Module 5: Numbers and Patterns

Question 1

Numbers

Answer 1

universally understood symbols that are often used to convey mathematical ideas

Question 2

Positional Numeral System

Answer 2

The value of a numeral depends on its position

Question 3

Earliest numerical system

Answer 3

Tally marks

Question 4

Why did early civilizations opt to adapt a positional notation in their numerical systems?

Answer 4

A positional notation makes it easier to write or represent larger numbers.

Question 5

Why were older positional notation systems that lacked a symbol for 0 difficult to use?

Answer 5

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.

Question 6

Where did the digits we commonly use today originate from?

Answer 6

North African Maghreb region of the Arab Empire (Hindu-Arabic numeral system)

Question 7

Characteristics of the Egyptian numeral system (arnd. 3000 BC)

Answer 7

Base 10 hieroglyphics, additive, non-positional

Question 8

Characteristics of the Babylonian numeral system (arnd. 3500 B.C.)

Answer 8

Sumerian numerals, cuneiform, sexagesimal system (Base 60) and decimal subsytem (Base 10), positional, left to right

Question 9

What does acrophinic mean in the context of the early Greek numeral system?

Answer 9

the symbol for the numeral is the 1st letter of its name

Question 10

Characteristics of the early Greek numeral system (arnd 1000 B.C.)

Answer 10

acrophonic, base 10, positional, additive

Question 11

Characteristics of the Greek numeral system developed around 400 B.C.

Answer 11

alphabetic or ionic, contains 27 letters that stood for numbers, positional

Question 12

Characterisitics of the early Roman numeral system (arnd. 900 B.C.)

Answer 12

Base 5, positional, additive and subtractive, left to right

Question 13

Characteristics of the early Mayan numeral system (arnd 2000 B.C)

Answer 13

vigesimal system (base 20), quinary subsystem (base 5), positional, top to bottom

Question 14

Characteristics of the early Chinese numeral system (1st form)

Answer 14

base 10, non positional, left to right

Question 15

Characteristics of the early Chinese numeral system (2nd form)

Answer 15

base 10, positional, left to right, used counting boards

Question 16

Inventor of the Abacus

Answer 16

Akkadians

Question 17

Characteristics of the Baybayin numeral system (arnd. 2000 B.C)

Answer 17

Base 10

Question 18

Who was the Indian mathematician that ‘invented’ 0 around 600 AD?

Answer 18

Brahmagupta

Question 19

Characteristics of the Hindu-Arabic numeral system (arnd 6th to 7th century AD)

Answer 19

base 10, positional, left to right

Question 20

Characteristics of the duodecimal numeral system

Answer 20

base 12 (dozenal), positional, left to right

Question 21

Characteristics of the octal numeral system

Answer 21

base 8, positional, left to right

Question 22

Characteristics of the binary numeral system

Answer 22

base 2, positional, left to right

Question 23

Who promoted Hindu-Arabic numerals in Europe?

Answer 23

Leonardo of Pisa A.K.A Fibonacci

Question 24

What was the title of Fibonacci’s book wherein he promoted Hindu-Arabic numerals in Europe?

Answer 24

Liber Abaci

Question 25

What is the Arabic term for 0 according to Fibonacci's Liber Abaci

Answer 25

zephirum

Question 26

What are some factors of banned numbers?

Answer 26

political and religious controversies, symbolic representations, ideological threats and differences, etc.

Question 27

The set of natural numbers is denoted by what symbol

Answer 27

N

Question 28

The set of natural numbers, together with the number zero, forms the set of whole numbers which is denoted by what symbol

Answer 28

W

Question 29

The set of natural numbers N, is also referred to as the set of __________, which is denoted by the symbol ____

Answer 29

positive integers; 𝐙+

Question 30

The set of rational numbers is denoted by what symbol

Answer 30

Q

Question 31

The set of irrational numbers is denoted by what symbol

Answer 31

𝐐′

Question 32

The set of real numbers which is the union of the set of rational and irrational numbers is denoted by what symbol

Answer 32

R

Question 33

What are the four axioms that satisfy the real number system

Answer 33

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

Question 34

Closure axiom

Answer 34

For any pair of real numbers, a and b, their sum (a+b) and product (ab) are also real numbers

Question 35

Commutativity axiom

Answer 35

For any pair of real numbers, their sum and product give the same result even if the elements combined are swapped

Question 36

Associativity Axiom

Answer 36

a + ( b + c ) = ( a + b ) + c and a ⋅ ( b ⋅ c ) = ( a ⋅ b ) ⋅ c

Question 37

Distributivity of Multiplication over Addition axiom

Answer 37

a ⋅ ( b + c ) = ( a ⋅ b ) + (a ⋅ c) and ( b + c ) ⋅ a = ( b ⋅ a ) + (c ⋅ a)

Question 38

Existence of the Identity Elements for Addition and Multiplication axiom

Answer 38

a+0 = 0+a = a and a(1) = 1(a) = a

Question 39

Existence of Inverse Elements under Addition and Multiplication axiom

Answer 39

a + (-a) = (-a + a) = 0 and a(a^-1) = a^-1(a) = a, provided that a is not equal to 0

Question 40

True or False: PEMDAS is an axiom, therefore operations must be done left to right.

Answer 40

FALSE

Question 41

What is an imaginary number?

Answer 41

a real number multiple of i, where i = √−1

Question 42

What is a complex numebr?

Answer 42

a combination of a real number and an imaginary number of the form z= a +bi where a,b are real numbers and i = √−1

Question 43

What is a prime number?

Answer 43

a natural number whose factors are one and itself

Question 44

What is a composite number?

Answer 44

a natural number that has other factors in addition to one and itself

Question 45

TRUE OR FALSE: One (1) is a prime number.

Answer 45

FALSE; it it neither prime nor composite

Question 46

How is a modular set of n elements (subset of whole numbers) represented?

Answer 46

𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1}

Question 47

What is the formula for the additive inverse in modular sets?

Answer 47

a + b ≡ 0 (mod n)

Question 48

What is the formula for the multiplicative inverse in modular sets?

Answer 48

ab ≡ 1(mod n)

Question 49

What is cryptography?

Answer 49

The study of encoding and decoding messages

Question 50

What is the numebr pattern shown: 1,3,5,7,9,11,...

Answer 50

positive odd numbers

Question 51

What is the number pattern shown: 2,7,12,17,22,...

Answer 51

arithmetic progression

Question 52

What number pattern is shown: 1, 10, 100, 1000, 10000, ...

Answer 52

powers of 10

Question 53

What number pattern is shown: 2, 6, 18, 54, 162

Answer 53

geometric progression

Question 54

What number pattern is shown: 1,4,9,16,25

Answer 54

perfect squares

Question 55

What number pattern is shown: 1,8,27,64,125,...

Answer 55

perfect cubes

Question 56

What number pattern is shown: 1,3,6,19,15,21

Answer 56

triangular numbers

Question 57

What are patterns used for?

Answer 57

retention and prediction

Question 58

Who invented Pascal's Triangle?

Answer 58

Blaise Pascal

Question 59

Fibonacci series

Answer 59

1,1,2,3,5,8,13,21,32,55,89,144,233,...

Question 60

What is the golden ratio?

Answer 60

a+b/a = a/b = 1.618

Question 61

an intristic property of a mathmatical object which causes it to remain invariant under certain classes of transformations

Answer 61

Symmetry

Question 62

What are the different kinds of symmetry?

Answer 62

reflectional, rotational, translational, glide

Question 63

What is the angle of rotation

Answer 63

the smallest angle that would preserve the figure when the figure is rotated

Question 64

What is the order of rotation?

Answer 64

the number of positions that preserves the figure when it is repeatedly rotated until it reaches one revolution (360/angle of rotation)

Question 65

What is a frieze pattern?

Answer 65

a strip with a symmetric pattern, repetitive in one direction

Question 66

How many possible Frieze patterns can be created out of a design that is of one color

Answer 66

7

Question 67

What frieze pettern displays only translational symmetry?

Answer 67

HOP

Question 68

What frieze pettern displays only translational and glide symmetries

Answer 68

STEP

Question 69

What frieze pettern displays only translational and vertical reflection symmetries?

Answer 69

SIDLE

Question 70

What frieze pettern displays only translational and rotational by a half turn symmetries?

Answer 70

SPINNING HOP

Question 71

What frieze pettern displays only translational, glide, and rotational by a half turn symmetries?

Answer 71

SPINNING SLIDE

Question 72

What frieze pettern displays only translational and horizontal reflection symmetries?

Answer 72

JUMP

Question 73

What frieze pettern displays transational, horizontal and vertical reflections, and rotational symmetries?

Answer 73

SPINNING JUMP