1.2 subsets of integers dfs

transcript

Chapter I

Algebra ReviewMATH-020

Dr. Farhana Shaheen1

Chapter I

Algebra Review1. The Real Number System2. Sets3. Inequality & Interval Notation4. Integer Exponents5. Ratios, Proportions, and Percentages6. Simple and Compound Interest

Chapter I

Algebra Review1. The Real Number System 1.1 Real Numbers 1.2 Subsets of Integers

1. 1 Real Number System

REVIEW: REAL NUMBER SYSTEM

Integers• Integers are just like Whole Numbers; however, they contain

negative numbers as well. • Negative Numbers are numbers smaller than 0. • Just like Whole Numbers, Integers do not contain Fractions.• Examples: -8, -5, 0, 4, 17, 23

INTEGERS • A Number Line is used to arrange all numbers along a line.

The points on the right are greater than the points on the left. The numbers on the Number Line are infinite, meaning they never end and keep increasing.

• Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }• Positive Integers = { 1, 2, 3, 4, 5, ... } = Natural Numbers• Negative Integers = { ..., -5, -4, -3, -2, -1 }• Non-Negative Integers = { 0, 1, 2, 3, 4, 5, ... } = Whole Numbers

SET OF POSITIVE AND NEGATIVE INTEGERS

1.2 SUBSETS OF INTEGERS

Special Subsets of Integers• Set of Integers Z = {….-3,-2,-1, 0, 1, 2,..} a subset of Real Numbers. It has many specific subsets like: Even Numbers: 0, 2, 4, 6, 8, … Odd Numbers: 1, 3, 5, 7, 9, … Prime Numbers; 1, 3, 5, 7, 11, 13, … Composite Numbers: 4, 15, 16, 28, …… Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, ……..

Even and Odd Integers

Even numbers• Even numbers are integers that can be completely divided

by 2. • Even numbers can be illustrated as {…-10,-8,-6, -4, -2, 0, 2, 4,

6, 8, 10…}

• Note: Zero is considered an even number.

Odd numbers• Odd numbers are integers that cannot be totally divided by 2.

• -Odd numbers can be illustrated as {…-9,-7,-5, -3, -1, 1, 3, 5, 7,

9…}• The collection of all even and odd numbers form the set of

integers.

Even and Odd integers

How to determine whether a number is Even or Odd

• To determine whether a number is even or odd, notice the number at the ones place. Like to check if the numbers 12, 15, 36, 43, 69, 88, 101, 204 are even or odd, check the ones digit.

• An even number ends in 0, 2, 4, 6, or 8.• An odd number ends in 1, 3, 5, 7, or 9

Even Odd 8 3

322 123

8234 1765

23432 9324917

Rules of Adding/ Subtracting/ Multiplying Integers

Adding Evens and Oddseven + even = even 2 + 4 = 6even + odd = odd 4 + 1 = 5odd + odd = even 3 + 3 = 6 Subtracting Evens and Oddseven - even = even 8 – 6 = 2even - odd = odd 10 – 9 = 1odd - odd = even 5 – 3 = 2 Multiplying Evens and Oddseven x even = even 4 x 2 = 8even x odd = even 5 x 2 = 10odd x odd = odd 5 x 5 = 25

Examples:

Primes and Composite Numbers

Prime Number • A Prime Number is a positive integer which is only divisible by

1 and the number itself. For example, 7 is a prime number because it is only divisible by 1 and 7.

• The number 6 is not a prime as it can be divided by 2 and 3.

• Note: A prime number is a whole number greater than 1.• Some examples of Prime Numbers are 2, 3, 5, 7, 11, 13, 17, 19,

23,...

Properties of Primes• Number 1 is neither a prime nor a composite number.• Number 2 is the ONLY Even prime number and the smallest

prime too.• All other prime numbers are odd.• All prime numbers are positive numbers.• Prime numbers are infinite.

How to find Prime Numbers

List of Prime Numbers... 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, 103, 107, 109, 113, 127, 131, ...

• We can use Sieve of Eratosthenes to find Prime Numbers. • To find all primes numbers less than 100, write all numbers

starting from 2.• Start with 2, and cancel all multiples of 2.• Then start with 3 and cancel all multiples of 3.• Similarly go for 5, 7, 11. The remaining numbers are all

Primes.23

Sieve of Eratosthenes

Sieve of Eratosthenes• The sieve of Eratosthenes is one of the most efficient ways to

find all of the smaller primes (below 10 million or so). It is named after Eratosthenes of Cyrene, an ancient Greek mathematician; although none of his works have survived, the sieve was described and attributed to Eratosthenes in the Introduction to Arithmetic by Nicomachus.

Prime Number

Composite Number• A Composite Number can be properly divided by numbers

other than 1 or itself.• For example, 6 is a composite number because it can be

divided by 1, 2, 3, and 6. It can be divided by more numbers (2 and 3) than just by 1 or itself.

•Note: The numbers 0 and 1 are neither prime nor composite.

Composite Numbers• A number which has divisors other than 1 and itself are called Composite

Numbers.• Example: 15= 1x3x5

• A composite number is defined as any number, greater than 1, that is not prime.• Every composite number can be written as a product of primes. For example: 54= 2.3.3.3 Note: Except for 0 and 1, a number is either a prime number or a composite number.

Example• Determine whether each number is a prime or Composite

numbers:• 0, 2, 4, 11, 37,78,79, 90, 1.

• Prime numbers:

• Composite numbers:

Example

Exercise

PRIMES: ALLAH’s FAVOURITE NUMBERS

Where do we find these numbers in our daily life?• In our body?• In nature?• In our daily routine?• In the Islamic rituals that we follow?• Or…In Cryptography??? • (Greek kryptos hidden, graphein to write)

Let us see where we find Prime numbers in human body…

Are Prime numbers important in our life?

Look at your face:• One head, but two parts of brain• One nose but two nostrils• 2 eyes, 2 ears, 2 cheeks• One mouth but 2 lips

Count your hands, feet, fingers and toes

2 hands with 5 fingers each2 feet with5 toes each

• You have ... • 2 hands each of which has • 5 fingers, each of which has • 3 parts separated by • 2 knuckles • Believe it or not, there are 19 bones in each of your hands.• Is this just a coincidence???

Look at your own hand:

Number of bones in leg• 30 in one leg. (excluding the hip)

1- femur (thigh bone)1- patella (knee cap)1- tibia (shin)1- fibula7- tarsals (ankle bones)5- metatarsals (foot bones)14- bones of the toes

• (all except the great toe have 3 bones)

Number of Ribs in Human Body• 12 pairs of ribs in human body:

7 are directly connected to the Sternum,(center part)3 are connected

through cartilage2 are floating

and not connected.

Size of Ribs• Twelve pairs of flexible, arch like ribs form the lateral portions of the thoracic cage. They increase in length from the first to the seventh and then decrease again from the eighth to the twelfth.(first 7 increasing and last 5 decreasing)

29 bones in Skull and 29 in Spine• There are 29 bones in the human skull. The cranium has 8; the face has 15 (including the lower jaw); the ears have 6. There are 29 bones in spinal cord.

Fore and hind legs of quadrupeds are different

Number 3

Repeated Three times

Drink water in three breaks

• Anas, may Allah be pleased with him, reported: The Messenger of Allah, sallallaahu ‘alayhi wa sallam, used to breathe three times in the course of a drink (he used to drink in three gulps).

Knock three times--- (Hadeeth)

• Allah's Apostle Prophet Muhammad (S.A.W.)said,• "If anyone of you asks the permission to enter thrice, and the

permission is not given, then he should return.'

5 Pillars of Islam

• The Qur'an presents them as a framework for worship and a sign of commitment to the faith. They are:

• (1) The shahada (Kalma),• (2) Daily prayers (salat),• (3) Alms giving (zakāt),• (4) Fasting during Ramadan (sawm)• (5) The pilgrimage to Makkah (Hajj) at least once in a

lifetime.

5 times prayers

5 times prayers: Fajr, Zohar, Asar, Maghrib, Isha• We recite Subhan Rabbiyal Azeem (3 times/ 7 times) and

Subhana Rabbiyal Aala (3 times/ 7 times)• Total number of Fardh

Prayers = 17 (prime#)

Reciting Adhkaar- 3,7 or 11 times• Most of the Adhkaar that we recite daily, morning or night, or

even after each prayer are in Prime numbers.• For example, reciting Allah-o-Akbar and Astaghferullah 3 times

after each Fardh Prayers; • Allah humma ajirna minanna’r 7 times in morning and evening.

SIGNIFICANCE OF NUMBER 7IN NATURE AND LIFE

Violet; Indigo; Blue; Green; Yellow; Orange; Red.

Prime Number 7 in Nature and Life

• 7 days in a week

• 7 notes in musical scale 56

7 wonders of the ancient world

THE NEW SEVEN WONDERS of the WORLD

About Seven Skies in Quran

• Seven skies is mentioned a lot in the Quran. What is meant by those skies? “And We have made, above you seven tracts; and We are never unmindful of (Our) Creation.”• In seven verses, the number of the skies is

mentioned as “seven”; in two of those verses it is stated that the seven skies are in the form of “layers”. (al-Mulk, 67/3; Nuh, 71/15) “Do you not consider how Allah has created seven heavens in layers” (Surah Nuh 71:15) 60

7 Heavens: The atmosphere has 7 layers• Today, it is known that the world's atmosphere consists of different layers that lie on top of each other. Based on the criteria of chemical contents or air temperature, the definitions

made have determined the atmosphere of the earth as seven layers. As mentioned in

Qur'an (67:3) “He Who created the seven heavens in layer”.

7 Seas

The Seven Seas as referenced in Medieval Arabian literature:

The Persian Gulf ("Sea of Fars") The Gulf of Khambhat ("Sea of Larwi") Bay of Bengal ("Sea of Harkand”) The Strait of Malacca ("Sea of Kalah”) The Singapore Strait ("Sea of Salahit”) The Gulf of Thailand ("Sea of Kardanj”), and

The South China Sea ("Sea of Sanji”). 62

• Al-Quran- • Surat Luqman 31• Verse 27, Part 21]

البحار سبعة ، العالم بحار• The "Arabian seven seas" البحار سبعة ، العالم must بحار

also have considered other important seas nearby which were navigated by Arabian and Phoenician seafarers:• the Adriatic Sea• the Arabian Sea• the Black Sea• the Caspian Sea• the Indian Ocean• the Mediterranean Sea• the Red Sea

7 Oceans

Feast of the Seven Fishes

• Some people believe that the feast is named for the Seven hills of Rome that surround the city.

Seven hills of Rome• Tradition holds that the seven hills were first occupied by small settlements and not grouped or recognized as a city called Rome.

Haft Sheen (7 S- Food)

• NOU ROUZ the Persian New Year is the only scientific New Year celebrated in the world today. Nou Rouz is symbolized by the Nou Rouz Table with (haft) seven items with S. • Some call it Haft (seven) Seen and decorate their

table with seven items whose names start with the letter “Seen” in Persian. All these items have a positive philosophic meaning. Others use seven items that start with the letter “SHeen”. Still others use seven trays (seenie) and put a variety of items.

•Number 7 in The Main Practices of Islam

Ablution (Wudu) has 7 steps• 1.Pour water over hands and wash them thrice • 2. Rinse mouth thrice • 3.Wash nose by putting water in it and then blowing it out • 4.Then wash face thrice• 5. Wash forearms up to the elbows thrice• 6. Pass wet hands over head, both ears and back side of neck • 7. Wash feet up to the ankles thrice

Prostrating to Allah• Seven parts of our body touch the ground…• Head (Forehead & nose)• Both hands• Both knees• Both feet

During Umrah

Tawaf Ka’bah: The circumfering of the Ka'bah 7 times:

• 7 rounds of Sai’

7 Requirements of Hajj

• (If one neglects a requirement he must compensate for it with

a ransom (fidyah) an animal sacrificed in Makkah. • Assuming Ihram from the Meeqat• Staying in Arafaat until the sun has set• Staying in Muzdalifah• Staying in Mina on the nights preceding days of Tashreeq • Stoning the pillars (Rami) • Shaving or cutting the hair• The farewell Tawaf (Al-wada)

Stoning of the Pillars

• (On 10th of ZilHajj):• Upon reaching Mina, hajjis hasten to stone the large pillar

(Jamratul Aqabah) ONLY, with 7 pebbles (throwing one by one), saying “Allahu Akbar” with each pebble thrown (7 times in total).

What to Sacrifice on Eid-ul-Adha?

• Generally, slaughter of goats, sheep, rams, cows, and camels is offered. • It is permissible for seven persons to share the

sacrifice of a cow or a camel on the condition that no one’s share is less than one seventh and their intention is to offer Qurbani (sacrifice).

Ayyam al-Tashreeq

• 11th , 12th, 13th of Zilhajj (Days of Drying Meat)• During these 3 days, (Ayyam at-Tashreeq), the pilgrims

stone the 3 pillars, from the time period between after zawaal (sun at highest point- no shadow) until the night. They stone all the 3 Jamarats, for which they need 21 (7x3)pebbles per day. • Method: Throw 7 stones (one by one), saying Allahu

Akbar each time: first to the small (Jamratul Sughra), then the medium (Jamratul Wusta), and finally, the large (Jamratul Aqabah) pillar.

Hadith about eating 7 Ajwa dates daily:• Sa’ad told of hearing Allah’s Messenger say, “He who has a

morning meal of seven ‘ajwa dates will not suffer harm that day through toxins or magic.” (Bukhari, Hadith 5327 and Muslim, Hadith 3814)

People of Cave (Ashaab e Kahaf)• They were 3, and 4th was the dog• They were 5, and 6th was the dog• They were 7, and 8th was the dog (as narrated in Quran)

Number 19 in Quran• The Qur'an consists of 114 (19 x 6) Suras.• Another mathematical miracle of the Qur'an is the manner in

which the number 19 is numerologically encoded in verses. This number is stressed in the words of the Qur'an: "There are nineteen (angels) in charge of it." (Qur'an, 74:30), and is encoded in various places in the Book.

NUMBER 19 in Quran

• The first Sura to be revealed, Surat al-'Alaq, consists of 19 verses and 285 (19 x 15) letters.• A total of 113 Suras in the Qur'an start with

Bismillah. Sura not to start with one is the ninth, Surat at-Tawba. Surat an-Naml is the only Sura to have two times Bismillah. One of these is at the beginning and the other in verse 30. Counting from Surat at-Tawba, which does not begin with Bismillah, Surat an-Naml follows 19 Suras on.

NUMBER 19 in Quran• The sum of all the occurrences of the name "Allah" in all the

verses whose numbers are multiples of 19 (i.e., verses 19, 38, 57, 76, etc.) is 133, or 19 x 7.

• The word "Rahman" (All-Merciful) appears 57 (19 x 3) times in the Qur'an.

• The verse “Which of your Lord’s marvels can you deny” has been repeated 31 times in Surah Al-Rahman which is a prime number. The sum of all the verses numbers where it is repeated, again gives a prime number 1433, and adding these digits gives again 1+4+3+3=11, a prime.

Prime numbers in Number Theory• When you pick up your cell phone you are using prime

numbers. A lot of algorithms have been developed for communication security and for transmission and compression of information. They depend on number theory and ultimately on prime numbers.

• So it is truly exploring this universe of numbers, which for a mathematician is very real and has surprising applications in the physical world.

1.2 subsets of integers dfs

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