Psst…
…you like mathematicians?
Yeah, you do. Here’s a timeline showing when many major math figures lived and what their main contributions were.
(Leibniz FTW.)
Petition for General Mills to make a cereal called “Bernoulli-O’s”
Happy birthday, Johann Bernoulli!
Johnann is one of the eight math whizzes of the Bernoullis, a Swiss family that somehow kept birthing amazing mathematicians into the world over a few generations.
This particular Bernoulli spent a lot of his time studying (and teaching!) infinitesimal calculus way back when calculus was at its very beginnings. He tutored both L’Hopital and Euler in math and was specifically thanked in the very first calculus textbook (written by L’Hopital). He also worked with his brother Jakob on a lot of problems, though there was a good deal of friction and the two often fought.
Johann is extra badass, though, because he was a good friend of LEIBNIZ and a student of his calculus. He was also one of the few who took Leibniz’ side and defended him when the whole Newton-Leibniz calculus controversy began. He actually took several problems and showed that they could be solved using Leibniz’ methods, but not Newton’s. A pretty cool guy, if you ask me.
STRUM STRUM I’M A DRUM
Alright you dude-machines, it’s time for a rant (and lots of caps lock).
(I’ve ranted about this like 238 times before, but it’s still important, so you have to deal with it.)
It really bothers me when people divorce mathematical theorems/proofs/lemmas/what-have-yous from the people who came up with them.
Like, I get it. The math on its own is obviously important. DUH.
But it really bugs the crap out of me when people are like, “why do we care about Such-And-Such who came up with the Such-And-Such Theorem? Just give us the math, yo!”
UM HELLO YOU WOULDN’T EVEN HAVE THE SUCH-AND-SUCH THEOREM IF IT WEREN’T FOR SUCH-AND-SUCH SO GIVE SUCH-AND-SUCH THEIR MOMENT IN THE SUN OKEY-DOKEY?
ALSO. I think knowing who/when/why/how someone freaking COMES UP WITH A THEOREM (or lemma or proof or whatev) can not only help someone better understand the reasoning/logic behind the theorem, but can also help put it into context with other possibly non-math events and maybe make it more relevant/understandable. Remember when I talked about how Kepler doubted the accuracy of the volume measure of a wine maker’s wine barrel and how that helped lead him on the path to figuring out a more accurate method of measuring the volume of such an oddly-shaped object? Not only is that an interesting tidbit of knowledge, but it helps give some context/background for the beginnings of calculus. It’s not necessary to understand the math, but I think it helps from making the math seem so removed from “real life” as it has a tendency to be if it’s taught as a bunch of formulas and Greek letters and “this will be on the test so memorize it” pieces of info.
C-to-the-hebyshev
We’ve been using Chebyshev’s Inequality quite a bit in both of my classes, and this morning I realized that I have yet to figure out more about Chebyshev and then annoy you all with a blog post about him.
‘Cause, you know, that’s just what I do.
Pafnuty Lvovich Chebyshev, who has at least eight correct transliterations of his last name, was a Russian mathematician born in 1821. He was one of nine children and, due to a stunted leg, spent a hell of a lot of time studying as a boy. He pretty much did math from the get-go, eventually becoming a professor and mentoring, among other people, Andrey Markov.
Chebyshev is probably most famous for the above-mentioned Chebyshev’s Inequality, which states that for a random variable X with standard deviation sigma, the probability that the outcome of X is no less than a*sigma standard deviations away from the mean is no more than 1/a2. The Inequality is used in the proof of the Weak Law of Large Numbers (which we proved at the start of our probability class here!).
Chebyshev is considered a founder of Russian mathematics and won the Demodov Prize, a prestigious scientific prize awarded to members of the Russian Academy of Sciences, in 1849. He died in St. Petersburg in 1894.
Math Man!
Guess whose birthday* it is today?
Hint 1: He was a Swiss mathematician born in 1667
Hint 2: He tutored l’Hopital in mathematics
Hint 3: Most of his family members were mathematicians as well
Give up?
It’s JOHANN BERNOULLI!
So why is he awesome?
Not only did he tutor l’Hopital—which eventually led to l’Hopital publishing the first formal book on calculus**–but he also tutored Euler when Euler was young. In fact, he was the one who convinced Euler’s father that he had the makings of a great mathematician, thus steering him away from a life of a pastor.
For a majority of his life, he was in a highly competitive relationship with his equally mathematically talented brother, Jakob. When his brother died of tuberculosis, Bernoulli’s jealousy actually shifted to his son (another mathematician!) and they had a few good disputes about who came up with what papers and ideas.
One super awesome thing about Bernoulli, though, was that he was one of the few who stayed on Leibniz’ side of the whole calculus dispute with Newton. He showed his support by demonstrating several problems that could be solved using Leibniz’ methods, but not Newton’s. Go Bernoulli!
Yay.
*He was born on July 27th by old style dates; by new style, he was born on August 6th.
**The book was basically all of Bernoulli’s teachings written up formally, which ticked Bernoulli off quite a bit even though l’Hopital mentioned him in the book.
Hey baby, let me indent your contour
Today was the last day of Complex Variables before the final next Wednesday. SAD!
A huge topic we’ve been focusing on for the past few weeks has been the residue theorem. Because I’ve been OBSCENELY BUSY, I haven’t had a chance to look up the dude behind the theorem (and behind pretty much everything we’ve talked about in this class): AUGUSTIN-LOUIS CAUCHY!!
So that’s what I did tonight. Cauchy was born in Paris in 1789. He worked under Laplace and was pretty familiar with Lagrange during the period when Napoleon was in power. Because of Lagrange’s urging, he entered secondary school in 1802 (at age 13) and it quickly became apparent that the dude was SUPER smart. He started winning all these academic competitions and basically out performed everyone else at the school.
When he was 36, he published one of his most important and well-known theorems: Cauchy’s Integral Theorem. A year later, in 1826, he gave the first formal definition of a residue of a function and shortly thereafter gave the residue theorem in another paper.
But that’s not all! Here are some other things Cauchy is responsible for:
- He was the first to rigorously prove Taylor’s Theorem
- He gave the necessary and sufficient condition for a limit to exist—the form of this condition is the one that’s still taught today!
- He created a test for absolute convergence
- He defined the concept of stress in elasticity
- He gave an explicit definition of infinitesimals in terms of sequences tending to zero.
And more! According to a biography written by Hans Freudenthal, more concepts and theorems have been named after Cauchy than for any other mathematician.
WOO!
WEIRD DREAM
(I think this is a sign that I’ve been doing too much of my math studying while watching Achievement Hunter.)
In the dream, I was hired by Nintendo to make a new version of Mario that could be played on the iPad. So I designed this game called “Math-io.” All the characters (apart from Math-io himself) were now mathematicians.
The best part? They were all mathematicians whose names rhymed with the original characters. Like Yoshi became Cauchy, Luigi became Fubini, etc. (I actually can’t remember the rest, but I was super proud in the dream that they all rhymed).
Actually, the premise of the game was exactly the same as your typical Mario, just with mathematicians.
Any takers, Nintendo?
Would a passionate speech about horology be considered a glockenspiel?
I think it would be super cool if someone came up with a cookbook in which all recipes were stupid reconfigurations of mathematicians’/statisticians’ names or mathematical objects.
Examples:
- Fibognocci
- Tukey Sandwiches
- Vennison
- Bonferroni and Cheese
- Putnaan (“Putnam” and “naan”…anyone?)*
- Gabriel’s Corn
- Mandelbratwurst
- Fig Newto—OH WAIT
I’d buy a cookbook like that.
*Yes, I know Putnam wasn’t a mathematician himself, but he’s got that competition named after him, so yeah. It counts.
Walk into the club like what up I got an infinite series
Today we had our second test in Complex Variables. The test involved figuring out quite a few Maclaurin series for functions involving i. I’ve been ridiculously busy and thus haven’t had a chance to check out the dude behind the Maclaurin series…until now!
So. A Maclaurin series is simply a Taylor series centered at zero. According to the almighty Wikipedia, this type of series is named after Colin Maclaurin, a Scottish mathematician that lived from 1698 – 1746 (so during Leibniz/Newton time and a little bit after). The reason centered-at-zero Taylor series are named after him is because he used them extensively in his Treatise of Fluxions when describing and characterizing points of inflection, minima, and maxima of smooth functions.
This dude was super smart. He entered college at 11 YEARS OLD and got a Masters degree three years later. He became a professor at age 19 and actually got a personal recommendation from Newton to be appointed deputy to James Gregory, the mathematics prof at Edinburgh, and then once he surpassed Gregory’s position, Newton was so impressed that he actually offered to pay his salary for him.
He also had a crapton of children (well, 7 children, which I guess probably wasn’t a crapton back then) and died of complications from edema.
Theorem Fun
Welcome to another edition of “let’s learn about the person behind the theorem!”
Today’s edition: Joseph Liouville, after whom the Liouville’s Theorem is named. The theorem is associated with complex analysis and states that every bounded entire function must be constant. We unfortunately didn’t have time to get into the proof/applications during today’s class, but for now, let’s just look at Mr. Liouville, shall we?
Liouville was born in Saint-Omer, France, in 1809. Apparently he was very organized in terms of getting stuff done with math and played a key role in both founding some mathematical journals (like the French mathematical journal Journal de Mathématiques Pures et Appliquées) as well as recognizing some important works that were, at the time, unpublished.
He himself did quite a lot in various fields of mathematics: number theory, complex analysis, topology, mathematical physics—he was also the first to prove the existence of transcendental numbers in 1844 (though the term “transcendental” was first used by—guess who?—Leibniz, back in 1682 and then refined into today’s definition by Euler).
He also has a crater on the moon named after him!
The Six Degrees of Leibniz
I submit that on Wikipedia, you can get from the page of any mathematician to Leibniz’ page in 6 clicks or less (even without clicking through the “Mathematician” or “Mathematics” links that show up in like the first sentence of every mathematician’s Wiki page).
Fun Examples for Fun
Starting mathematician: George Polya
Click 1: Probability Theory
Click 2: Probability
Click 3: Christiaan Huygens
Click 4: Gottfried Leibniz
Starting mathematician: George Boole
Click 1: Differential Equation
Click 2: Derivative
Click 3: Gottfried Leibniz
Starting mathematician: John Venn
Click 1: Set Theory
Click 2: Principia Mathematica
Click 3: Philosophiæ Naturalis Principia Mathematica
Click 4: Gottfried Leibniz
Starting mathematician: Sewall Wright
Click 1: Philosophy
Click 2: Gottfried Leibniz
Starting mathematician: Henri Poincaré
Click 1: Bernhard Riemann
Click 2: Riemann Integral
Click 3: Integral
Click 4: Gottfried Leibniz
De Moivre!
Today in Complex Variables we learned about the de Moivre formula. So, like I do every time I learn a formula named after some dude, I had to look up the dude to see who he was.
Abraham de Moivre lived from 1667 – 1754 (another one of those long-lived mathematicians) and was friends with Newton, Halley, and Stirling, among others.
Originally from France, he moved to London and, while there, became pretty obsessed with Newton’s newly-published Principia Mathematica and basically memorized the material.
In fact, he took Newton’s binomial theorem and was able to generalize it to the multinomial theorem. This work (plus the fact that he was friends with Newton, I’m sure) got him membership in the Royal Society in 1697.
[And—I have to mention it, I’m sorry—he was one of the dudes on Newton’s little crony committee that was put together to hear the plagiarism charges against Leibniz in 1712.]
De Moivre’s famous formula originated (apparently) from this derivation in 1707:
which he later generalized to this form:
Euler proved it using his own Euler’s formula, so that pretty much cinched it. The reason de Moivre’s formula is so important is because it creates quite a nice connection between complex numbers and trigonometry.
Nifty, eh?
Green & Stokes
So in my continuing saga of “Let’s Make Stupid Jokes About Everything” (aka, “My Life”) and in the same vein as that Neil & Prey dream I had awhile back, I think someone should propose a detective/mystery show called Green & Stokes. It’d be like NUMB3RS crossed with Law & Order crossed with Columbo, except with exponentially more puns.
They’d work for the LAMD (Los Angeles Math Department) or something, because cities would have their own math departments in whatever universe that would allow Green and Stokes to be mathematicians AND detectives AND live during the same time period.
And the episode names could each be a pun on some other famous mathematician’s name (or other dumb puns).
- “Rolle with the Punches”
- “Out with the Old, in with the Newton”
- “Bourbaki and the Case of the Empty Set”
I DON’T EVEN KNOW.
This is why I need school to start again.
Edit: holy crap, I forgot how crappy gifs can be when they’re exported from Flash (especially when you don’t know what you’re doing), but here’s the theoretical show’s opening animation nonetheless:
Edit 2: fixed it (sort of; it’s still dumb)
OH CRAP SORRY PASCAL
So I hadn’t checked my little mathematician birthdays database in awhile and decided to check it yesterday. It turns out I missed Pascal’s birthday by a day. He was going to get my blog yesterday but I was distracted by freaking out about my final. I’m still freaking out about my final, but I have nothing else to blog about related to it. So Pascal shall get my blog today instead!
Though he only lived the 39 years between 1623 and 1662, Blaise Pascal was an incredibly accomplished mathematician and inventor. He was educated by his father and was still in his teens when he began to explore advanced topics on his own.
He was interested in math (particularly geometry) early on; when he was 16 he wrote an essay on conic sections that was so advanced that Descartes read it and thought that Pascal’s father had actually written it for him.
Little Pascal was also very interested in the idea of a mechanical calculator, and he was strongly motivated to produce the first working prototype of what was called the “Pascaline” in 1642 to help ease his father’s work as tax commissioner for the king of France. The calculator could do addition and subtraction* and was a great help to his father, but because of its cost it failed to be a commercial success.
Probably Pascal’s most famous contribution to mathematics is Pascal’s Triangle and the closely-related Pascal’s rule which states how the triangle is to be constructed. The triangle displays the binomial coefficients resulting from the binomial theorem along with other really cool properties (might have to do a blog just on his triangle here in a bit…). The development of this triangle led to conversations with Fermat, and the two collaborated together to develop probability theory.
In addition to his contributions to math, Pascal also gave the world the hydraulic press, the syringe, and did a whole ton of experiments with vacuums and hydrodynamics (he’s got the SI unit of pressure named after him as well, though that obviously happened much later). Some of his most famous demonstrations of the effect of elevation on atmospheric pressure involved carrying barometers to the tops of churches to see what happened to the mercury levels.
Cool dude, huh? See? 17th-century Europe!!
BLOG COMPLETE!
*The Pascaline was what Leibniz was trying to improve on with his Step Reckoner by including also the functions of multiplication and division.**
**Yes, I have to mention him in every post.
Goin’ Green
I’M DOING IT AGAIN!
Today we learned about Green’s Theorem. So who is this Green fellow?
[Edit: Okay, originally I was just going to talk about Green ‘cause while Green’s Theorem is just a special case of Stokes’ Theorem, we haven’t learned the latter yet. But turns out both Green and Stokes are named George and that’s hilarious, so we must press on and speak of both.]
So who are these two fellows?
George Green lived from 1793 – 1841. Like what seems to be a large proportion of mathematicians at the time, he was British. He lived in Nottingham. There are two reasons why these are interesting facts:
1. Nottingham, at the time, wasn’t really burning it up intellectually. Only about 25-50% of children received any sort of education, and Greene himself attended an academy for only one year when he was 8. It took him until age 36 to gather enough money (and free time) to afford a higher education (and he died when he was 47, so unfortunately he didn’t have too much time to enjoy it).
2. Despite the setbacks as far as formal education goes, Green was a very smart dude. He was largely self-taught (obviously) and once he finally got to Cambridge he pretty much kicked ass. What’s most interesting, though, are his studies in math. Historians aren’t exactly sure how Green reached the understanding of calculus that was necessary for developing his theorem. He likely used the “Mathematical Analysis,” which was a form of calculus Leibniz developed, but this was during the post Newton-Leibniz controversy over calculus and England pretty much forbade the use of everything calculus-related that originated from Leibniz. ‘Cause they had Newton’s calculus. Never mind that Newton’s notation was inferior and didn’t lend itself to the applications/developments that Leibniz’ notation did and that forbidding the better notation/methods from the England set the country back in mathematical advancements for like a century.* But somehow Green got a hold of it and made his improvements and came up with his theorem and was generally awesome. (LEIBNIZ POWER! Okay, I’m done).
And what about George Stokes? Who was he? Stokes’ life overlapped the end of Green’s life (1819 – 1903). Stokes was Irish and rocked the fields of fluid dynamics, optics, and mathematical physics. He actually did quite a variety of things, so I’m just going to list a few.
- He came up with a way to calculate the terminal velocity of a sphere falling in a viscous fluid (Stokes’ Law!)
- He expressed a mathematical description of rainbows using a divergent series, something that wasn’t really understood just yet by most.
- He was secretary and then president of the Royal Society.
- He wrote a paper in which he tried to describe the variation in gravity across the earth’s surface.
- And, of course, Stokes’ Theorem in math.
YAY GEORGES!
Sorry, I’m going to keep doing these little mathematician snippets until…well, until I feel like stopping. So ha.
*I’m not bitter.
Who is Fubini?
I dig my calc III teacher. He’s awesome. But I wish he’d do what I wish all math teachers would do when they introduce a theorem or lemma or rule: tell us a little bit about the person responsible for it, especially if the theorem/lemma/rule is named after the dude.
Like today we talked a lot about Fubini’s Theorem. We used it in like three examples. I used it on the homework I did right after class.
All the while without knowing who the heck this Fubini guy was.
So I checked him out this afternoon. Guido Fubini was an Italian mathematician who lived from 1879 to 1943. He was pretty into geometry and calculus for most of his life and moved around in different professorships in Europe before accepting an invitation to teach at Princeton in 1939 (partially to get away from the Nazis; he was Jewish).
So what the heck is this theorem, anyway?
Well. Let’s just look at rectangular domains first (because that’s all we’ve learned so far, haha…we’re doing non-rectangular domains tomorrow). So let’s look at a pretty double integral to start.
(P.S. I’m loving this chapter on double integrals already, simply because it means I have to write more integral signs. I FREAKING LOVE THAT SYMBOL.)
Say some rectangular region R is defined by the intervals [a,b] x [c,d]. If a function of two variables z= f(x,y) is continuous over R, then we can write the volume of the solid that lies below the surface z = f(x,y) and above the rectangle R as:
Or:
Iterated integrals!
Cool? Cool. So what does Fubini’s Theorem state? Again, assuming z = f(x,y) is continuous over R and R is a rectangular region, Fubini’s Theorem allows us to switch the order of integration while still getting the same correct result at the end:
Which is pretty snazzy (there’s a few other statements in the theorem; I just chose this conclusion as the example to show here).
But what I found most interesting about this theorem is that while double integration has been around for quite a long time, this theorem was proved sometime during Fubini’s lifetime–sometime in the late 1800s or early 1900s. (I can’t find an exact date for it, but that’s mainly because my internet’s deciding to be a bitch right now). Which makes sense, I guess, considering there exist cases where this doesn’t hold and so it may not have been an “obvious” thing or may not have been easily provable…but still. Interesting passage of time before we got to this theorem.
HUZZAH CALCULUS!
Shoutout to Newton
Let’s clear up a misconception today.
Despite how much I bitch about him, I do not dislike Sir Isaac Newton.
I don’t know how anyone really could dislike the guy. I mean come on. Anyone who can contribute that much to science and society deserves the utmost respect, even if he wasn’t the easiest guy to get along with. Personality does not beget worth—it’s what you do that counts.
Anyway.
He actually is one of my favorite scientists, and it baffles me just how much he did. It’s really incredible. I would have liked to know him. I just bitch about him a lot because of the whole calculus thing, ‘cause my main man Leibniz got screwed over and that makes me want to invent a time machine so that I could go back and maek things right and then make out with him for the rest of eternity upsets me.
But I do not dislike Sir Isaac Newton.
So shoutout to England’s greatest scientist! I’d make a horrible pun here in your honor, but I can’t think of any right now.
ZOMG KLEIN AND KOLMOGOROV
(Again, sorry for spamming y’all with mathematicians.)
(Also, I’m using this as a distraction from the fact that I’m a complete moron who will never amount to anything ever.)
HOKAY.
SO.
Two badass math fellows were born on this day.
1. Felix Klein (isn’t that a badass name?)
Klein was a German mathematician born on this day in 1849. He is most famous for the non-orientable surface named after him: The Klein bottle!
Picture (from Wiki):
A Klein bottle is similar to a Mobius strip except unlike the Mobius, it is a closed manifold rather than an open one. And while a Mobius strip can be imbedded in 3-D space (which is why we can make them with just a piece of paper and some tape!), a Klein bottle cannot.*
2. Andrey Kolmogorov
Kolmogorov, a Soviet mathematician born on this day in 1903, was known for quite a lot of things, but for me the thing that jumps out is the nonparametric statistical test he had a hand in developing.
The Kolmogorov-Smirnov test is one used when your data just aren’t being normal but you still need to examine them somehow. In particular, the test is used when you want to determine if two datasets differ significantly. It gets around the issue of nonparametric-ness by making no assumption about the distribution of data in either set.
I’d go into how, but it’s late and I have to get up early tomorrow and if I start talking about stats I won’t shut up for days, so make me promise I’ll do a blog on the K-S test later.
But yeah. Two awesome math dudes today! Woo!
*That doesn’t stop people from making Klein-esque models. If you want a Klein bottle model, go here.
N-N-N-NAPIER
I should just change my (semi-)weekly science blogs to “In This Blog Claudia Blah Blahs about a Mathematician” because that’s pretty much what I do weekly anyway.
(It’s ‘cause of that damn birthdays site, man.)
Today’s feature: John Napier of Scotland!
Yeah, he was a cool dude. Did some stuff, you know, just a few small things like DISCOVERING LOGARITHMS.
Napier studied math as a hobby (his main focus was theology) but, wisely, turned more towards math upon discovering logarithms and subsequently publishing a book about them in 1614. He created tons of calculating tables that were used to make calculations involving e much easier. He also invented an abacus-like device that could be used to quickly calculate products and quotients of numbers. This tool was called Napier’s bones because it involved the use of 10 long rods printed with numbers. The rods, back in the day, were made of ivory and thus looked like long bones.
He also did work with decimal notation, refining previous notational standards set in place by Simon Stevin.
Despite natural logs being my natural enemy (HA GET IT no seriously my brain cannot handle them), I’ve gotta admit that discovering freaking logarithms is pretty damn snazzy.
Not “discovering calculus” snazzy, but snazzy nonetheless.
END!
N-N-N-NAPIER
EeEEeeeEEEEEeeeeeeee
Happy birthday, Pierre-Simon Laplace!
Considered the “Newton of France”, Laplace is another one of those guys who just did EVERYTHING. He did a lot with probability—both Frequentist and Bayesian—and he’s even got a distribution named after him.
Anyway.
I know at least three of my readers dig The Oatmeal. Today I read a comic of his that I’d never seen before. I advise you to read it as well if you haven’t yet come across it.
Who is this Taylor guy, anyway?
We’ve started Taylor series in calc this week. Which is cool; I think I’m understanding what they are/how they work/why they’re important. But one thing I don’t know is who this Taylor fellow is.*
TO THE WIKIPEDIA-MOBILE!
So it looks like Brook Taylor was an early 18th century British mathematician. He did some work on the then newly described calculus, coming up with work Lagrange popularized in the late 1700s and, of course, came up with Taylor’s Theorem and Taylor series.
Here’s a cool demonstration of Taylor series approximation for various trig functions.
Edit: OH GODS HE STUDIED UNDER KEILL. Why, Taylor, whyyyyy?!
Edit 2: he was on the Royal Society committee set to hear Newton’s claims against Leibniz, too! Whaaaaat.
*I wish more teachers gave at least brief little intros on the people who come up with all this cool stuff. Especially when we’re learning about something expressly NAMED after someone.
More birthday stuff
I can’t stay away from that website with the mathematician birthday/deathday data.
So I decided to look at the data a little differently this time. Each square represents either a death or a birth on the day of the year. The data are broken up by month by a small white space.
Note the “most eventful” and “least eventful” days. And look at October and all its within-group variance.
I think I’m going to do some stats on this data. Because that’s what I do. But I can’t do it for like another week, since next week involves LOTS of studying/homework/panic attacks.
BUT LATER!
More birthday fun!
This is fantastic info right here.
I don’t know why, but birthdays and the distribution of birthdays is really fascinating to me. I’m going to have to analyze this somehow.
And the birth and death statistics page is awesome, too. Interesting, interesting stuff.
Edit: compiled ‘em!
It’s kind of cool that I found this today, as today was the birthday of Rene Descartes! Happy birthday, Rene!
Edit 2: Not only did we lose l’Hopital on my birthday, but we lost Bertrand Russell as well? Man, throw in the birth of Ayn Rand* and February 2nd has not been kind to the world.
*James Joyce’s birth almost makes up for this.
Eigenblogger 