Six Generations Family Tree of the Curie Family

Marie Curie's life holds a particular fascination for me - it signifies a life full of struggle and deep commitment to achieving excellence amidst adversity.  Six Nobel Prizes have been shared within the Curie Family, with Marie Curie being the first and only woman who was awarded two Nobel Prizes in different subjects.  Marie Curie's life is a poignant example of how women were actively and openly discriminated in prevailing male-dominated environment of the early 20th century.   

In such situations, it is natural to ask who her other family members were, what were their achievements and who her descendants are. At home in France, because she was a woman, Marie Curie was denied obvious honours like membership of the French Science Academy. Despite the many roadblocks, Marie Curie not only excelled as a scientist but also won the affection and admiration of the public and the politicians throughout the world.  

In the following, I shall provide Marie Curie's family tree spanning six generations.  I refer you to my blogs (1, 2)  for a detailed discussion of Curie family's scientific achievements. 

For a concise version of Curie Family Tree, please click here.

The organisation of the family tree starts with the central characters (Marie Curie and her immediate family - parents and children).  The following slides then explore the relatives in more detail.  I feel this makes it easier to see the generations without overcrowding.  I acknowledge help from several published accounts of the Curies with a lot of help from the Wiki - these may be reached here (1, 2, 3, 4, 5). 

In the following six slides, each generation is highlighted by a different colour. 

Please click on a slide to see its full page image.  Press Escape key to return to the blog.

A word about the organisation of the slides.  

Slide 1 shows the central characters - Marie and Pierre Curie and their two daughters & their husbands.  The two generations won all six Nobel Prizes - more than any other family in history.

Slide 2 shows Marie Curie's family - her siblings, parents and grandparents.  It is worth mentioning that her sister Dr Bronislawa Dluska was working in Paris and encouraged Marie to come to Paris for studies and provided financial support too to make it possible.

Slide 3 likewise shows Pierre Curie's family - his brother, parent and grandparents.  Pierre and Jacques worked very closely together and did pioneering work in piezoelectricity and magnetic properties of materials.

Slide 4 deals with Eve Curie's husband's family.  Eve Curie had a remarkable career as a diplomat, journalist and author.  She also wrote the first biography of Marie Curie.

Slide 5  describes Irene and Frederic Joliot Curie's children Helen & Pierre Joliot, and grandchildren.  Family tree of Helen Joliot's husband (Michel Langevin) is included because of the historic interest in Marie Curie and Paul Langevin around 1910.

Slide 6  shows Frederic Joliot's family tree with his parent and grandparents researched. 

It has been an exciting experience to update the Curie Family tree.  My talks on the Curies go in detail about their research and personal lives - these may be reached by clicking on the following two links 1  and 2.

The information included in this blog has been teased out of many many websites - the links to all are not easy to list.  I acknowledge all the help I have obtained from Wiki and other genealogy websites.  If anyone wishes to be acknowledged individually then please let me know at ektalks@yahoo.co.uk and I shall be delighted to do so.

Thanks for reading. 

Sum of Powers of Numbers - A Derivation of Bernoulli/Faulhaber Formula

Sum of Powers of Natural Numbers - Exciting to Explore Different Ways of Doing It

'While there is one correct answer, there can be multiple ways to solve a problem, encouraging creativity and different perspectives' 

I have always found numbers fascinating - they are versatile, form patterns, help develop creative thinking and more. Satisfaction of solving a problem is a wonderful feeling, greatly enhanced by finding yet another way to do so. 

I remember that one of the first problems at school was summing the first N natural numbers -  the answer S = N(N+1)/2 was amazing; you can set N = 100 and instantly get the sum of all the numbers from 1 to 100.  

In this blog, I wish to discuss several different ways of summing the powers of natural numbers.  The motivation to write this blog came from the way many people reacted to my previous blog about the amazing properties of the number 2025 - in particular the fact that the square of the sum of numbers from 1 to 9 is equal to the sum of their cubes. 

      2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +9)^2 

        2025 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 

I can fully understand the confused looks as the result is so counterintuitive, yet true.  Of course, Wiki has a way to make it look obvious as shown in the next slide:  

The figure shows the result for N = 5, but it is easy to peal off squares or add more to demonstrate it for other value of N.  

Sum of Natural Numbers:   First, I shall discuss the sum of first power of natural numbers. We can write the sum in two different ways:

S = 1 + 2 + 3 +...................+ (N -1) + N  ...    eq.1   (N terms here)

S = N + (N -1) + ..................+ 2 + 1       ... eq. 2   (N terms)

Adding the two equations will give us

2S = (N+1) + (N+1) + ..........(N+1) + (N+1)    (N terms)

Or     2S = N(N+1)   giving us   S = N(N+1)/2     ... eq.3

I find it fascinating how maths works:  Multiply eq.1 by 2 to obtain

2S = 2 + 4 + 6 + ...................+ 2N = N(N+1)    ... eq.4  (N terms)

Subtract 1 from each digit in the above equation

      1 + 3 + 5 + ................+(2N -1) = N(N+1) - N    ... eq.5

The LHS has N terms - that is why we subtract N on the RHS

Hence,    1 + 3 + 5 + ....  +(2N -1) = N^2 + N - N = N^2     ...  eq.6

What eq.6 says is that the sum of the first N odd numbers is equal to N^2.   

Eq.4 tells us that  the sum of the first N even numbers is N(N+1) = N^2+N.

I shall now discuss various ways by which we can show that the square of the sum of numbers 1 to N is equal to the sum of cubes of numbers from 1 to N.  We shall also look at the sum of higher powers of natural numbers.

Final Word: We have discussed several methods of summing powers of natural numbers.  The methods have great heuristic value and may be used in a classroom situation to encourage students.

However, the best situation is if the sum of powers may be expressed as a closed expression for all powers - it would then be a matter of working out the expression for a chosen power quotient.  Such an expression is available and is called the Faulhaber's formula.  The derivation of the Faulhaber's formula generally requires advanced knowledge of mathematics.  In Part 2, I shall use a generating function approach to derive the Faulhaber's formula - a method that is tractable with higher school level mathematics.  

The New Year 2025: 2025 is an Amazing Number

My learned friend Prab Bhatt had talked to me about some interesting properties of the number 2025 - the new year that has just started. I share these with some additions:

2025 is a perfect square:  2025 = 45 x 45

In our times, years that are perfect squares happen about once a century.  The previous square year was 1936, and the next will be 2116 - 91 years later.  

Curiously, 2025 is made up of 20 and 25 which add to give 45 - square root of 2025!

2025 is the sum of three squares:  we can write 2025 as follows

                    2025 =  5² + 20² + 40² 

2025 is the product of two squares:  we can write 2025 as follows

                    2025 = 9² x 5² = 3² x 15²

2025 is the sum of cubes of all the digits from 1 to 9:  we can write 2025 as follows

The next year that is the sum of cubes of digits from 1 to 10 will be 3025 - 1000 years later.  Previously, 1296 was the year that was the sum of cubes of digits from 1 to 8!

2025 is the square of the sum of digits from 1 to 9

2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +9)²

This makes 2025 square of a triangular number, namely 45.  

While 2025 is not a triangular number itself, it is the sum of the 44th and 45th triangular numbers (990 and 1035).

2025 is interesting:  The sum of digits of 2025 is equal to 9.  

                               2 + 0 + 2 + 5 = 9

We can divide 2025 by 3 and the sum of digits of the quotient is again equal to 9. 

2025  ÷ 3 = 675.  Sum of digits of 675 is 6 + 7 + 5 = 18 

                              and the sum of digits in 18 is 9

And again:  675  ÷ 3 = 225.  Sum of digits of 225 is 2 + 2 + 5 = 9 

2025 and number 7:  If we break number 7 in its odd and even parts we have 

                       7 = 6 +1 = 4 + 3  = 2 + 5.  

The product of odd parts raised to the corresponding even part gives

Happy New Year!

Part 2: A Generating Function for Varg Fractions; Developing Generalized Varg Fractions

Recently, I had published a blog entitled 'Interesting Properties of Some Pythagorean Triples - Introducing Varg Fractions'  where I had introduced a class of fractions with some interesting properties.  

This article (Part 1) may be accessed here. In Part 1, we had used Pythagorean triples to generate Varg Fractions.  

What are Varg Fractions?  A varg fraction F is such that F+1 and F-1 are perfect squares. This implies that F²-1 = (F+1)(F-1) is also a perfect square. 

(If F <1, then 1+F, 1-F and 1-F² are perfect squares.)

As in Part 1, we shall only consider irreducible fractions such that the numerator (N) and the denominator (D) have no common divisor.  Also, since F+1 and F-1 are perfect squares, the denominator D of a varg fraction F must also be a perfect square.  The following slide (Slide 3 of Part 1) lists the first few varg fractions.

We can construct varg fractions using Pythagorean Triples, however, the method has some serious limitations (will become clear later in this article).  Here, I discuss a generating function for varg fractions that is easier to use and also allows generalisation of varg fractions that is not possible with Pythagorean triples.

Generating Function for F>1:  since square roots of F+1 and F-1 are both rational numbers, their sum is also a rational number (a rational number may be expressed as the ratio of two integers i.e. as a fraction).  

Slide 1 explains the details:

For Part 1, Click here.