (360/angle of rotation)
What is a frieze pattern?
a strip with a symmetric pattern, repetitive in one direction
How many possible Frieze patterns can be created out of a design that is of one color
7
What frieze pettern displays only translational symmetry?
HOP
What frieze pettern displays only translational and glide symmetries
STEP
What frieze pettern displays only translational and vertical reflection symmetries?
SIDLE
What frieze pettern displays only translational and rotational by a half turn symmetries?
SPINNING HOP
What frieze pettern displays only translational, glide, and rotational by a half turn symmetries?
SPINNING SLIDE
What frieze pettern displays only translational and horizontal reflection symmetries?
JUMP
What frieze pettern displays transational, horizontal and vertical reflections, and rotational symmetries?
SPINNING JUMP
