#euclids division algorithm

[edusquaremaths]

NCERT Class 10

Chapter: Real Numbers

Exercise 1.1

Question 3: An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

[goldiers1]

Euclid - Conquering Math with Ancient Wisdom

  Euclid is a historical figure who has been revered for centuries, and for good reason. Born around 300 B.C., he is widely known as one of the most influential figures in mathematics and his work has had an enduring impact on modern education. In this blog post, we will explore Euclid’s contributions to mathematics and how they have helped shape our understanding of math today.  

Euclid's Elements of geometry - Title Page - Double Spread. Photo by University of Glasgow Library. Flickr.  

The Birthplace of Math

Euclid was born in ancient Greece, a culture that was determined to develop and refine mathematical principles. Euclid is credited with creating the Elements, an important collection of thirteen books that laid out the foundations of mathematics as we know it today. This work served as a cornerstone for modern math and has been studied throughout the ages.  

Euclid Elements of Geometry Printed by Erhard Ratdolt, Venice, 1482. Photo by University of Glasgow Library. Flickr.  

Euclid’s Mathematical Contributions

The Elements contains many of Euclid’s discoveries, including geometry, the study of shapes, and angles. He also proved that there is an infinite number of prime numbers (numbers divisible only by themselves and 1). This discovery has had a lasting impact on mathematics, as it allowed for the development of advanced algorithms that are used in modern computers. In addition to his discoveries, Euclid also developed several important axioms and postulates, which serve as a foundation for mathematics. These concepts were so influential that they have been adopted by mathematicians everywhere and are still being used today.  

Euclid's Elements: A Revolutionary Text

Euclid wrote a thirteen-volume collection of definitions, axioms, and theorems titled The Elements—a book that has become an integral part of mathematical study. This exhaustive text covers plane geometry, number theory, and optics among other topics; many of these are still studied today by students across the world. It is hard to overestimate the importance of this work; it not only provided a solid foundation for geometric studies but also laid the groundwork for later mathematicians to build upon his teachings.  

Accolades for Euclid

Euclid’s accomplishments were widely praised during his lifetime and long after he passed away. He was even given the title “The Father of Geometry” due to his seminal contributions to this field. His influence has been so great that he is often compared to other legendary figures such as Aristotle or Plato; some even believe he was a student of Plato himself! His works have been referenced for centuries by scholars looking to better understand mathematics and its implications in various fields such as physics or engineering.  

Experimental Mathematics. Photo by Ed Brambley. Flickr.  

Applying Ancient Wisdom to Modern Math

Euclid’s teachings still hold relevance in modern mathematics. In fact, many of his discoveries are now being applied to modern technology and other scientific fields. For instance, Euclidean geometry is used to understand the properties of space-time in relativity theory and it has also been used to develop computer algorithms. The influence of Euclid’s work can be seen in many areas today and it serves as a reminder that, even after centuries, ancient wisdom still has much to teach us.  

Conclusion

Euclid remains one of the most renowned figures in mathematics today due to his monumental contributions over two thousand years ago. Through his book Elements, he provided us with a solid foundation on which later mathematicians could continue their studies and pushed humanity forward in our understanding of math and its applications in other fields. For those interested in learning more about mathematics, Euclid’s works remain essential reading materials that will help them gain critical insight into this fascinating subject matter.   Sources: THX News, Wikipedia & Saint Andrews. Read the full article

[4denthusiast]

The trial and error in the first one is basically avoidable. Working in the integers mod 7,

60n+1 = 0

4n+1 = 0

4n = -1

n = -1/4

Then since the multiplicative inverse of 4 mod 7 is 2 (which is probably most easily found just by checking all the cases, but that's much quicker than doing the calculation with 60), n=-2 (mod 7) and the smallest possible positive value of n is 5, giving 60*5+1=301. This is basically the first step of Euclid's algorithm for modular division, but if the numbers were big enough that the full algorithm were necessary, that might take a bit too much memory to do in your head.

I bought this book from goodwill for a dollar and a lot of the games are hit or miss but overall I'm enjoying it, but there is a section of mental games (just to be played in your head, no pen or paper), and this one game is so unbelievably hard??

Note that you are supposed to do these entirely in your head, writing nothing down.

I wasn't timing myself because I was working on these while I was falling asleep last night but I think the second one took me over an hour.

1. A certain number leaves a remainder of one when it is divided by 2, 3, 4, 5 or 6 but leaves no remainder when it is divided by 7. What is the smallest possible value of the number?

2. A certain number, when divided by 2, 3, 4, 5, 6, 7, 8, 9 and 10, leaves remainders of 1, 2, 3, 4, 5, 6, 7, 8 and 9 respectively. What is the smallest possible value of the number?

And there are 20 of these questions. Totally worth the dollar, hours of fun (if you are the type of nerd I am)

[lawsofemotion]

Updated Number Theory! I finished the notes for Chapter 1. However, there are no practice problems or 1.4. The practice problems will come later next year. Have a happy Christmas and holidays!

[ashishkumarletslearn]

#RealNumbers #Class10Maths

In this lecture I have discussed the Meaning & Method to find HCF / GCD and Euclid's Division Algorithm of Chapter 1 Real Numbers Class 10 Maths.  

[qhd4]

I never liked Euclid's algorithm all that much, the results that come from it are much more interesting then iterated division, plus the proofs that use it are ugly imo, a bunch of tedious calculation.

Jacobson's Basic algebra 1 gives such a nice argument for why algebraic numbers have minimum polymials. Consider a field extension E/F (i.e. a field E that contains a field F). For any u ∈ E we have in E the subring F[u] and the subfield F(u). These are defined as the smallest subring and subfield of E respectively that contain both F and u. If E = F(u) for some u ∈ E, then E/F is called a simple field extension.

Let F(u)/F be an arbitrary simple field extension. Consider the polynomial ring F[X]. By the universal property of polynomial rings (which is essentially what one means when they say that X is an indeterminate), there is a unique ring homomorphism F[X] -> F(u) that sends any a ∈ F to itself and that sends X to u. If the kernel of this homomorphism is the zero ideal, then F[u] is isomorphic to F[X], so u is transcendental over F. If the kernel is non-zero, then by definition there are non-zero polynomials p ∈ F[X] such that p(u) = 0 in F(u), so u is algebraic over F. Because F[X] is a principal ideal domain, there is a polynomial p that generates the kernel. In other words, p divides all polynomials q such that q(u) = 0 in F(u). Two polynomials (over a field) generate the same ideal if and only if they differ by a constant factor, so there is a unique monic minimum polynomial in F[X] for u.

[allrounderknowledgehub]

Chapter : Real Numbers

Exercise : 1.1

Question 1. Use Euclid's division algorithm to find HCF of

(i) 135 and 225

(ii) 196 and 38220

(iii) 867 and 255

Solution 1.

(i) 135 and 225

Since 225>135, we can apply Euclid's division lemma to 225 and 135

225 = 135×1 + 90

Since the remainder 90≠0 , we can apply Euclid's division lemma to 135 and 90

135 = 90×1 + 45

Again we can see that the remainder 45≠0 we will again apply Euclid's division lemma to 90 and 45

90 = 45×2 + 0

We get remainder = 0

So in last step divisor is 45 so the HCF of 135 and 225 is 45.

To check full answer click here

https://allrounderknowledgehub.blogspot.com/2021/05/ncert-solutions-class-10th.html?m=1

[wisdomrays]

TAFAKKUR: Part 241

MUSLIM CONTRIBUTIONS TO MATHEMATICS: Part 1

When we talk about Muslim contributions to mathematics we are usually referring to the years between 622 and 1600 ce. This was the golden era of Islam when it was influential both as a culture and religion, and was widespread from Anatolia to North Africa, from Spain to India.

Mathematics, or "the queen of the sciences" as Carl Friedrich Gauss called it, plays an important role in our lives. A world without mathematics is unimaginable. Throughout history, many scholars have made important contributions to this science, among them a great number of Muslims. It is beyond the scope of a short article like this one to mention all the contributions of Muslim scholars to mathematics; therefore, I will concentrate on only four aspects: translations of earlier works, and contributions to algebra, geometry, and trigonometry. In order to understand fully how great were the works of scholars in the past, one needs to look at them with the eye of a person of the same era, since things that are well-known facts today might not have been known at all in the past.

There has never been a conflict between science and Islam. Muslims understand everything in the universe as a letter from God Almighty inviting us to study it to have knowledge of Him. In fact, the first verse of the Qur'an to be revealed was:

Read! In the Name of your Lord, Who created… (Alaq 96:1).

Besides commanding us to read the Qur'an, by mentioning the creation the verse also draws our attention to the universe. There are many verses which ask Muslims to think, to know, to learn and so on. Moreover, there are various sayings of the Prophet Muhammad, peace be upon him, encouraging Muslims to seek knowledge. One hadith says, "A believer never stops seeking knowledge until they enter Paradise" (al-Tirmidhi).

In another hadith, the Prophet said, "Seeking knowledge is a duty on every Muslim" (Bukhari). Hence it is no surprise to see early Muslim scholars who were dealing with different sciences.

TRANSLATIONS

Prophet Muhammed (pbuh) said, “Knowledge is the lost property of a Muslim; whoever finds it must take it” ; hence Muslims started seeking knowledge. One way they did this was to start translating all kinds of knowledge that they thought to be useful. There were two main sources from which Muslim scholars made translations in order to develop the field of science, the Hindus and the Greeks. The Abbasid caliph al-Mamun (804–832) had a university built and ordered its scholars to translate into Arabic many works of Greek scholarship. Between 771 and 773 CE the Hindu numerals were introduced into the Muslim world as a result of the translation of Sithanta from Sanskrit into Arabic by Abu Abdullah Muhammad Ibrahim al-Fazari. Another great mathematician, Thabit ibn Qurra, not only translated works written by Euclid, Archimedes, Apollonius, Ptolemy and Eutocius, but he also founded a school of translation and supervised many other translations of books from Greek into Arabic. While Hajjaj bin Yusuf translated Euclid’s Elements into Arabic, al-Jayyani wrote an important commentary on it which appears in the Fihrist (Index), a work compiled by the bookseller Ibn an-Nadim in 988. A simplified version of Ptolemy’s Almagest appears in Abul-Wafa’s book of Tahir al-Majisty and Kitab al-Kamil. Abu’l Wafa Al-Buzjani commented on and simplified the works of Euclid, Ptolemy and Diophantus. The sons of Musa bin Shakir also organized translations of Greek works.

These translations played an important role in the development of mathematics in the Muslim world. Moreover, the ancient Greek texts have survived thanks to these translations.

ALGEBRA AND GEOMETRY

The word "algebra" comes from "Al-Jabr", which is taken from the title of the book Hisab Al-Jabr wal Muqabala by Muhammad ibn Musa al-Khwarizmi (780–850). Al-Khwarizmi, after whom the "algorithm" is named, was one of the great mathematicians of all times. Europe was first introduced to algebra as a result of the translation of Khwarizmi's book into Latin by Robert Chester in 1143. The book has three parts. The first part deals with six different types of equations:

(ax2 = bx) ; (ax2 = b) ; (ax = b) ; (ax2 + bx = c) ; (ax2 + c = bx) ; (bx + c = ax2)

Khwarizmi gives both arithmetic and geometric methods to solve these six types of problems. He also introduces algebraic multiplication and division. The second part of Hisab Al-Jabr deals with mensuration. Here he describes the rules of computing areas and volumes. Since Prophet Muhammad, peace be upon him, said, “Learn the laws of inheritance and teach them to people, for that is half of knowledge," the last and the largest part of this section concerns legacies, which requires a good understanding of the Islamic laws of inheritance. Khwarizmi develops Hindu numerals and introduces the concept of zero, or “sifr” in Arabic, to Europe. The word “zero” actually comes from Latin “zephirum,” which is derived from the Arabic word “sifr.”

The three sons of Musa bin Shakir (about 800–860) were perhaps the first Muslim mathematicians to study Greek works. They wrote a great book on geometry, Kitab Marifat Masakhat Al-Ashkal (The Book of the Measurement of Plane and Spherical Figures), which was later translated into Latin by Gerard of Cremona. In the book, although they used similar methods to those of Archimedes, they move a step further than the Greeks to consider volumes and areas as numbers, and hence they developed a new approach to mathematics. For example, they described the constant number pi as “the magnitude which, when multiplied by the diameter of a circle, yields the circumference.”

A well-known poet, philosopher and astronomer Omar Khayyam (1048–1122) was at the same time a great mathematician. His most famous book on algebra is Treatise on the Demonstration of Problems of Algebra. In his book besides giving both arithmetic and geometric solutions to second degree equations he also describes geometric solutions to third degree equations by the method of intersecting conic sections. He also discovered binomial expansion [26]. His work later helped develop both algebra and geometry.

Thabit bin Qurra (836–901) was an important mathematician who made many discoveries in his time. As mentioned in the Dictionary of Scientific Biography he “played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.”

To give an idea of his importance, we will just give here, without details, one of his theorems on amicable numbers. Two natural numbers m and n are called “amicable” if each is equal to the sum of the proper divisors of the other:

for n > 1, let pn=3.22n–1 and qn=9.22n–1–1. If pn–1 , pn and qn are prime numbers, then a=2n pn–1 pn and b=2nqn are amicable.

[edusquaremaths]

NCERT Class 10

Chapter: Real Numbers

Exercise 1.1

Question 1(ii), (iii)

Use Euclid’s division algorithm to find the HCF of 196 and 38220.

Use Euclid’s division algorithm to find the HCF of 867 and 255.

[kvltoris]

i like that the wikipedia page for "division algorithm" vaguely implies euclid wrote while programs

[ncertsolutionsbook]

What is Euclid Division Algorithm

What is Euclid Division Algorithm Euclid’s Division Lemma: For any two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0 ≤ r < b. For Example (i) Consider number 23 and 5, then: 23 = 5 × 4 + 3 Comparing with a = bq + …

[learnenglishcbsein]

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers Class 10 Maths Real Numbers Exercise 1.1

Question 1 Use Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 Solution: (i) 135 and 225

Method 1:

Method 2:

(ii) 196 and 38220 Method 1:

Method 2:

(iii) 867 and 255 Method 1:

Method 2:

Question 2: Show that any positive odd integer is of the form 6q…