SIGNIFICANCE OF PRIME NUMBERS IN ISLAM

Dr. Farhana Shaheen

Yanbu University College

2

STORY OF NUMBERS

• The story of numbers begins with Natural numbers = {1, 2, 3, 4, 5, ……} initiated by counting, Whole Numbers, Integers, Real Numbers, and ends up to the set of Complex Numbers, the largest existing numbers, comprising of Real and Imaginary numbers

(x+iy, where , x, y are real).

3

4

Special Subsets of Integers

• Set of Integers Z = {….-3,-2,-1, 0, 1, 2,..}

a subset of Real Numbers, has many

specific subsets like:

Even Numbers: 0, 2, 4, 6, 8, …

Odd Numbers: 1, 3, 5, 7, 9, …

Prime Numbers, Composite Numbers, Fibonacci Numbers,

Lucas Numbers, etc.

5

What are Prime Numbers

• Numbers which have only two divisors (factors), 1 and the number itself.

• A prime number can be divided, without a remainder, only by itself and by 1.

• For example, the only factors of 17 are 17 and 1.

• 2, 3, 5, 7, 11, 13, 17, 19, 23, … are all primes.• A prime number must be a whole number

greater than 1.

6

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

Except for 0 and 1, a number is either a prime number or a composite number.

7

Prime factorization of number 48 using Tree Diagram

8

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.

9

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.

10

Sieve of Eratosthenes

11

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.

12

Primality

• The property of being prime is called primality. Verifying the primality of a given number n can be done by trial division. The simplest trial division method tests whether n is a multiple of an integer m between 2 and . If n is a multiple of any of these integers then it is a composite number, and so not prime; if it is not a multiple of any of these integers then it is prime.

• Is the number 2311 a Prime?

n

13

Why are Prime numbers fascinating?

• Because it's easy to understand what they are and hard to prove that they behave any way. And it's hard to prove that they behave the way that they should. It's hard to prove that they don't behave the way that they shouldn't. So it's like a little puzzle that you look at and you think, "Ahhh, that's easy." But the more you twist it and the more you twist it, it still looks easy, but you never get to the end. You never actually solve the problems.

14

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)

15

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

Are Prime numbers important in our life?

16

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

17

Count your hands, feet, fingers and toes

18

2 hands with 5 fingers each2 feet with5 toes each

.

19

• 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:

20

There are three things involved in body movement:

1. The Brain 2. Muscles 3. Skeleton.

• 1. The brain sends messages to the muscle to let it know it’s time for work.

• 2. Muscles contract• 3. Pulls bones attached

to them.

21

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)

22

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.

23

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)

24

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.

25

Fore and hind legs of quadrupeds are different

PRIMES: ALLAH’s FAVOURITE NUMBERS

27

Repeated Three times

28

5 Pillar 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.

29

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#)

30

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.

31

SIGNIFICANCE OF NUMBER 7

IN NATURE AND LIFE

32

7 Colours in Rainbow

33

Prime Number 7 in Nature and Life

• 7 days in a week

• 7 notes in musical scale

34

7 wonders of the ancient world

35

THE NEW SEVEN WONDERS of the WORLD

36

37

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)

38

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”.

39

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”).

40

البحار سبعة ، العالم بحار

• The "Arabian seven seas" البحار سبعة ، العالم بحارmust 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

41

7 Oceans

42

• Number 7 in The Main Practices of Islam

43

Ablution (Wudu) has 7 steps

• 1.Pour water over hands and wash them thrice • 2. Rinse mouth thrice • 3.Washed 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

44

During Umrah

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

• 7 rounds of Sai’

45

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)

46

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).

47

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).

48

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.

49

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)

50

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)

51

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.

52

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.

53

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.

54

• 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.

55

Prime Numbers in Cryptography

• You can search for the importance and significance of Prime numbers in Cryptography, the art of sending secret messages, the establishment of a technical system used in maintaining the secrecy of electronic communications.

56

Prime Numbers and Cryptography

• Cryptography (Greek word kryptos hidden, graphein to write) is to turn messages into gibberish so that they cannot be read by persons other than the intended recipient.

• If you send your credit card number and PIN over the internet to an online book store, the book store should be able to read it, but no one else should.

57

Cryptography: How it works?

• A cryptographer starts with plain text such as "ATTACKATDAWN" and produces cipher text such as "OTMMGKLHDTIR." This is encryption. Decryption is the reverse: it produces plain text from cipher text.

• Large primes are used in this process of encryption and decryption.

58

Prime numbers and Cryptography

• Every time you buy something on the internet with your credit card, you use prime numbers to keep your personal information secure. How? Using RSA cryptography.

59

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.

60

61

QUESTIONS