MTE 3101: KNOWING NUMBERS

Tutorial 1A

What is meant by the statement “The Mayan system of numeration was place valued”?

What are the features of our modern Hindu-Arabic system of numeration that make it superior to other systems and led to its adoption world wide.

Why is it important to learn about early number systems? Why is its development an important part of numerations systems?

What is meant by the statement “The Mayan system of numeration was place

valued”?

The Mayan number system was developed by the ancient Maya civilization of

Central America. Similar to the number system we use today, the Mayan system

operated with place values. To achieve this place value system they developed the

idea of a zero placeholder. The Maya seem to be the first people who used a place

value system and a symbol for zero. Beyond these similarities there are some

significant differences between the Mayan number system and our modern system.

The Mayan system is in base 20 (vigesimal) rather than base 10 (decimal). This

system also uses a different digit representation. The Mayan numbers are based on

three symbols:

In the Mayan system, there are two kinds of elements in each place: dots and bars. A full place looks like this and can be expressed as 19 in our system:

Adding one dot to the place shown above would make it worth 20, which is expressed as a dot with a below it.

Each dot in the new place is worth 20, and since five dots equal one bar, each bar is worth 100 in that place.

The value of a dot in the third place is 400 (20 x 20), and a bar in that place is worth 2,000.

In each successive place, one dot is worth 20 times as much as a dot in the previous place.

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Mayan numbers have any number of places, and numbers with as many as six or seven places have been identified in hieroglyphs discovered at the sites of Mayan cities.

Because the dots are worth 1 and the bars are worth 5, there should never be more than four dots in a single place. When adding Mayan numbers, every group of five dots becomes one bar.

Likewise, the maximum value of one place is 19, so that four bars, or 20, is too large a number to fit in one place. The four bars are carried and equal one dot in the next highest place.

Let's add, in Mayan, the numbers 37 and 29:

 First draw a box around each of the places of the numbers so you (and I) won't get confused.

 Next, put all of the elements from both numbers into a single set of places (boxes), and to the right of this new number draw a set of empty boxes corresponding to each place of the number to the left:

 You are now ready to start carrying. Begin with the place that has the lowest value, just as you do with Arabic numbers. Start at the bottom place, where each dot is worth 1. There are six dots. Five dots make one bar, so draw a bar through five of the dots, leaving you with one dot which is under the four-dot limit. Put this dot into the bottom place of the empty set of boxes you just drew:

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 Now look at the bars in the bottom place. There are five, and the maximum number the place can hold is three. Four bars are equal to one dot in the next highest place, so draw a circle around four of the bars and an arrow up to the dots' section of the higher place. At the end of that arrow, draw a new dot. That dot represents 20 just the same as the other dots in that place.

Not counting the circled bars in the bottom place, there is one bar left. One bar is under the three-bar limit; put it under the dot in the set of empty places to the right.

 Now there are only three dots in the next highest place, so draw them in the corresponding empty box.

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What are the features of our modern Hindu-Arabic system of numeration that

make it superior to other systems and led to its adoption world wide.

A- compounding of powers of ten

B- ability to perform addition

C- Ability to repeat numbers

D- Invention of zero

First, Hindu-Arabic numbers use placement within a number to indicate a higher

value. For example, in the number 256, the "5" indicates five tens and the "2"

indicates two hundred units. The same numerals in a different order represent a

totally different number, as for example, 562, which represents five hundreds, six

tens and two singles. Roman numbers make little use of the order in which numerals

are presented.

Second, Hindu-Arabic numerals include a symbol for zero, while the Roman system

completely lacks that. The zero is used as a place holder in such numbers as 1028,

indicating one thousand, no hundreds, two tens and eight singles. This place holder

allows aligning of several numbers and makes addition and subtraction easier, and

multiplication and division so much easier that calculations can be done with Hindu

numerals that are simply impossible with Roman numerals.

The fundamental numerals and diverse attributes of Hindu-Arabic numeration

system include the digits. There are 10 symbols and they can be used by mish

mashing them to represent all possible numbers. They are grouping by tens (decimal

system). Grouping into sets of 10 is a basic principle is the basic of this system.

Base of the system is the number of objects grouped together therefore this system

is called as a base ten system.

Place value (hence positional) is each of the various places in the numeral has its

own value. The place-value of the numerals is determined by the place-value

attributes of the system. A chip abacus is used to symbolise numbers written in place

value.

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Additive and multiplicative-the value of a Hindu-Arabic numeral is found by

multiplying each place value by its corresponding digit and then adding all the out-

come products. The numeral’s expanded form or expanded notation is expressing a

numeral as the sum of its digits times their respective place value. This numeration

system requires fewer symbols to represent numbers than did initial system and it is

also far superior when performing computations.

There are procedures to be followed when naming the numerals in English. Number

0 till 12, all have exclusive names. For number 13 till 19, “teens” are added to their

earlier names with the ones place named first. The number 20 till 99 are

combinations of earlier names but reversed from the teens in that the tens place is

named first. The numbers 100 till 999 are combinations of hundreds and previous

names.

Why is it important to learn about early number systems? Why is its

development an important part of numerations systems?

A numeral system (or system of numeration) is a writing system for expressing

numbers, that is a mathematical notation for representing numbers of a given set,

using graphemes or symbols in a consistent manner. It can be seen as the context

that allows the numerals "11" to be interpreted as the binary symbol for three, the

decimal symbol for eleven, or as other numbers in different bases.

Ideally, a numeral system will:

Represent a useful set of numbers (e.g. all whole numbers, integers, or rational numbers)

Give every number represented a unique representation (or at least a standard representation)

Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual decimal representation of whole numbers gives every whole

number a unique representation as a finite sequence of digits. However, when

decimal representation is used for the rational or real numbers, the representation

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may not be unique: many rational numbers have two numerals, a standard one that

terminates, such as 2.31, and another that recurs, such as 2.309999999... .

Numerals which terminate have no non-zero digits after a given position. For

example, numerals like 2.31 and 2.310 are taken to be the same, except for some

scientific contexts where greater precision is implied by the trailing zero.

How important are numbers, have you ever thought about it, but if you just focus on

past events or sequences you will realise that certain numbers are always talking to

you.

US has considered number 13 as important, the flag has 13 stripes, when it became

independent country, it had 13 states originally and even the annoucement letter

through which independence was announced was signed by 13 persons, its motto

E.Sluribusllnum has 13 characters and when George Washington raised rebulion

standard he was saluted with 13 guns.

Roman philosopher Pythagorus was a great mathematician and believed that " All

the forms, Art and thoughts are ruled by numbers". Romans also believe that the

mysteries of the world are hidden in numbers.

The influence of numbers on Indian Astrology is well known.

A numerologist closely works with the name of the person, date of birth to arrive at a

persons basic number before making any predictions.

If one arrives at the basic number, from the date of birth and arriving at a number

from a persons name one can lead a successful and happy life, thru numerology one

can know about a perticular number, color, day, month and year which will be

auspicious.

It is quite intriguinng what wonders numbers can do, it is possible to know about the

things hidden in a persons mind, and are capable of bringing out the innermost

feelings of a persons heart, also tell us about lost things and from where can it be

found, but for that one needs to know about hidden mystries of numbers and to

concentrate on the issue in point.

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In ones life we come accross lots of people and somtimes there are few whom you

would not like to meet again and on the other hand some with whom you have

developed a life long affinity, its because of the unity of letters of that name attract

you as your name too would have the similar waves.

Numbers can do wonders.

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Tutorial 1B

Some students say that studying different number bases is a waste of time. Explain the value studying base four, or example , in helping to understand our decimal system.

Quaternary is the base-4 numeral system. It uses the digits 0, 1, 2 and 3 to

represent any real number.

It shares with all fixed-radix numeral systems many properties, such as the ability to

represent any real number with a canonical representation (almost unique) and the

characteristics of the representations of rational numbers and irrational numbers.

See decimal and binary for a discussion of these properties.

In mathematical numeral systems, the base or radix is usually the number of unique

digits, including zero, that a positional numeral system uses to represent numbers.

For example, for the decimal system (the most common system in use today) the

radix is 10, because it uses the 10 digits from 0 through 9.

Examples of numeral systems:

The decimal system, the most used system of numbers in the world, is used

in arithmetic. Its ten digits are "0-9".

The binary numeral system, widely used in computing, is base two. The two

digits are "0" and "1".

The octal system, which is base 8, is also often used in computing. The eight

digits are "0-7".

Also in widespread use in computing is the hexadecimal system. It is base 16,

and the 16 digits are "0-9" followed by "A-F".

In base four, each digit in a number represents the number of copies of that power of

four. That is, the first digit tells you how many ones you have; the second tells you

how many fours you have; the third tells you how many sixteens (four-times-fours)

you have; the fourth tells you how many sixty-fours (four-times-four-times-fours) you

have; and so on. The methodology for conversion between decimal and base-four

numbers is just like that for converting between decimals and binaries, except that

binary digits can be only "0" or "1", while the digits for base-four numbers can be "0",

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"1", "2", or "3". (As you might expect, there is no single solitary digit in base-four

math that represents the quantity "four".)

Convert 35710 to the corresponding base-four number.

I will do the same division that I did before, keeping track of the remainders.

(You may want to use scratch paper for this.)

Then 35710 converts to 112114.

Convert 80710 to the corresponding base-four number.

Note: Once I got "3" on top, I had to stop, because four cannot divide into 3.

Reading the numbers off the division, I get that 80710 converts to 302134.

Convert 302134 to the corresponding decimal number.

I will list out the digits, and then number them from the RIGHT, starting at

zero:

digits:  3  0   2  1  3

numbering:  4  3   2  1  0

Each digit stands for the number of copies I need for that power of four:

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3×44 + 0×43 + 2×42 + 1×41 + 3×40

    = 3×256 + 0×64 + 2×16 + 1×4 + 3×1

    = 768 + 32 + 4 + 3

    = 807

As expected, 302134 converts to 80710.

What is a real-world use for number base 2 and 16?

The binary numeral system, or base-2 number system represents numeric values

using two symbols, 0 and 1. More specifically, the usual base-2 system is a

positional notation with a radix of 2. Owing to its straightforward implementation in

digital electronic circuitry using logic gates, the binary system is used internally by all

modern computers.

In mathematics and computer science, hexadecimal (also base-16, hexa, or hex) is a

numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most

often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or a

through f) to represent values ten to fifteen.

Its primary use is as a human-friendly representation of binary coded values, so it is

often used in digital electronics and computer engineering. Since each hexadecimal

digit represents four binary digits (bits) — also called a nibble — it is a compact and

easily translated shorthand to express values in base two.

In digital computing, hexadecimal is primarily used to represent bytes. Attempts to

represent the 256 possible byte values by other means have led to problems.

Directly representing each possible byte value with a single character representation

runs into unprintable control characters in the ASCII character set. Even if a standard

set of printable characters were devised for every byte value, neither users nor input

hardware are equipped to handle 256 unique characters. Most hex editing software

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displays each byte as a single character, but unprintable characters are usually

substituted with a period or blank.

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Tutorial 2A

State in your own words what the number systems is. Do you think it is important? Why?

A number system is the set of symbols used to express quantities as the

basis for counting, determining order, comparing amounts, performing calculations,

and representing value. It is the set of characters and mathematical rules that are

used to represent a number. Examples include the Arabic, Babylonian, Chinese,

Egyptian, Greek, Mayan, and Roman number systems. The ISBN and Dewey

Decimal System are examples of number systems used in libraries. Social Security

even has a number system.

How important are numbers, have you ever thought about it, but if you just

focus on past events or sequences you will realise that certain numbers are always

talking to you.

US has considered number 13 as important, the flag has 13 stripes, when it

became independent country, it had 13 states originally and even the annoucement

letter through which independence was announced was signed by 13 persons, its

motto E.Sluribusllnum has 13 characters and when George Washington raised

rebulion standard he was saluted with 13 guns.

Roman philosopher Pythagorus was a great mathematician and believed that

" All the forms, Art and thoughts are ruled by numbers". Romans also believe that

the mysteries of the world are hidden in numbers.

The influence of numbers on Indian Astrology is well known.

A numerologist closely works with the name of the person, date of birth to

arrive at a persons basic number before making any predictions.

If one arrives at the basic number, from the date of birth and arriving at a

number from a persons name one can lead a successful and happy life, thru

numerology one can know about a perticular number, color, day, month and year

which will be auspicious.

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It is quite intriguinng what wonders numbers can do, it is possible to know

about the things hidden in a persons mind, and are capable of bringing out the

innermost feelings of a persons heart, also tell us about lost things and from where

can it be found, but for that one needs to know about hidden mystries of numbers

and to concentrate on the issue in point.

In ones life we come accross lots of people and somtimes there are few

whom you would not like to meet again and on the other hand some with whom you

have developed a life long affinity, its because of the unity of letters of that name

attract you as your name too would have the similar waves.

Numbers can do wonders.

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Tutorial 3

1. What is a prime number?

In mathematics, a prime number (or a prime) is a natural number which has exactly

two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers

are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,

83, 89, 97, 101.

An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC.[2]

The number 1 is by definition not a prime number. The fundamental theorem of

arithmetic establishes the central role of primes in number theory: any nonzero

natural number n can be factored into primes, written as a product of primes or

powers of primes (including the empty product of factors for 1). Moreover, this

factorization is unique except for a possible reordering of the factors.

The property of being prime is called primality. Verifying the primality of a given

number n can be done by trial division, that is to say dividing n by all smaller

numbers m, thereby checking whether n is a multiple of m, and therefore not prime

but a composite. For big primes, increasingly sophisticated algorithms which are

faster than that technique have been devised.

There is no known formula yielding all primes and no composites. However, the

distribution of primes, that is to say, the statistical behaviour of primes in the large

can be modeled. The first result in that direction is the prime number theorem which

says that the probability that a given, randomly chosen number n is prime is

inversely proportional to its number of digits, or the logarithm of n. This statement

has been proved at the end of the 19th century. The unproven Riemann hypothesis

dating from 1859 implies a refined statement concerning the distribution of primes.

Despite being intensely studied, many fundamental questions around prime numbers

remain open. For example, Goldbach's conjecture which asserts that any even

natural number bigger than two is the sum of two primes, or the twin prime

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conjecture which says that there are infinitely many twin primes (pairs of primes

whose difference is two), have been unresolved for more than a century,

notwithstanding the simplicity of their statements.

Prime numbers give rise to various generalizations in other mathematical domains,

mainly algebra, notably the notion of prime ideals.

Primes are applied in several routines in information technology, such as public-key

cryptography, which makes use of the difficulty of factoring large numbers into their

prime factors. Searching for big primes, often using distributed computing, has

stimulated studying special types of primes, chiefly Mersenne primes whose primality

is comparably quick to decide. As of 2009, the largest known prime has about 13

million decimal digits.

2. Describe a process for finding a prime factorization

In number theory, the prime factors of a positive integer are the prime numbers that

divide into that integer exactly, without leaving a remainder. The process of finding

these numbers is called integer factorization, or prime factorization.

For a prime factor p of n, the multiplicity of p is the largest exponent a for which pa

divides n. The prime factorization of a positive integer is a list of the integer's prime

factors, together with their multiplicity. The fundamental theorem of arithmetic says

that every positive integer has a unique prime factorization.

To shorten prime factorization, numbers are often expressed in powers, so

This may not be much of a shortening, but it is in

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For a positive integer n, the number of prime factors of n and the sum of the prime

factors of n (not counting multiplicity) are examples of arithmetic functions of n that

are additive but not completely additive.

Determining the prime factors of a number is an example of a problem frequently

used to ensure cryptographic security in encryption systems; this problem is believed

to require super-polynomial time in the number of digits- it is relatively easy to

construct a problem that would take longer than the known age of the Universe to

calculate on current computers using current algorithms.

Two positive integers are coprime if and only if they have no prime factors in

common. The integer 1 is coprime to every positive integer, including itself. This is

because it has no prime factors; it is the empty product. It also follows from defining

a and b as coprime if gcd(a,b)=1, so that gcd(1,b)=1 for any b>=1. Euclid's algorithm

can be used to determine whether two integers are coprime without knowing their

prime factors; the algorithm runs in a time that is polynomial in the number of digits

involved.

The function ω(n) represents the number of distinct prime factors of n, while Ω(n)

represents the total number of prime factors. If , then .

For example, 24 = 23.31, so: ω(24) = 2 and Ω(24) = 3 + 1 = 4.

The prime factors of 6 are 2 and 3 (6 = 2 × 3). Both have multiplicity 1.

5 has only one prime factor: itself (5 is prime). It has multiplicity 1.

100 has two prime factors: 2 and 5 (100 = 22 × 52). Both have multiplicity 2.

2, 4, 8, 16, etc. each have only one prime factor: 2. (2 is prime, 4 = 22, 8 = 23,

etc.)

1 has no prime factors. (1 is a unit)

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What does GCF mean and what is the procedure for finding the GCF of a set of

numbers?

In mathematics, the greatest common divisor (gcd), also known as the greatest

common factor (gcf), or highest common factor (hcf), of two or more non-zero

integers, is the largest positive integer that divides the numbers without a remainder.

The greatest common factor, or GCF, is the greatest factor that divides two numbers.

To find the GCF of two numbers:

1. List the prime factors of each number.

2. Multiply those factors both numbers have in common. If there are no common

prime factors, the GCF is 1.

What does LCM mean and what is the procedure for finding the LCM of a set of numbers?

In arithmetic and number theory, the least common multiple or lowest common

multiple (lcm) or smallest common multiple of two integers a and b is the smallest

positive integer that is a multiple both of a and of b. Since it is a multiple, it can be

divided by a and b without a remainder. If either a or b is 0, so that there is no such

positive integer, then lcm(a, b) is defined to be zero.

The definition is sometimes generalized for more than two integers: The lowest

common multiple of integers a1, ..., an is the smallest positive integer that is a

multiple of a1, ..., an.

The least common multiple (LCM) of two numbers is the smallest number (not zero)

that is a multiple of both.

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Compare a set of number with their GCF and LCM. Make as many statements(a) about the set of numbers compared to their GCF and LCM.(b) about their GCF as compared to their LCM.

Set of numbers:

6, 12, 24,

GCF:

Prime factors of 6 : 2 x 3

Prime factors of 12 : 2 x 2 x 3

Prime factors of 24 : 2 x 2 x 2 x 3

So, the GCF is, 2 x 3 where the common factor is 2 and 3.

GCF = 6

LCM:

Multiples of 6

0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

Multiples of 12

0, 12, 24, 36, 48, 60, 72

Multiples of 24

24, 48, 72

So, the LCM is 24

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(a) about the set of numbers compared to their GCF and LCM.

All prime factors of the numbers has 2 and 3

6 has one 2, 12 has double 2 and 24 has triple two.

Every numbers on the set has only one three

There are no other prime numbers except 2 and 3 for the prime factors of

the numbers.

The multiples of 24 contain in multiple of 6 and 12.

The least common factor is 24 because it is the smallest same factor on

the numbers.

Multiples of 6 and 12 has their own LCM which is 12.

Multiples of 12 contain in multiples of 6.

(b) about their GCF as compared to their LCM.

GCF is smaller (6) than LCM (24)

GCF is the factor of LCM

24 can be divided by 6 which the result will not have remainder.

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