Article status: Draft
Time Estimate for Reading: 15 min
Learning Objectives: Euclid's Division Lemma, Class X, NCERT, Mathematics
Effort Required: low
Pedagogy Model: Symbolification of mathematics
Prior Math Tools: Secondary school level Arithmetic (long division)
It happened to me. i always thought or was told, mathematics deals with numbers. Open any mathematics book, there are less and less of numbers and more and more of symbols. brain sees it, but mind does not accept it. a mental block. easiest way is to runaway from mathematics.
I kept running away from mathematics until Isaac Asimov (The greatest science fiction author) and Prof Shankar Ramamurthy (Yale) stopped me. Isaac Asimov introduced me to the concept of formula-analysis (we have used this in our articles on science and discovery) and Prof Shankar Ramamurthy introduced me to the power of symbolification. The 'ification' may be new to you. we like it that way. just that it sounds and rhymes good. There are more of this in our Tamil language.
What better way to start than asking, "why x is the unknown?".
Invest a little of your time. 3 min and 57 sec. watch this TED video by Terry Moore.
Having got to terms with x, let us move ahead and try to deal with more symbols. Let us begin with Euclid again. The Euclid's Division Lemma.
Euclid's division Lemma.
Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
a = bq + r
and
0 ≤ r < |b|,
where |b| denotes the absolute value of b
Feel like running away (i used to. but not now). please resist your temptation. it is not that difficult. At a first sight, it may look incomprehensible, just give it a try. There is nothing wrong in giving it a try.
All that Euclid has done is symbolification of division.
take a few examples of division.
5/5; the quotient (q) is 1, remainder (r) is 0,5 the numerator (n) and 5 the denominator (d).
11/5; the quotient is 2, remainder is 1.
4/5; the quotient is 0 and remainder is 4
10/5; the quotient (q) is 2, remainder (r) is 0,10 the numerator (n) and 5 the denominator (d).
We chose these examples such that we had the 3 scenarios. equal to, less than, greater than. we will be using this in most of physics.
n = d | r =0
n > d | r ≠ 0
n < d | r ≠ 0
Observations:
- we find that remainder can never be greater than denominator.
- take the example 11/5. 11 (n) = 5(d) * 2 (q) + 1(r)
Now let us retain these symbols and take away the numbers
therefore, n = d*q + r; where 0 <= r <= |d|
That's Euclid's division Lemma, a = bq+r. just replace a with n and b with d.
The question is why do we take so much pain? why cant we leave it as such?
There are quite some advantages in symbolification.
1. It is like a rule that is followed by all integers.
2. The Euclidean division lemma is extended into an algorithm to find out the Greatest Common Denominator. It is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 147 = 252 − 105.
The range of applications are more. please explore the reference section.
Welcome to the world of symbolification. you may want to call it algebra. it makes life easier.
Note:
We will be adding a few solved examples and exercise problems from NCERT text book later.
References:
https://en.wikipedia.org/wiki/Euclidean_division
https://en.wikipedia.org/wiki/Euclidean_algorithm
https://en.wikipedia.org/wiki/Euler%27s_totient_function
understanding the euclidean algorithm
https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm
https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/the-fundamental-theorem-of-arithmetic-1
Exercise
https://www.khanacademy.org/quetzalcoatl/deprecated-exercises/deprecated-bucket-5/v/the-fundamental-theorem-of-arithmetic
Method of generating primes
https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/sieve-of-eratosthenes-prime-adventure-part-4
Primes density vs natural logarithm
https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/prime-number-theorem-the-density-of-primes
A critical review of textbooks by Mr Badri Seshadri
Time Estimate for Reading: 15 min
Learning Objectives: Euclid's Division Lemma, Class X, NCERT, Mathematics
Effort Required: low
Pedagogy Model: Symbolification of mathematics
Prior Math Tools: Secondary school level Arithmetic (long division)
It happened to me. i always thought or was told, mathematics deals with numbers. Open any mathematics book, there are less and less of numbers and more and more of symbols. brain sees it, but mind does not accept it. a mental block. easiest way is to runaway from mathematics.
I kept running away from mathematics until Isaac Asimov (The greatest science fiction author) and Prof Shankar Ramamurthy (Yale) stopped me. Isaac Asimov introduced me to the concept of formula-analysis (we have used this in our articles on science and discovery) and Prof Shankar Ramamurthy introduced me to the power of symbolification. The 'ification' may be new to you. we like it that way. just that it sounds and rhymes good. There are more of this in our Tamil language.
What better way to start than asking, "why x is the unknown?".
Invest a little of your time. 3 min and 57 sec. watch this TED video by Terry Moore.
Having got to terms with x, let us move ahead and try to deal with more symbols. Let us begin with Euclid again. The Euclid's Division Lemma.
Euclid's division Lemma.
Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
a = bq + r
and
0 ≤ r < |b|,
where |b| denotes the absolute value of b
Feel like running away (i used to. but not now). please resist your temptation. it is not that difficult. At a first sight, it may look incomprehensible, just give it a try. There is nothing wrong in giving it a try.
All that Euclid has done is symbolification of division.
take a few examples of division.
5/5; the quotient (q) is 1, remainder (r) is 0,5 the numerator (n) and 5 the denominator (d).
11/5; the quotient is 2, remainder is 1.
4/5; the quotient is 0 and remainder is 4
10/5; the quotient (q) is 2, remainder (r) is 0,10 the numerator (n) and 5 the denominator (d).
We chose these examples such that we had the 3 scenarios. equal to, less than, greater than. we will be using this in most of physics.
n = d | r =0
n > d | r ≠ 0
n < d | r ≠ 0
Observations:
- we find that remainder can never be greater than denominator.
- take the example 11/5. 11 (n) = 5(d) * 2 (q) + 1(r)
Now let us retain these symbols and take away the numbers
therefore, n = d*q + r; where 0 <= r <= |d|
That's Euclid's division Lemma, a = bq+r. just replace a with n and b with d.
The question is why do we take so much pain? why cant we leave it as such?
There are quite some advantages in symbolification.
1. It is like a rule that is followed by all integers.
2. The Euclidean division lemma is extended into an algorithm to find out the Greatest Common Denominator. It is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 147 = 252 − 105.
The range of applications are more. please explore the reference section.
Welcome to the world of symbolification. you may want to call it algebra. it makes life easier.
Note:
We will be adding a few solved examples and exercise problems from NCERT text book later.
References:
https://en.wikipedia.org/wiki/Euclidean_division
https://en.wikipedia.org/wiki/Euclidean_algorithm
https://en.wikipedia.org/wiki/Euler%27s_totient_function
understanding the euclidean algorithm
https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm
https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/the-fundamental-theorem-of-arithmetic-1
Exercise
https://www.khanacademy.org/quetzalcoatl/deprecated-exercises/deprecated-bucket-5/v/the-fundamental-theorem-of-arithmetic
Method of generating primes
https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/sieve-of-eratosthenes-prime-adventure-part-4
Primes density vs natural logarithm
https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/prime-number-theorem-the-density-of-primes
A critical review of textbooks by Mr Badri Seshadri
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