Patrick, Paul, and I got a chance to hang out on Venice Beach and talk about our experiences teaching and learning. We all have some common experiences: learning science and math didn’t always come easily to us, and I think one of the reasons we are good teachers is because we know how students can struggle.

By: Tyler DeWitt.

]]>For puzzle #4b, what do the colors mean on this second, similar map?

To find the answers, simply scroll down.

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Keep scrolling….

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Solution:

In the first map, consider the number of letters in the name of each state. Is this number prime or composite?

In the second map, consider the number of characters, rather than letters, in each state’s name. This number is different for states with two-word names, due to the single character, a blank space, needed to separate the two words. Again: prime, or composite?

]]>In the mathematical field of dynamical systems an **attractor **is a set of numerical values toward which a system** tends to evolve,** for a wide variety of starting conditions of the system.

So I wonder, what are my numerical values normally?

Dynamical systems in the physical world tend to arise from dissipative systems:

if it were not for some driving force, the motion would cease.(Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) Thedissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior.The subset of the phase space of the dynamical system corresponding to the typical behavior is theattractor, also known as the attracting section or attractee.

This means I am not the attractor, because the two situations which I have encountered these past two days are not typical behavior for my life, hence I must be an initial transient for drama?

A confusing start to a blogpost but what I’m trying to get at is that I find it interesting when life is seemingly free from drama and then two days in a row I happen to run into situations where people are/have been in danger.

**1. Yesterday I got off** the subway at Danderyds Sjukhus. 5 people standing around one man who is laying on the floor, shaking, eyes flickering, not responding. I walk up to them. Apparently he fell down on the floor. Another subway stops and a woman gets off and walks towards the crowd saying *“Is anyone here a doctor?” “NO!”* We all answer.* “I am. Let me see how he is doing.” *She proceeds with a medical routine, flashlight on his pupils, etc. Then he wakes up. His arm is bleeding. He looks dazed and she asks him to stick his tongue out. Which he does. He seems fine. Then he sits up and the ambulance is on its way.

**2. Tonight, in the car, on my way home after running in the forest. **A man stands on the verge of the road next to his car which has the warning signals turned on. He is on the phone, ripping his hair. I stop and ask him what has happened. *“I crashed into the sign” “How are you?*” I asked *“Oh no no I’m good, just shocked, shaking”.* I got out and helped him call a wrecker because motor-oil was pouring out of the car. He talked to family and friends a little. I made him laugh and tried to make him feel less shaken up. When he seemed better and the wrecker was sure to come, I left.

**Hence the start of this blogpost, is it possible to explain events in life through mathematics and physics? What has shifted in my universe? I wonder if one could calculate situations in reverse then eventually one would be able to predict the future? **

Ok, so basically in my sixth form we have a maths society called *Sigma Society*, or *Sigma Soc* for those of us who frequent it often, and each summer a new Upper Sixth student takes over the position of ‘chair’ as the old Upper Sixth leaves school to go on to university etc., and this year I’ll be going into Upper Sixth and I’M THE CHAIR!

Realistically speaking, it’s not the most coveted of positions: of all the 250 or so girls in sixth form, there were only maybe 7 of us who were sure to be at every talk or lecture held by the society, and then maybe 10 or so more who would come to a few of them. But to be honest, that isn’t the point. The point is that this is a chance for me to take control of what is, by nature, my favourite school society and turn it into something that others may love and be proud of.

The first hurdle is the lack of a budget. Given that most speakers (understandably) request that you at least pay their travel expenses, and we only have the money for a pack of custard creams and cup of squash, this could be a bit* of a problem. However, my hope is that we can find some good local mathematicians and also tap into the school alumnae to see if there’s anyone willing to come and talk to us.

I’m also planning on organising a couple of trips to see public lectures, as a small group of us went to see an amazing physics lecture on knots last year at the Open University and it would be great to see more things like that. And finally we’ll have the joint lectures with our brother school’s maths society, Pythagoreans (how come they get the cool names?!), so all in all we should be able to scrape together a nice programme for the year.

Alongside these lectures and talks, I figured we could pull together our own discussions if everyone came up with a topic they liked and found out a bit about it to explain it to the group, then we talked about it and people chipped in with questions and their own knowledge of it. It would be nice if we could spend some time just batting about ideas and playing with things like Möbius strips and hexaflexagons and unexpected probabilities rather than being under pressure to learn the course content the whole time.

I’m looking forward to a good year for Sigma Society. I know there are some enthusiastic students coming up into the Lower Sixth and I hope we keep them inspired and curious to the best of our abilities.

*read “actually maybe rather a lot uh-oh”

]]>As many of you know, I am a bit obsessed with Alfred North Whitehead, who was himself a mathematician and philosopher. While reading his biography this week, I stumbled across the way he defines Geometry, around 1906-1907. He calls geometry the “science of cross classification. Victor Lowe says “The remark is typical of Whitehead’s outlook. His definition of geometry accords with the view that as pure mathematics it has no determinate subject matter in the usual sense of the term, but deals merely with types of relation, *and applies to any entities whose interrelations satisfy the formal axioms*.” In this sense, then, geometry is simply how certain elements relate to certain elements within certain mathematical schemes. Whitehead observes in his *Adventures of Ideas* that other sciences of cross classification “have not been developed, partly because no obvious applications have obtruded themselves, and partially because the abstract interest of such sciences has not engaged the interest of any large group of mathematicians.”

This isn’t the first time that a formal scientist or mathematician takes the art world for granted when making absolute statements like this. I wish that Whithead had been informed of Lessing, or had stayed alive long enough to understand Brecht. Because to me, Dramturgy is exactly another science of cross-classification, the study of aesthetic *relations.* Dramaturgy is, in short, a geometry. Its subject matter is often much more complex, and the results it attempts to attain are not quantifiable . Neither are the means. But we do attempt to study and arrange our points and lines in regard to the larger whole. We do not necessarily use logical symbols or attempt to work out rational relations in our head. Its much more intuitive than that. But nevertheless there is a great similarity here between the two sciences. We both ask how point A is related to point B, and where each exists in space. Dramaturgy simply displaces with the exactitude of mathematics and asks what that relation *means*.

The invention goes as follows.

Say the music has only two types of Notes. Those are: “Long” notes of Music takes 2 syllables, “Short” notes of Music has 1 syllable.

In this case, for a fixed number of syllables (say 8 syllables), how many number of different sequence possible with Long notes & Short notes?

Let us start with 1.

For length = 0 syllable(s) , the number of sequence possible is, N_{0}=1 [**empty sequence]**

For length = 1 syllable(s) , the number of sequence possible is, N_{1}=1 **[ 1 ]**

For length = 2 syllable(s) , the number of sequence possible is, N_{2}=2 **[ 11, 2 ]**

For length = 3 syllable(s) , the number of sequence possible is, N_{3}=3 **[ 111, 21, 12 ]**

For length = 4 syllable(s) , the number of sequence possible is, N_{4}=5 **[ 1111, 211, 112, 121, 22 ]**

and so on.

The above sequence is exactly same as **Fibonacci number (F _{n} = F_{n-1} + F_{n-2})**

We can generalize the sequence as below:

**say, the allowed notes (syllables) are k _{1}, k_{2}, k_{3} …. k_{m}**

**So, the number of sequences possible with length of “n” syllables F _{n} = F_{n-k1} + F_{n-k2} + F_{n-k3} + … + F_{n-km}**

**Fear Factor**

Many secondary school students have developed some element of fear about these two subjects. This is always as a result of long term negative information they get every year about the poor and low level performance in WAEC and NECO English and Mathematics results. Psychologytoday.com defines fear as a vital response to physical and emotional danger—if we didn’t feel it, we couldn’t protect ourselves from legitimate threats.

This quote can be translated in the context of the threat posed by these two subjects to candidates preparing for exam. Since most of these students have lost hope and are psychologically defeated due to fear, this has affected their level of interest and passion in these subjects and major percentage of students have chosen to pass these exams through malpractice or external help.

**Lack of Good Teachers in These Subjects **

Another important factor that has contributed to high level of failure in WAEC and NECO English and Mathematics examinations is lack of quality, seasoned and good teachers in these subjects.

The level of corruption in our system especially in public sector has made it possible for people who are not well suited for a job to occupy the position due to personal connection or network. Many teachers who have been employed and are teaching right now have not even taken a course in teaching practice and majority care less about their students and their inability to learn and understand the subject.

**Lack of Passion**

When you do not have passion for what you do, it will be very difficult to be efficient and productive. This goes a long way to explain the lack of passion many students have developed in English and Mathematics subjects. Loss of passion is the same as loss of interest and this phenomenon is always caused by factors like a bad style of teaching, fear factor, negative hype about the subjects and more.

Many English and Mathematics teachers have failed to boast their students’ morale in these subjects and have failed to teach these subjects in ways that make it fun and interactive for students to enjoy while learning.

**Lack of Reading and Practice **

The last thing many students love doing is to open their English or Mathematics textbooks and practise their skills themselves. Many students consider it to be boring especially when there is lack of interest. How on earth do you expect a student who has not taken out time to learn and practise English and Mathematics on their own to come to exam after three years of senior secondary and pass WAEC or NECO English and Mathematics subjects without any external help?

For many students, lack of money in their families have resulted to their inability to pay for extra-mural classes after school in order to learn more about the subjects they have hard time to comprehend.

**Final Take**

There are many other factors apart from the above listed factors, but at the end of the day it is the students that bear the hurt of any consequences that may arise. There is need for students to demand more from their schools and teachers in order to tackle these problems.

What many students have failed to understand is the power of determination and hard work. Do you know that if you are able to pass chemistry, you can also work that hard to pass English and Mathematics?

You should not play the blaming card on anybody in as much as we all know how bad things have gone but you have to take hold of your future and set the path you wish to follow.

*Note: This article is authored by ToscanyAcademy and originally published on toscanyacademy.com*

“And beside this, giving all diligence,to your faith virtue; and to virtue knowledge; And to knowledge temperance; and to temperance patience; and to patience godliness; And to godliness brotherly kindness; and to brotherly kindness charity. For if these things be in you, and abound, they make you that ye shall neither be barren nor unfruitful in the knowledge of our Lord Jesus Christ.addBut he that lacketh these things is blind, and cannot see afar off, and hath forgotten that he was purged from his old sins. Wherefore the rather, brethren, give diligence to make your calling and election sure: for if ye do these things, ye shall never fall”(II Peter 1:5-10).

So how are you doing with “Bible Mathematics?” Have you added these attributes to your life?

**Faith** — The basis for all we believe and do (**Hebrews 11:1**). Without it, one cannot please God (**Hebrews 11:6**).

**Virtue** — The ability to always do what is right especially when others are not looking. Easier said than done (**II Samuel 11**).

**Knowledge** — The data which one desperately needs to please the Lord. It proceeds from His mouth (**Proverbs 2:6**) and is acquired through diligent study (**II Timothy 2:15**).

**Temperance** — The ability to control urges and desires. Most struggle with this now and then in various ways (**I Corinthians 7:5**).

**Patience** — The capability of “enduring” tough and difficult times. We must “endure” to the end (**Mark 13:13**).

**Godliness** — The talent of being “separate” and “different” from the world. We must “transform” ourselves (**Romans 12:1-2**).

**Brotherly Kindness** — The love shared between brethren is special. It is a bond which was never meant to be broken, and should always be cherished (**John 13:35**).

**Charity** — The sincere interest we have for all. The love which causes us to want everyone to go to Heaven. This is the love God had for mankind (**John 3:16**).

On this day, be “adding” these actions and attitudes to your life. Beyond that, be sure they are “abounding.” In so doing, you are making your “calling” and “election” sure. Be certain to do your “Bible Mathematics” today!

— lrp2

]]>In an ironic twist of fate, the word that I misspell most often is MATHEAMTICS. Sort of. It’s not that I mis*spell* it as much as I mis*type* it. For unobvious reasons, I tend to transpose the M and A in the third syllable. And statistically speaking, it stands to reason that I’d screw it up often — it’s a word I type frequently.

Spelling is important, as authors Robert Magnan and Mary Lou Santovec claim in their e-book *1001 Commonly Misspelled Words*. Spelling is so very important, in fact, they’ve taken great care in writing a description of their book for Amazon. Here’s an excerpt:

It seems that Magnan and Santovec should increase their list to 1003 words and add *memory* and *correct* (misspelled as “memeory” and “corret” in the excerpt above). And then at the end of the paragraph, it seems they’re trying to make a joke by spelling *knowledgeable* incorrectly but then correcting the spelling between the dashes; yet both instances are spelled the same — correctly, in fact — which I suspect was autocorrect fixing the deliberate misspelling before this blurb went to print.

One has to wonder if these mistakes were intentional, to keep the reader on her toes and emphasize how important spelling is. Sort of like the deliberate mistake in this proof that 1 = 2:

(For the infinitely geeky, there’s this follow-up, posted by Anders Kaseorg on Quora:

Suppose there are *n* proofs that 1 = 2. From this we derive that there are *n* + 1 – 1 = *n* + 2 – 1 = *n* + 1 proofs that 1 = 2. Therefore, by induction, there are infinitely many proofs that 1 = 2.)

But, I digress. My purpose in writing this post was to provide a list of hints for how to spell the most frequently misspelled math words. As it turns out, many of the math words that you’d think would be hard to spell — words with several syllables or lots of letters, like *parabola* or *triangle* or *differentiation* — are actually spelled exactly like they sound. Most of the really hard-to-spell math words are the names of mathematicians, like de Moivre, Poincaré, Weierstrauss, Euler, and Euclid.

The following are math words pulled from a variety of lists of commonly misspelled words.

**forth/fourth/forty** : there’s a *u* in the ordinal number, but not in the multiple of 10. To help you remember the difference, keep in mind that 40 is the largest number that, when spelled out, has all its letters in alphabetical order — and that won’t be the case if a *u* is included.

**twelfth** : there’s a little “elf” in *twelfth*, even if you incorrectly say “twelth” without the *f*.

**ninth / ninety** : fifth and fifty are parallel, in that both change *v* to *f* and drop the *e*. Sadly, ninth and ninety aren’t. Sorry, I don’t have any tips for remembering this one… except maybe that it’s on this list, which will help you remember that they’re spelled differently.

**existence** : one *i*, three *e*‘s, no *a*‘s.

**height** : the three dimensions are *length*, *width*, and *height*, not *heighth*, regardless of how my father pronounced it. There’s no *h* at the end.

**independent** : ants live in colonies, which isn’t very independent. That’ll help you remember that independent ends in –*ent*, not –*ant*.

**neighbor** : the *ei* takes on a long *a* sound, despite the *i* before *e* rule, and then there’s a silent *gh*. Yeah, lots of opportunities for goofing up this one. No one would fault you for spelling it *nayber*.

**operator** : when AT&T introduced the 1-800-OPERATOR promotion in the mid-1980’s, it was a disaster. The majority of would-be callers spelled *operator* with an *e* instead of an *o* in the last syllable.

**perseverance** : perhaps less mathy than the other words on this list, but Common Core includes it in Math Practice 1. There’s an *a* in the last syllable, and there’s no *r* before the *v*.

**principal/principle** : remember that Al wants to collect interest on his *principal* investment, but Lee likes the pigeonhole *principle*.

**similar** : another one that doesn’t have –*er* at the end.

Love the comparison with Berkeley Earth Surface Temperatures.

Originally posted on Open Mind:

Almost all of us live on land, not the ocean. And, most of us live in the northern hemisphere, not the southern. For the benefit of most of us, let’s take a closer look at how temperature has changed, in the northern hemisphere, on land.

View original 1,100 more words

The activity Unit Rates with Ratios of Fractions provides many real world opportunities for students to find unit rates when one or both quantities are fractions. A variety of tools and representations are available to help students with this concept: a walking bug, double number lines, tables, and more.

The activity extends students’ conceptual understanding of unit rate and reinforces dividing both parts of the rate by the denominator to result in 1 unit in the denominator. The result– a unit rate! The activity further leads them to understand that of course dividing by the denominator is equivalent to multiplying by its reciprocal, and further leads to a shortcut–simply multiply the numerator by the reciprocal of the denominator. Students come to realize that the only difference between a unit rate with fractions and one with whole numbers is in working with the different types of numbers.

**Problem 7. **Compute

**Problem 8. **Consider all words of length in the Latin alphabet. Define the *weight *of a word as where is the number of letters not used in this word. Prove that the sum of the weight of all words is .

**Problem 9. **An complex matrix is called *t-normal *if where is the transpose of . For each , determine the maximum dimension of a linear space of complex matrices consisting of -normal matrices.

**Problem 10. **Let be a positive integer and let be a polynomial of degree with integer coefficients. Prove that

- Average= Sum of observations/ Number of observations
- Average Speed: Suppose a man covers a certain distance at x kmph and an equal distance at y kmph.

Then, the average speed during the whole journey is 2xy/(x+y) kmph

*Mean, Median, Mode, and Range*

Mean, median, and mode are three kinds of “averages”.

**Mean:** The mathematical average of a set of numbers.

Ex: The average of 2,6,5,4,9 is (2+6+5+4+9)/5= 5.2

**Median:** The median is the middle value. To find the median, the numbers have to be listed in numerical order.

Ex: To find the median of the previous set of numbers, we have to write them in order: 2,4,5,6,9. Among the 5 numbers, the middle one will be (5+1)/2= 3^{rd}number, which is 5.

Let’s take another set of numbers: 6,8,11,15,18,20. Here the median is (6+1)/2=3.5. It means that to find the median we have to find the average of the 3^{rd} and 4^{th} number: (11+15)/2= 13. So the median is 13.

**Mode:** The mode is the number that is repeated more often than any other in a given set of numbers.

Consider the following set: 3,3,6,7,8,10,7,4,3.

In this set, “3” appears the maximum number of times. So “3” is the mode.

**Range:** The “range” is just the difference between the largest and smallest values.

3, 18, 13, 14, 13, 16, 14, 21, 13

Here, the largest value is 21 and the smallest value is 3. So, range= 21-3= 18.

*Sample Problems*

- The average of first five multiples of 5 is:
- 3
- 20
- 15
- 10
- 5

- In his geometry class, Ben took 6 exams. His scores were 82, 82, 81, 85, 77, and 97. What was his average score on the exams?
- 86
- 80
- 81
- 84
- 95

- Jessica has taken 4 exams and her average score so far is 79. If she gets 100, a perfect score, on the remaining 2 exams, what will her new average be?
- 86
- 80
- 78
- 84
- 74

- The average of 50 numbers is 30. If two numbers, 35 and 40 are discarded, then the average of the remaining numbers is nearly:
- 56
- 54
- 68
- 56
- None of these

- The average score of a cricketer for ten matches is 37.5 runs. If the average for the first seven matches is 40.2, then find the average for the last three matches.

a. 30.3

b. 32

c. 31

d. 31.2

e. 33.3 - There were 60 students in a hostel. Due to the admission of 4 new students the expenses of the mess were increased by 51 per day while the average expenditure per head diminished by Tk. 2. What was the original expenditure of the mess?
- 2700
- 2685
- 2635
- 2895
- Insufficient data

- Sadaf’s marks were wrongly entered as 75 instead of 57. Due to that the average marks for the class got increased by half (1/2). The number of students in the class is:
- 41
- 36
- 40
- 70
- 62

- The average weight of 10 boys in a class is 60.25 kg and that of the remaining 5 boys is 50.5 kg. Find the average weights of all the boys in the class.
- 55 kg
- 48 kg
- 57 kg
- 49 kg
- 53 kg

- If the average marks of three batches of 55, 60 and 45 students respectively is 50, 55, 62, then the average marks of all the students is:
- 53.5
- 59
- 62
- 52.35
- 55.25

- The average age of husband, wife and their child 3 years ago was 28 years and that of wife and the child 5 years ago was 25 years. The present age of the husband is:
- 35 years
- 33years
- 50 years
- 55 years
- 30 years

*Answers*

- C

Explanation:

The first five multiples of 5 are 5,10,15,20,25. Their average= (5+10+15+20+25)/5= 15 - D

Explanation:

Average= Sum of observations/ Number of observations = - A

Explanation:

Total number= 79*4+100*2= 516

Average =516/6= 86

- C

Explanation:

Total sum of 48 numbers = (50 × 30) – (35 +40)

= 1500 – 75

= 1425

Average = 1425/48 = 29.68

- D

Explanation:

Score of last three matches= (37.5*10)- (40.2*7)= 93.6

Average = 93.6/3 =31.2 - B

Explanation:

Let, per head expenditure before= x, after (x-2)

New daily expenditure= 65*(x-2)

Previous daily expenditure= 60*x

64*(x-2)-60x= 51

>4x-64=51

>x= 115/5 =44.75

Original daily expenditure= 23* 60= 2685 - B

Explanation:

Difference between sum of numbers = 75-57=18

18/x=1/2

>x=18*2=36 - C

Explanation:

Average = (10*60.25+5*50.5)/15= 57 kg - E

Explanation:

Average = (55*50+60*55+45*62)/ (55+60+45) = 8840/ 160 = 55.25

- B

Explanation:

Sum of the present ages of husband, wife and child = (28 x 3 + 3 x 3) years = 93 years.

Sum of the present ages of wife and child = (25 x 2 + 5 x 2) years = 60 years.

So, Husband’s present age = (93 – 60) years = 33 years.

Revise lesson 4 of Maths for class test on Monday.

]]>Share this Rating. Title: A Beautiful Mind (2001) 8.2 10. Want to share IMDb’s rating on your own site? Use the HTML below. 21122011 Vidéo incorporée This feature is not available right now. Please try again later. Directed by Michel Gondry. With Jim Carrey, Kate Winslet, Gerry Robert Byrne, Elijah Wood. When their relationship turns sour, a couple undergoes a procedure to have. WITH A NEW PREFACE BY THE AUTHOR In her bestselling classic, An Unquiet Mind, Kay Redfield Jamison changed the way we think about. Image (ĭm′ĭj) n. 1. a. A representation of the form of a person or object, such as a painting or photograph. b. A sculptured likeness. 2. Physics An optically. The number π is a mathematical constant, the ratio of a circle’s circumference to its diameter, commonly approximated as 3.14159. It has been represented by the. In aesthetics, the sublime (from the Latin sublīmis) is the quality of greatness, whether physical, moral, intellectual, metaphysical, aesthetic, spiritual, or artistic. Get Lost in the Internet’s Mind-Bending, Math-Inspired Art That enigmatic smile, drawn here with a single line. This illustration, commissioned by Bill Cook at. Worksheets for teachers and homeschoolers. Over 2000 free worksheets and growing! Our sponsors help to keep the worksheets free! Math worksheets, Dolch words,. Bienvenue au Carrefour Dentaire de Montréal. Avez-vous une urgence dentaire ou cherchez-vous un dentiste de confiance? Le Carrefour Dentaire de Montréal a tout ce.

]]>Complete Ex5.9 as discussed in class.Bring your Text book and NB2 tomorrow for Maths Class.

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Mark asked how we know the redshift of the CMB if it has no emission or absorption lines, which is the usual way to determine redshifts of e.g. stars and galaxies. I decided that the answer deserves its own blogpost – so here it is.

As I explain in more detail in my book on the CMB, the origin of the CMB is from the time that the Universe had cooled enough so that hydrogen atoms could form from the sea of protons and electrons that existed in the early Universe. Prior to when the CMB was “created”, the temperature was too high for hydrogen atoms to exist; electrons were prevented from combining with protons to form atoms because the energy of the photons in the Universe’s radiation (given by where is the frequency) and of the thermal energy of the electrons was high enough to ionise any hydrogen atoms that did form. But, as the Universe expanded it cooled.

In fact, the relationship between the Universe’s size and its temperature is very simple; if represents the size of the Universe at time , then the temperature at time is just given by

This means that, as the Universe expands, the temperature just decreases in inverse proportion to its size. Double the size of the Universe, and the temperature will halve.

When the Universe had cooled to about 3,000K it was cool enough for the electrons to finally combine with the protons and form neutral hydrogen. At this temperature the photons were not energetic enough to ionise any hydrogen atoms, and the electrons had lost enough thermal energy that they too could not ionise electrons bound to protons. Finally, for the first time in the Universe’s history, neutral hydrogen atoms could form.

For reasons that I have never properly understood, astronomers and cosmologists tend to call this event *recombination*, although really it was *combination*, without the *‘re’* as it was happening for the first time. A term I prefer more is *decoupling*, it is when matter and radiation in the Universe decoupled, and the radiation was free to travel through the Universe. Before decoupling, the photons could not travel very far before they scattered off free electrons; after decoupling they were free to travel and this is the radiation we see as the CMB.

It was shown by Richard Tolman in 1934 in a book entitled *Relativity, Thermodynamics, and Cosmology* that a blackbody will retain its blackbody spectrum as the Universe expands; so the blackbody produced at the time of decoupling will have retained its blackbody spectrum through to the current epoch. But, because the Universe has expanded, the peak of the spectrum will have been stretched by the expansion of space (so it is not correct to think of the CMB spectrum as having cooled down, rather than space has expanded and stretched its peak emission to a lower temperature). The peak of a blackbody spectrum is related to its temperature in a very precise way, it is given by Wien’s displacement law, which I blogged about here.

In 1990 the FIRAS instrument on the NASA satellite COBE (COsmic Background Explorer) measured the spectrum of the CMB to high precision, and found it to be currently at a temperature of (as an aside, the spectrum measured by FIRAS was the most perfect blackbody spectrum ever observed in nature).

It is thus easy to calculate the current redshift of the CMB, it is given by

and “voilà”, that is the redshift of the CMB. Simples ;)

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