Maths\Facts\Tricks\Trivia\Fun.Numbers Fun Facts
Magic 1089
Here's a cool mathematical magic trick. Write down a three-digit number whose digits are decreasing. Then reverse the digits to create a new number, and subtract this number from the original number. With the resulting number, add it to the reverse of itself. The number you will get is 1089!
For example, if you start with 532 (three digits, decreasing order), then the reverse is 235. Subtract 532-235 to get 297. Now add 297 and its reverse 792, and you will get 1089!
Presentation Suggestions:
You might ask your students to see if they can explain this magic trick using a little algebra.
The Math Behind the Fact:
If we let a, b, c denote the three digits of the original number, then the three-digit number is 100a+10b+c. The reverse is 100c+10b+a. Subtract: (100a+10b+c)-(100c+10b+a) to get 99(a-c). Since the digits were decreasing, (a-c) is at least 2 and no greater than 8, so the result must be one of 198, 297, 396, 495, 594, 693, 792, or 891. When you add any one of those numbers to the reverse of itself, you get 1089!
Mind-Reading Number Trick
Think of a number, any positive integer (but keep it small so you can do computations in your head).
1. Square it.
2. Add the result to your original number.
3. Divide by your original number.
4. Add, oh I don't know, say 17.
5. Subtract your original number.
6. Divide by 6.
The number you are thinking of now is 3!
How did I do this?
Presentation Suggestions:
Ham it up with magician's patter. Step 4 could be anything you want---someone's age, or their favorite number--- just ask the crowd for suggestions. (This will change the final outcome of Step 6, but see below for how.)
The Math Behind the Fact:
Clearly no matter what you start with, the answer should come out the same (zero wasn't allowed because of Step 3). We can see why this trick works by using a little bit of high school algebra! If you follow the instructions starting with the variable X instead of an actual number, you will see that X is eliminated by Step 5.
Using this idea, you can make up your own mental math trick right on the spot! (Just don't do anything too obvious, like tell people to add 5, subtract their original number, and say "the number you are thinking of is 5".)
Red-Black Pairs Card Trick
Here's a terrific mathematical card trick that will impress your friends. When you do this trick, the effect of the card trick will look like this:
You have a deck of cards, and you ask for a volunteer who knows how to do a riffle shuffle. You then cut the deck and then give the volunteer the halves of the deck and ask him to do one riffle shuffle and return the deck to you. Now say "There's no way I could know anything about the deck right now, right? Well, I was born with the amazing ability to feel the redness and blackness of cards with my fingertips. However, my talent is not that refined. I can only feel red and black cards in pairs." As you say this, put the deck of cards behind your back (so that you cannot see them) and then, at regular intervals, you fish around in the deck and pull out pairs of cards and show them to the audience. These pairs will all have exactly one black and one red card!
Presentation Suggestions:
Before performing the trick, order the deck alternating colors, all the way through, red-black-red-black-... etc. (When you flash the deck before their eyes, they really won't notice this pattern if you do it quickly.)
After this, there is really only one thing you need to remember to ensure that the trick works: you must cut the deck (not the spectator), and you must do it in such a way that the bottom of each half of the deck is a different color. Then, no matter how the spectator riffle shuffles the deck, the cards will always drop in red-black or black-red pairs. See below for explanation.
Then, all you have to do after the deck is returned and you put it behind your back is to pull out the top 2 cards. It will be either red-black or black-red! Then pull out the next 2 cards, which again will be red-black or black-red. You can continue in this fashion to the end of the deck, if you like!
Of course, you should make it look as if you are trying really hard to find the cards (even though what you are really doing is very easy). Spectators will wonder if you are pulling one card off the top and one card off the bottom; but you can pull the deck out and show them that this is not the case.
The Math Behind the Fact:
The reason the trick works at the point of the riffle shuffle is both simple and stunning: if you cut the deck so that the cards at the bottom of each half are different colors, then the first card that gets "dropped" in the shuffle will be a different color then the second card that gets dropped, no matter which half of the deck they come from. As an example, if the first card that gets dropped is black, then after that both halves will have red cards at the bottom, so no matter which card falls next it will be red! After this, both halves again have different colored cards at bottom and we are back to the situation at start. So all the cards will fall off in either red-black or black-red pairs.
The message of this trick is that one shuffle is not enough to randomize a deck of cards-- you really can know something about the deck after one shuffle... but only if you stack the deck in a particular way first!
Multiply 37,037 by any single number (1-9), then multiply that number by 3. Every digit in the answer will be the same as that first single number. For example: 37,037*5=185,185. 185,185*3=555,555.
If you multiply 111,111,111 by 111,111,111 you get 12,345,678,987,654,321.
Pi has been calculated to 2,260,321,363 digits. The billionth digit in Pi is 9.
Here is a simple method of telling someone their age, the trick is not difficult to do or solve but it's quick and good fun!.
1) Hand the punter a calculator (or pen & paper).
2) Mentally estimate their age but don't tell them.
3) Ask them to enter their age into the calculator (without you seeing of course).
4) Tell them that "as there are 12 months in the year, add 12 to their age".
5) Say that "as there are 7 days in a week, and 52 weeks in a year, add 752 to the number already in the calculator".
6) Ask them to tell you the last digit.
7) All you do is subtract 4 from this number and that is the last digit of their age.
8) If the number they tell you is less than 4, just add 10 before you subtract the 4 (e.g. they say 3, you add 10 then take 3 = 9).
9) You have already estimated their age so just knowing the exact last digit is usually enough to accurately be able to tell them their age.
e.g. you think they look 25.
Their sum is 23+12+752=787, they tell you 7, you take off 4 which is 3, you mentally adjust their age down to 23 (i.e. they will not look either 13 or 33 so must be 23)
Math Riddles
Which weighs more? A pound of gold or a pound of feathers?
Both weigh the same.
How is the moon like a dollar?
They both have 4 quarters.
What is alive and has only 1 foot?
A leg.
When do giraffes have 8 feet?
When there's two of them.
How many eggs can you put in an empty basket?
Only one, after that the basket is not empty.
What coin doubles in value when half is deducted?
A half dollar.
What is the difference between a new penny and an old quarter?
24 cents.
If you can buy eight eggs for 26 cents, how many can you buy for a cent and a quarter?
8.
Why should you never mention the number 288 in front of anyone?
Because it is too gross (2 x 144 - two gross).
Where can you buy a ruler that is 3 feet long?
At a yard sale.
If there were 9 cats on a bridge and one jumped over the edge, how many would be left?
None - they are copycats.
If you take three apples from five apples, how many do you have?
You have three apples.
What has 4 legs and only 1 foot?
A bed.
How many times can you subtract 6 from 30?
Once; after that it is no longer 30 (Don't try this in a test!)
If one nickel is worth five cents, how much is half of one half of a nickel worth?
$0.0125
How many 9's between 1 and 100?
20.
Which is more valuable - one pound of $10 gold coins or half a pound of $20 gold coins?
One pound is twice of half pound.
It happens once in a minute, twice in a week, and once in a year? What is it?
The letter 'e'.
How can half of 12 be 7?
Cut XII into two halves horizontally. You get VII on the top half.
When things go wrong, what can you always count on?
Your fingers.
Why are diapers like 100 dollar bills?
They need to be changed.
A street that is 40 yards long has a tree every 10 yards on both sides. How many total trees on the entire street?
10, 5 on each side.
What goes up and never comes down?
Your age.
What did one math book say to the other math book?
Wow, have I got problems!
Why is the longest human nose on record only 11 inches long?
Otherwise it would be a foot.
hope you enjoyed these math riddles.
Magic Of Maths
1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321
Some Tips and Tricks
Here are some tricks that may help you remember your times tables. Everyone thinks differently, so just ignore any tricks that don't make sense to you.
Every entry has a twin, which may be easier to remember. For example if you forget 8×5, you might remember 5×8. This way, you only have to remember half the table.
to multiply by Trick
2 add the number to itself (example 2×9 = 9+9)
5 The last digit always goes 5,0,5,0,..,
is always half of 10× (Example: 5x6 = half of 10x6 = half of 60 = 30)
is half the number times 10 (Example: 5x6 = 10x3 = 30)
6 if you multiply 6 by an even number, they both end in the same digit. Example: 6×2=12, 6×4=24, 6×6=36, etc
9 is 10× the number minus the number. Example: 9×6 = 10×6 - 6 = 60-6 = 54
The last digit always goes 9,8,7,6, ..
if you add the answer's digits together, you get 9. Example: 9×5=45 and 4+5=9. (But not with 9×11=99)
10 put a zero after it
11 up to 9x11: just repeat the digit (Example: 4x11 = 44)
for 10x11 to 18x11: write the sum of the digits between the digits (Example: 15x11 = 1(1+5)5 = 165) Note: this works for any two-digit number, but if the sum of the digits is more than 9, you will have to "carry the one".
12 is 10× plus 2×
Remembering Squares Can Help
This may not work for you, but it worked for me. I like remembering the squares (where you multiply a number by itself):
1×1=1 2×2=4 3×3=9 4×4=16 5×5=25 6×6=36
7×7=49 8×8=64 9×9=81 10×10=100 11×11=121 12×12=144
And this gives me one more trick. if the numbers you are multiplying are separated by 2 (example 7 and 5), then multiply the number in the middle by itself and subtract one. See this:
5×5 = 25 is just one bigger than 6×4 = 24
6×6 = 36 is just one bigger than 7×5 = 35
7×7 = 49 is just one bigger than 8×6 = 48
8×8 = 64 is just one bigger than 9×7 = 63
etc
Always End With 1089
Add two number together and always end with 1089.
Here is How:
Pick a three digit number. The three numbers used must be different. i.e. 123 Reverse that number. 123 becomes 321
Take the smallest three digit number from the largest.
321 - 123 = 198 Take the answer and reverse that number. 198 becomes 891
Add that number to the answer of the subtraction. 891 + 198 = 1089 The answer will be 1089!
So Why does it do this?
Find out here
A Little Footnote
But I ended with 198 not 1089. This happens if you pick a number like 546. Try to pick consecutive numbers like 123, 345, 765 etc. If it does happen, it is not a problem. Just repeat the stages again starting from 198.
Percentages (%)
Percentage means parts per 100
When you say "Percent" you are really saying "per 100"
So 50% means 50 per 100
(50% of this box is green)
And 25% means 25 per 100
(25% of this box is green)
Using Percentages
Because "Percent" means "per 100" you should think "this should always be divided by 100"
So 75% really means 75/100
And 100% is 100/100, or exactly 1 (100% of any number is just the number, unchanged)
And 200% is 200/100, or exactly 2 (200% of any number is twice the number)
Use the slider on the left, and try some different numbers (example, what is 60% of 80?)
A Percentage can also be expressed as a Decimal or a Fraction
A Half can be written...
As a percentage: 50%
As a decimal: 0.5
As a fraction: 1/2
Some Worked Examples
Calculate 25% of 80
25% = 25/100 (25/100) × 80 = 20
So 25% of 80 is 20
A Skateboard is reduced 25% in price in a sale. The old price was $120. Find the new price
Find 25% of $120
25% = 25/100 (25/100) × $120 = $30
25% of $120 is $30
So the reduction is $30
Take the reduction from the original price $120 - $30 = $90
The Price of the Skateboard in the sale is $90
The Word
"Percent" comes from the latin Per Centum. The latin word Centum means 100, for example a Century is 100 years
Who Owns the Crocodile
There are 5 girls in a long row in maths.
Each girl has a favourite chocolate bar, colour, pet, hobby, and would like to go on a certain holiday.
All the girls like different things.
Your task is to solve the following clues - "who owns the crocodile"
Jo likes the Wispa Bite
The person with the hamster likes swimming
Hannah eats Dairy Milk
Jessica is on the left of Georgina
Lucy is the first on the left
The first person on the right likes swimming
The person who eats Milky Bars owns a horse
The person in the middle eats Dairy Milk
Jessica likes green
The person on the left of the middle wants to go to Tobago
The person who wants to go to the Maldives likes lilac
The person who likes Wispa Bites sits next to the person who wants to go to Florida
The person who likes pink wants to go to Florida
the person who sits first on the left likes lilac
The girl that likes blue owns a puppy
The person who likes skiing sits next to the person who has a hamster
The girl on the right of the girl who likes tennis likes horse riding
The girl next to the girl who likes Milky Bars likes Boost
The girl who likes purple wants to got to Canada
The girl who likes Crunchies owns a rabbit
The girl who likes skiing sits next to the girl who plays ten-pin bowling
Jessica wants to go to Australia
Census
A census taker approaches a house and asks the woman who answers the door "How many children do you have, and what are their ages?"
Woman: "I have three children, the product of their ages are 36, the sum of their ages are equal to the address of the house next door."
The census taker walks next door, comes back and says "I need more information."
The woman replies "I have to go, my oldest child is sleeping upstairs."
Census taker: "Thank you, I now have everything I need."
What are the ages of each of the three children?
The Problem:
A census taker approaches a house and asks the woman who answers the door,"How many children do you have, and what are their ages?"
Woman: "I have three children, the product of their ages are 36, the sum of their ages are equal to the address of the house next door."
The census taker walks next door, comes back and says, "I need more information."
The woman replies, "I have to go, my oldest child is sleeping upstairs."
Census taker: "Thank you, I now have everything I need."
What are the ages of each of the three children?
The Solution:
The reason the census taker could not figure out the children's ages is because, even with knowing the number on the house next door, there were still two possibilities.
The only way that the product could be 36 and still leave two possibilities is if the sum equals 13. These possibilities being 9, 2 and 2 and 6, 6 and 1.
When the home owner stated that her "Oldest" child is sleeping she was giving ths census taker the fact that there is an "oldest." The children's ages are therefore 9,2 and 2.
Notes
I sometimes get asked why other possible solutions are not shown. Answer: because there aren't any more! A key element of this puzzle is that the census taker needed more information!
For example, another way to factor 36 is 12, 3 and 1. The sum is 16. If the number next door had been 16, the census taker would not have needed to come back for more information.
Another factoring is 6, 3 and 2. The sum is 11. Once again, the census taker would not have needed to come back if that had been the number next door.
And nor will 4, 3 and 3 work because its sum of 10 is also unique.
Hope that makes sense
TRIPLETS/////
Triplets
Puzzles -> Logic Puzzles
Three sisters are identical triplets. The oldest by minutes is Sarah, and Sarah always tells anyone the truth. The next oldest is Sue, and Sue always will tell anyone a lie. Sally is the youngest of the three. She sometimes lies and sometimes tells the truth.
Victor, an old friend of the family's, came over one day and as usual he didn't know who was who, so he asked each of them one question.
Victor asked the sister that was sitting on the left, "Which sister is in the middle of you three?" and the answer he received was, "Oh, that's Sarah."
Victor then asked the sister in the middle, "What is your name?" The response given was, "I'm Sally."
Victor turned to the sister on the right, then asked, "Who is that in the middle?" The sister then replied, "She is Sue."
This confused Victor; he had asked the same question three times and received three different answers.
Who was who?
Triplets - Solution
Puzzles -> Logic Puzzles
The Puzzle: Three sisters are identical triplets. The oldest by minutes is Sarah, and Sarah always tells anyone the truth. The next oldest is Sue, and Sue always will tell anyone a lie. Sally is the youngest of the three. She sometimes lies and sometimes tells the truth.
Victor, an old friend of the family's, came over one day and as usual he didn't know who was who, so he asked each of them one question.
Victor asked the sister that was sitting on the left, "Which sister is in the middle of you three?" and the answer he received was, "Oh, that's Sarah."
Victor then asked the sister in the middle, "What is your name?" The response given was, "I'm Sally."
Victor turned to the sister on the right, then asked, "Who is that in the middle?" The sister then replied, "She is Sue."
This confused Victor; he had asked the same question three times and received three different answers.
Who was who?
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(Scroll down if you really want
to see the Solution ...)
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The Solution . . .
The first one cannot be Sarah, because that would make the first one a liar. The second one cannot be Sarah for the same reason. So, the third sister must be Sarah. This means the middle one is Sue and the only one left is Sally
Quotes
...beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
-St. Augustine, De Genesi ad Litteram
A mathematician is a machine for turning coffee into theorems
-''Paul Erdös
Jokes
Several old jokes common amongst the scientific disciplines illustrate the difference between the mathematical mind and that of other disciplines. One goes as follows:
An engineer, a physicist, and a mathematician are all staying at a hotel one night when a fire breaks out. The engineer wakes up and smells the smoke; he quickly grabs a garbage pail to use as a bucket, fills it with water from the bathroom, and puts out the fire in his room. He then refills the pail and douses everything flammable in the room with water. He then returns to sleep.
The physicist wakes up, smells the smoke, jumps out of bed. He picks up a pad and pencil and makes some calculations, glancing frequently at the flames. He then measures exactly 15.6 liters of water into the garbage pail, and throws it on the flames, which are extinguished. Smiling, he returns to sleep.
Finally the mathematician wakes up. He too grabs a pad and begins furiously writing; glancing at the flames; and then writing more. After a while he gets a satisfied look on his face; entering the bathroom, he produces a match, lights it, and then extinguishes it with a bit of running water. "Aha! A solution exists," he murmurs - and returns to his slumbers.
Another one involves an astrologer, a chemist, and a mathematician on a bus during their first visit to Scotland. They see a black sheep grazing alone in a pasture as they drive by.
The astrologer excitedly exclaims, "Ah, this shows Scottish sheep are black!"
The chemist didactically corrects him: "No, no, it just shows some Scottish sheep are black."
The mathematician then says, "Actually, we can only be sure there is one Scottish sheep that is half black."
A shoeseller meets a mathematician and complains that he does not know what size shoes to buy. “No problem,” says the mathematician, “there is a simple equation for that,” and he shows him the Gaussian normal distribution. The shoeseller stares some time at het equation and asks, “What is that symbol?” “That is the Greek letter pi.” “What is pi?” “That is the ratio between the circumference and the diameter of a circle.” Upon this the shoeseller cries out: “What does a circle have to do with shoes?!”
“First and above all he was a logician. At least thirty-five years of the half-century or so of his existence had been devoted exclusively to proving that two and two always equal four, except in unusual cases, where they equal three or five, as the case may be.” — Jacques Futrelle, “The Problem of Cell 13″
Most mathematicians are familiar with — or have at least seen references in the literature to — the equation 2 + 2 = 4. However, the less well known equation 2 + 2 = 5 also has a rich, complex history behind it. Like any other complex quantitiy, this history has a real part and an imaginary part; we shall deal exclusively with the latter here.
Many cultures, in their early mathematical development, discovered the equation 2 + 2 = 5. For example, consider the Bolb tribe, descended from the Incas of South America. The Bolbs counted by tying knots in ropes. They quickly realized that when a 2-knot rope is put together with another 2-knot rope, a 5-knot rope results.
Recent findings indicate that the Pythagorean Brotherhood discovered a proof that 2 + 2 = 5, but the proof never got written up. Contrary to what one might expect, the proof’s nonappearance was not caused by a cover-up such as the Pythagoreans attempted with the irrationality of the square root of two. Rather, they simply could not pay for the necessary scribe service. They had lost their grant money due to the protests of an oxen-rights activist who objected to the Brotherhood’s method of celebrating the discovery of theorems. Thus it was that only the equation 2 + 2 = 4 was used in Euclid’s “Elements,” and nothing more was heard of 2 + 2 = 5 for several centuries.
Around A.D. 1200 Leonardo of Pisa (Fibonacci) discovered that a few weeks after putting 2 male rabbits plus 2 female rabbits in the same cage, he ended up with considerably more than 4 rabbits. Fearing that too strong a challenge to the value 4 given in Euclid would meet with opposition, Leonardo conservatively stated, “2 + 2 is more like 5 than 4.” Even this cautious rendition of his data was roundly condemned and earned Leonardo the nickname “Blockhead.” By the way, his practice of underestimating the number of rabbits persisted; his celebrated model of rabbit populations had each birth consisting of only two babies, a gross underestimate if ever there was one.
Some 400 years later, the thread was picked up once more, this time by the French mathematicians. Descartes announced, “I think 2 + 2 = 5; therefore it does.” However, others objected that his argument was somewhat less than totally rigorous. Apparently, Fermat had a more rigorous proof which was to appear as part of a book, but it and other material were cut by the editor so that the book could be printed with wider margins.
Between the fact that no definitive proof of 2 + 2 = 5 was available and the excitement of the development of calculus, by 1700 mathematicians had again lost interest in the equation. In fact, the only known 18th-century reference to 2 + 2 = 5 is due to the philosopher Bishop Berkeley who, upon discovering it in an old manuscript, wryly commented, “Well, now I know where all the departed quantities went to — the right-hand side of this equation.” That witticism so impressed California intellectuals that they named a university town after him.
But in the early to middle 1800’s, 2 + 2 began to take on great significance. Riemann developed an arithmetic in which 2 + 2 = 5, paralleling the Euclidean 2 + 2 = 4 arithmetic. Moreover, during this period Gauss produced an arithmetic in which 2 + 2 = 3. Naturally, there ensued decades of great confusion as to the actual value of 2 + 2. Because of changing opinions on this topic, Kempe’s proof in 1880 of the 4-color theorem was deemed 11 years later to yield, instead, the 5-color theorem. Dedekind entered the debate with an article entitled “Was ist und was soll 2 + 2?”
Frege thought he had settled the question while preparing a condensed version of his “Begriffsschrift.” This condensation, entitled “Die Kleine Begriffsschrift (The Short Schrift),” contained what he considered to be a definitive proof of 2 + 2 = 5. But then Frege received a letter from Bertrand Russell, reminding him that in “Grundbeefen der Mathematik” Frege had proved that 2 + 2 = 4. This contradiction so discouraged Frege that he abandoned mathematics altogether and went into university administration.
Faced with this profound and bewildering foundational question of the value of 2 + 2, mathematicians followed the reasonable course of action: they just ignored the whole thing. And so everyone reverted to 2 + 2 = 4 with nothing being done with its rival equation during the 20th century. There had been rumors that Bourbaki was planning to devote a volume to 2 + 2 = 5 (the first forty pages taken up by the symbolic expression for the number five), but those rumor remained unconfirmed. Recently, though, there have been reported computer-assisted proofs that 2 + 2 = 5, typically involving computers belonging to utility companies. Perhaps the 21st century will see yet another revival of this historic equation.
REFRIGERATE ELEPHANTS
Analysis:
1. Differentiate it and put into the refrig. Then integrate it in the refrig.
2. Redefine the measure on the referigerator (or the elephant).
3. Apply the Banach-Tarsky theorem.
Number theory:
1. First factorize, second multiply.
2. Use induction. You can always squeeze a bit more in.
Algebra:
1. Step 1. Show that the parts of it can be put into the refrig. Step 2. Show that the refrig. is closed under the addition.
2. Take the appropriate universal refrigerator and get a surjection from refrigerator to elephant.
Topology:
1. Have it swallow the refrig. and turn inside out.
2. Make a refrig. with the Klein bottle.
3. The elephant is homeomorphic to a smaller elephant.
4. The elephant is compact, so it can be put into a finite collection of refrigerators. That’s usually good enough.
5. The property of being inside the referigerator is hereditary. So, take the elephant’s mother, cremate it, and show that the ashes fit inside the refrigerator.
6. For those who object to method 3 because it’s cruel to animals. Put the elephant’s BABY in the refrigerator.
Algebraic topology:
Replace the interior of the refrigerator by its universal cover, R^3.
Linear algebra:
1. Put just its basis and span it in the refrig.
2. Show that 1% of the elephant will fit inside the refrigerator. By linearity, x% will fit for any x.
Affine geometry:
There is an affine transformation putting the elephant into the refrigerator.
Set theory:
1. It’s very easy! Refrigerator = { elephant } 2) The elephant and the interior of the refrigerator both have cardinality c.
Geometry:
Declare the following:
Axiom 1. An elephant can be put into a refrigerator.
Complex analysis:
Put the refrig. at the origin and the elephant outside the unit circle. Then get the image under the inversion.
Numerical analysis:
1. Put just its trunk and refer the rest to the error term.
2. Work it out using the Pentium.
Statistics:
1. Bright statistician. Put its tail as a sample and say “Done.”
2. Dull statistician. Repeat the experiment pushing the elephant to the refrig.
3. Our NEW study shows that you CAN’T put the elephant in the refrigerator.
Theorem: 4 = 5
Proof:
-20 = -20
16 - 36 = 25 - 45
4^2 - 9*4 = 5^2 - 9*5
4^2 - 9*4 + 81/4 = 5^2 - 9*5 + 81/4
(4 - 9/2)^2 = (5 - 9/2)^2
4 - 9/2 = 5 - 9/2
4 = 5
Theorem: 3=4
Proof:
Suppose:
a + b = c
This can also be written as:
4a - 3a + 4b - 3b = 4c - 3c
After reorganizing:
4a + 4b - 4c = 3a + 3b - 3c
Take the constants out of the brackets:
4 * (a+b-c) = 3 * (a+b-c)
Remove the same term left and right:
4 = 3
Mathematics and Mathematicians
"Infinity is a fathomless gulf into which all things vanish"
- Marcus Aurelius
The number 10 is used as a convenient base to count with, but the Gauls of ancient France, the Mayas of Central America, and other peoples used a base of 20. The Sumerians, the Babylonians, and others after them used a base of 60—convenient because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. The use of base 60 survives in the division of hours into minutes and minutes into seconds, and the division of the circle into 360 (60 × 6) degrees
The earliest known unit of length was used around 2,300 B.C. by megalithic tomb builders in ancient Britain. The name of the unit is not known, but its length was about 2.72 feet
In the oldest surviving work about mathematics (the Rhind papyrus, written by the ancient Egyptian scribe Ahmes around 1650 B.C.), there is a section on arithmetic headed "Directions for Knowing all Dark Things".
Euclid is the most successful textbook writer of all time. His Elements, written around 300 B.C., has gone through more than 1,000 editions since the invention of printing
Syracuse's leading citizen in the third century B.C. was the greatest scientist and mathematician of ancient times, Archimedes, nine of whose famous treaties on geometry and hydrostatics survive. When the Roman consul Marcellus conquered Syracuse, he instructed his men that Archimedes was not to be harmed. But Archimedes was run through by a sword when he begged a Roman soldier not to destroy geometrical figures he had drawn in the sand.
The modern decimal position system, in which the placing of numerals indicates their value (units, tens, hundreds etc.), was the invention of the Hindus, around 800 A.D. Their invention of the sign for zero greatly simplified arithmetic computation. By comparison, the Roman numeral system containing no zero was awkward.
A billion in America is different from a billion in Great Britain. An American billion is a thousand million (1,000,000,000), but a British billion is a million million (1,000,000,000,000). Most of the other names for large numbers are different in the U.S. and the U.K. I typically use the American names in this site
King Richard I the Lion-Hearted passed the first law requiring standards for length and volume. These standards were made from iron and were kept by sherrifs and magistrates
Gerbert of Aurillac, who became Pope Sylvester II (999-1003), attempted to introduce Arabic numerals into Christian Europe. These numerals facilitate calculations much easier than the Roman numerals in use at the time, but their use never caught on for a few more centuries.
Leonhard Euler (1707-1783) was probably the most productive mathematician of all time, publishing enough mathematical papers to fill 90 volumes of books. Even though he became completely blind at the age of 60, he still published over 400 mathematical papers, most of which he dictated to a servant untrained in mathematics.
A theory put forward by Polish mathematicians Steven Banach and Alfred Tarski in the early 20th century states that there is a way of dividing a sphere into separate parts and rearrange them so that they fill all of a larger sphere without leaving any gaps.
The ounce is most familiar as a unit of weight equalling 28.35 grams (avoirdupois measure), but in troy measure, it equals 31.1 grams. As a unit of length, an ounce equals 0.016 inches, and in volume it equals 2/3 of a whiskey jigger.
The number 10 is used as a convenient base to count with, but the Gauls of ancient France, the Mayas of Central America, and other peoples used a base of 20. The Sumerians, the Babylonians, and others after them used a base of 60—convenient because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. The use of base 60 survives in the division of hours into minutes and minutes into seconds, and the division of the circle into 360 (60 × 6) degrees
In 1266, Henry III of England decreed that "an English penny, called a sterling, round and without any clippings, shall weight 32 wheatcorns in the middle of the ear. Twenty pence do make an ounce, and 12 ounces a pound". The English kings used troy (named for the French town Troyes) weight for currency measurements, and in 1527 it became the legal standard for minting coins.
A pound of feathers weighs more than a pound of gold. Precious metals like gold are weighed using troy weights, in which a pound consists of 12 ounces (5,760 grains). Objects such as feathers are weighed using avoirdupois weights, in which a pound consists of 16 ounces or 7,000 grains.
In 1852, the first official calculation of the height of Mount Everest was made. Six measurements were made, and all of them were different. When they averaged the six results (between 28,990 feet and 29,026 feet), the result was 29,000 feet exactly. Unwilling to publish a figure that looked like a mere estimate, the people who made the calculation arbitrarily added 2 feet to the value, giving a value of 29,002 feet
A "light year" is a measure of distance, not time. It is the distance that light travels in a year and is equal to about 9.5 trillion kilometres, or about 6 trillion miles.
The measurement of the statutory mile (as opposed to the nautical mile, which is slightly longer) derives from ancient Rome. As walking was then the primary mode of transportation, the Romans measured distances in paces, which were about five feet. So milia passsum, 1,000 paces, measured 5,000 feet. However, in 1575 the British Parliament added 280 feet to this measurement, declaring the mile to be 1,760 yards or 5,280 feet, so that it could be divided evenly into furlongs. One furlong is 660 feet long, giving 8 furlongs to the mile.
Until 1959, when the avoirdupois pound was established as 453.59237 grams, the United States and Great Britain had different values for the pound. The American version was 453.5924277 grams and the British pound 453.592338 grams. Similarly, the British and American inches differ slightly (with the American inch 2.540005 centimetres, and the British inch 2.539998 centimetres).
The carat was derived from the weight of a seed of the carob tree.
Statistics
"There are three kinds of lies: lies, damned lies, and statistics."
—Benjamin Disraeli
There are 318,979,564,000 ways of playing the first four turns in a game of chess. There are 169,518,829,100,544,000,000,000,000,000 ways of playing the first ten turns of a game of chess.
In the year 10,000 B.C., there were only 5 million people on earth. In 8,000 B.C., this figure had risen to only 8 million.
Currently there are more than 6 billion people on earth. Around 1900 there were only 1.6 billion people.
Girolamo Cardano, a sixteenth-century Italian physician and mathematician, asserted that each face of a die will turn up exactly once in any given six rolls, despite the fact that he was a notorious gambler and should have observed quite the opposite at the gambling tables.
The total number of different bridge hands possible is roughly 54 octillion, or 54,000,000,000,000,000,000,000,000,000
On August 18, 1913, on an unbiased roulette wheel at Monte Carlo, evens came up 26 times in a row. The probability of this occuring is 1 in 136,823,184.
The typical person breathes 370,000 cubic metres of air in their lifetime.
5.25 billion of the world's 6 billion people live north of the Equator.
In 1985, NASA estimated that the probability of an accident occurring to the space shuttle was 1 in 100,000. However, on January 28, 1986, only the 25th shuttle launch, Challenger exploded after take-off, killing all seven astronauts aboard, and on February 1, 2003, the 113rd mission, Columbia exploded on re-entry, again killing all seven astronauts. Earlier estimates by other groups had estimated the probability as being closer to 1 in 100, a probability that seems more reasonable.
Half of those who are killed by bombs are the people who were trying to make or set the bombs
The chances of winning a lottery in which six numbers are drawn from 49 is 1 in 13,983,816.
100 people a year choke to death on ball-point pens.
In 1662, John Graunt, a London merchant, published the first set of actuarial tables in his book Observations on the Bills of Mortality. In his list of deaths in London in 1632, seven people are listed as being murdered, 10 people as dying of cancer, and no mention is made of heart ailments. On the other hand, 13 people are listed as dying of "planet", 38 from "king's evil", and 98 from "rising of the lights". The saddest statistic, however, is that out of 9,535 deaths that year, 2,268 of them were of infants.
A 1947 study found that during the Second World War, only about 15 to 25 percent of the American infantry ever fired their rifles in combat.
Q: Why did the mathematician name his dog "Cauchy"?
A: Because he left a residue at every pole.
Q: What's sado-masochism?
A: The standard deviation of the mean.
Q: What do you get when you cross an elephant and a grape?
A: I dunno, but its magnitude is Elephant, grape, sin theta.
Q: What do you get when you cross a mountain climber with a grape?
A: You can't. A mountain climber is a scaler.
Q: What do farmers study in trigonometry?
A: Swine and cow-swine.
Q: What's the contour integral around Western Europe?
A: Zero, because all the Poles are in Eastern Europe.
Q: How many numerical analysts does it take to screw in a light bulb?
A: 0.9973 after the first three iterations.
Q: How many statisticians does it take to change a lightbulb?
A: Two plus or minus three.
Q: How many applied mathematicians does it take to screw in a lightbulb?
A: One, who gives it to two statisticians, thereby reducing it to an earlier riddle.
Q: How many topologists does it take to change a light bulb?
A: It really doesn't matter, since they'd rather knot.
The Most Pathetic Fact in Maths
If you multiply 1089 x 9 you get 9801. It's reversed itself! This also works with 10989 or 109989 or 1099989 and so on.
1 is the only positive whole number that you can add to 1,000,000 and you get an answer that's bigger than if you multiply it by 1,000,000
19 = 1 x 9 + 1 + 9 and 29 = 2 x 9 + 2 + 9. This also works for 39,49,59,69,79,89 and 99.
153, 370, 371 and 407 are all the "sum of the cubes of their digits". In other words 153=13+53+33
If you divide any square number by 8 you get a remainder of 0, 1 or 4.
2 is the only number that gives the same result added to itself as it does times by itself
If you multiply 21978 by 4 it turns backwards
There are 12,988,816 different ways to cover a chess board with 32 dominoes.
Sixty nine squared = 692 = 4761 and sixty nine cubed = 693 = 328509. These two answers use all the digits from 0 to 9 between them.
You can chop a big lump of cheese into a maximum of 93 bits with 8 straight cuts
In the English language "forty" is the only number that has all its letters in alphabetical order.
1 ÷ 37 = 0·027027027... and 1 ÷ 27 = 0·037037037...
132 = 169 and if you write both numbers backwards you get 312 = 961.
This also works with 12 because 122 = 144 and 212 =441.
1/1089 = 0·00091827364554637281... (And the numbers in the 9 times table are 9,18,27,36...)
8 is the only cube that is 1 less than a square.
To multiply 10,112,359,550,561,797,752,808,988,764,044,943,820,224,719 by 9 you just move the 9 at the very end up to the front. It's the only number that does this. (Thank goodness!)
The number FOUR is the only number in the English language that is spelt with the same number of letters as the number itself
1x9 +2 = 11 , 12x9 +3 = 111 , 123x9+4 = 1111 and so on.
Surface Area of a Sphere
The area of a disk enclosed by a circle of radius R is Pi*R2.
The formula for the circumference of a circle of radius R is 2*Pi*R.
A simple calculus check reveals that the latter is the derivative of the former with respect to R.
Similarly, the volume of a ball enclosed by a sphere of radius R is (4/3)*Pi*R3.
And the formula for the surface area of a sphere of radius R is 4*Pi*R2.
And, you can check that the latter is the derivative of the former with respect to R.
Coincidence, or is there a reason?
Presentation Suggestions:
Let your students tell you those geometry formulas if they remember them.
The Math Behind the Fact:
Well, no, it is not a coincidence. For the ball, a small change in radius produces a change in volume of the ball which is equal to the volume of a spherical shell of radius R and thickness (delta R). The spherical shell's volume is thus approximately (surface area of the sphere)*(delta R). But the derivative is approximately the change in ball volume divided by (delta R), which is thus just (surface area of the sphere).
So, if I tell you the 4-dimensional "volume" of the 4-dimensional ball is (1/2)*Pi2*R4, what is 3-dimensional volume of its boundary?
Volume of a Ball in N Dimensions
The unit ball in Rn is defined as the set of points (x1,...,xn) such that
x12 + ... + xn2 <= 1.
What is the volume of the unit ball in various dimensions? Let's investigate:
The 1-dimensional volume (i.e., length) of the 1-dimensional ball (the interval [-1,1]) is 2.
The 2-dimensional volume (i.e., area) of the unit disc in the plane is Pi.
The 3-dimensional volume of the unit ball in R3 is 4/3 Pi.
The "volume" of the unit ball in R4 is (Pi/2) * Pi.
Apparently, as the dimension increases, so does the volume of the unit ball.
What does this volume tend to as the dimension tends to infinity?
Intuitively, one may think that in higher and higher dimensions there's more and more "room" in the unit ball, allowing its volume to become larger and larger. Does the volume become infinite, or does it approach a sufficiently large constant as the dimension increases?
The answer is surprising and shows how our intuition is often misleading. Using multivariable calculus one can calculate the volume of the unit ball in Rn to be
V(n) = Pin/2 / Gamma(n/2 + 1),
where Gamma is the Gamma function that generalizes the factorial function (i.e., Gamma(z+1) = z!). For n even, say n=2k, the volume of the unit ball is thus given by
V(n) = Pik/k!.
Since k! tends to infinity faster than Pik, it follows that V(n) tends to 0 as n tends to infinity!
In higher and higher dimensions you can fit less and less stuff into the unit ball. Of course, by stuff we mean n-dimensional stuff, since the unit ball in Rn always contains all the lower dimensional unit balls!
Presentation Suggestions:
Try computing the volume of the unit ball in R3 and R4 using multivariable calculus. Then using a computer algebra package plot V(n) using the formula above. What dimension seems to have the maximal volume? Now plot V(n)1/n. Explain. Explore these same ideas with the surface area. See also Surface Area of a Sphere.
The Math Behind the Fact:
One may work with the formula for V(n) by applying Stirling's Formula, which approximates Gamma(x+1) by xx e-x (2 Pi x)1/2 for large x, to see why the surprising fact above is true.
Another heuristic is the following probabilistic argument. Pick n points independently and identically distributed (i.i.d.) from a uniform distribution in [-1,1], and form an n-tuple out of these numbers. The resulting vector represents a point picked randomly out of the unit box B=[-1,1]n, so the probability that such a point is in the unit n-ball is the ratio R(n) of the volume V(n) to the volume of the unit box, which is 2n.
Notice that if there are just two coordinates of this point that are greater than 1/Sqrt[2], then the point cannot be in the unit n-ball. As n grows, we choose more and more coordinates i.i.d. from the uniform distribution, and the smaller the probability is that just zero or one of those n coordinates are bigger than 1/Sqrt[2]. A little thought reveals that for large n, this probability decreases by about 1/Sqrt[2] for each new coordinate that is chosen. This shows that the ratio R(n) tends to 0 as n goes to infinity.
However, we hope to show that V(n)=2nR(n) tends to 0 as n goes to infinity. A refinement of the above argument will do the trick: if there are just five coordinates of this point that are greater than 1/Sqrt[5], then the point cannot be in the unit n-ball. For large n, as each new coordinate chosen, the probability than less than five coordinates are bigger than 1/Sqrt[5] drops by about 1/Sqrt[5]. So V(n) changes by about a factor 1/Sqrt[5] as n is incremented, for large n. On the other hand, the factor 2n changes by a factor of 2 as n is incremented, for large n. Hence 2n changes by a factor of 2/Sqrt[5] for large enough n, so whatever this quantity is, it eventually gets smaller and smaller
Two Envelopes Paradox
I have two envelopes, and inside each I have put some money. In fact, one envelope contains twice as much money as the other.
I'll let you select one envelope, which you can have after the game is over. But as soon as you select one, I offer you the option to switch envelopes. Should you switch?
You reason as follows: My envelope has $x, and with probability 1/2 the other envelope has either $x/2 or $2x dollars. Thus the expected value of the other envelope is (1/2)($x/2) + (1/2)($2x) which is $1.25x. This is greater than the $x in my current envelope. Therefore I should switch envelopes...
But if you do switch, a similar argument would instruct you to switch back... and therefore keep switching! What's going on here? Is there a flaw in the reasoning?
Presentation Suggestions:
You can make this paradox concrete by bringing two envelopes to class and put some (unknown amount of) money in them, one twice as much as the other.
The Math Behind the Fact:
The expected value calculation is flawed because the conditioning on the relative value $x is incorrect. You need to have some idea of what the prior distribution of money in the envelope is before you can do the calculation. For instance, if you knew that the two amounts were $5 and $10, then if you took the $5 envelope (i.e., if x=5), there is NO chance that the amount in the other envelope is $x/2; it must be $2x=$10. Similarly, if you took the $10 envelope, the other envelope must be $5. So conditioning on whether you took the $5 or $10 envelope, the expected value of the other envelope is actually (1/2)(10)+(1/2)(5)=7.5. However, if you observe what's in your envelope, then you can condition on what you see; the expected value of the other envelope is $10 if you see $5 in yours, or vice versa. If you see $5 you should switch, and if you see $10 you should not. So there is no paradox in this case.
However, surprisingly, there are some prior probability distributions of money in the envelopes for which it always makes sense to switch (whether or not you look at what's inside your envelope)!
For instance, suppose that amount of money in the two envelopes is($2k,$2k+1) with probability (2/3)k/3, for each integer k>=0. It is a fun exercise to check that no matter what you have in your envelope, the other envelope has higher expected value, and you should switch!
How to resolve this paradox is a perplexing philosophical question. (Some of you may object that the prior distribution has infinite mean, but this does not fully resolve the paradox, since in theory if such a distribution exists, one would still have to wrestle with the paradox of continually switching envelopes!)
Successive Differences of Powers
List the squares:
0, 1, 4, 9, 16, 25, 36, 49, ...
Then take their successive differences:
1, 3, 5, 7, 9, 11, 13, ...
Then take their successive differences again:
2, 2, 2, 2, 2, 2, ...
So the 2nd successive differences are constant(!) and equal to 2.
OK, now list the cubes, and in a similar way, keep taking successive differences:
0, 1, 8, 27, 64, 125, 216, 343, 512, ...
1, 7, 19, 37, 61, 91, 127, 169, ...
6, 12, 18, 24, 30, 36, 42, ...
6, 6, 6, 6, 6, 6, ...
Gee, the 3rd successive differences are all constant(!) and equal to 6.
What happens when you take the 4th successive differences of 4th powers? Are they constant? What do they equal? (They're all 24.) And the 5th successive differences of 5th powers?
Aren't derivatives similar to differences? What do you think happens when you take the n-th derivative of xn?
Presentation Suggestions:
Have students do these investigations along with you. If you assign the n-th derivative of xn on a previous homework, then you can make the connection between the two right away.
The Math Behind the Fact:
This pattern may seem very surprising. It can be proved by induction. Taking differences is like a discrete version of taking the derivative, where the space between successive points is 1.
This idea has a very practical application: given a sequence generated by an unknown polynomial function, you use the calculation of successive differences to determine the order of the polynomial! Then use the first N terms of the sequence with the first N terms of the polynomial to solve for the generating function.
Red-Black Card Trick
Here's a pretty easy card trick that you can do that can also be pretty surprising. Here's how the trick you do will appear to others:
Take a deck of cards, and give it to a spectator and ask her to shuffle the deck and return it to you face down. You take the cards, and (with a little showmanship but without looking at the fronts of the cards) separate them into two piles, and then say "Just by feeling the redness or blackness of the cards with my fingers, I've made two piles so that the number of red cards in the first pile is the number of black cards in the second pile."
Have your spectator turn over the cards and verify!
Presentation Suggestions:
Your spectator can shuffle the cards as many times as she likes--- it won't matter! When she gives the cards to you, all you are really doing (though don't make it obvious) is counting the cards into two piles so that there are 26 cards in each pile.
The Math Behind the Fact:
The reason this trick works is simple... if the number of red cards in the first and second piles is R and S, and the number of black cards in the first and second piles is A and B, then we know that R+S=26 (since the total number of red cards is 26) and S+B=26 (since the total number of cards in the second pile is 26). These two equations can be subtracted from one another to show that R-B=0, or R=B.
Multiplication by 111
If you liked the Fun Fact Multiplication by 11, you'll enjoy seeing how to take that idea one step farther. Here's a quick way to multiply by 111.
To multiply a two-digit number by 111, add the two digits and if the sum is a single digit, write this digit TWO TIMES in between the original digits of the number. Some examples:
23x111= 2553
41x111= 4551
The same idea works if the sum of the two digits is not a single digit, but you should write down the last digit of the sum twice, but remember to carry if needed. So
57x111= 6321
because 5+7=12, but then you have to carry the one twice.
If the number you are multiplying by 111 is a three-digit number, say ABC, then the answer will have five digits (though it may be six if there is a carry involved):
the first digit is A,
the second digit is A+B,
the third digit is A+B+C,
the fourth digit is B+C,
the fifth digit is C.
Again, you must remember to carry if any of these sums is more than one digit. Thus 123x111=13653, 241x111=26751, and for an example where carrying is needed, 352x111=39072. (Because of the carries, it may be easier to do the sums and write the answer down from right to left.)
Presentation Suggestions:
Do the Multiplication by 11 Fun Fact first.
The Math Behind the Fact:
Multiply using the traditional (long) method for multiplication, and you will find that the above shortcut works because it is doing exactly the same sums that you would have to do using the traditional method for multiplication.
Difference of Squares
You have all learned that
a2 - b2 = (a + b)(a - b)
But perhaps you haven't thought about how to use this to do fast mental calculations! See if you can guess how this trick can help you do the following in your head:
43 x 37
78 x 82
36 x 24
Let's do the first one. 43 x 37 = (40 + 3)(40 - 3) = 402 - 32 = 1600 - 9 = 1591.
Practice these, and you'll be able to impress your friends!
Presentation Suggestions:
See if the class can figure out what you are about to do.
The Math Behind the Fact:
The moral of the story is if you think CREATIVELY about everything that you learn in mathematics, even the easy stuff like algebra, you will find some surprising applications
Squares Ending in 5
Give me any 2 digit number that ends in 5, and I'll square it in my head!
452 = 2025
852 = 7225, etc.
There's a quick way to do this: if the first digit is N and the second digit is 5, then the last 2 digits of the answer will be 25, and the preceding digits will be N*(N+1).
Presentation Suggestions:
After telling the trick, have students see how fast they can square such numbers in their head, but doing several examples.
The Math Behind the Fact:
You may wish to assign the proof as a fun homework exercise: multiply (10N+5)(10N+5) and interpret! The trick works for larger numbers, too, although it may be harder to do this in your head. For instance 2052 = 42025, since 20*21=420.
Squaring Quickly
You may have seen the Fun Fact on squares ending in 5; Here's a trick that can help you square ANY number quickly.
It's based on the algebra identity for the difference of squares, but with a twist! Can you figure it out?
542 = 50 * 58 + 42 = 2916.
422 = 40 * 44 + 22 = 1764.
372 = 34 * 40 + 32 = 1369.
You have to pretty proficient at multiplying one digit numbers by two digit numbers in your head to do this trick well. But if you master this, then you can build upon it in some amazing ways:
1162 = 100 * 132 + 162 = 13,200 + 256 = 13,456.
Thinking CREATIVELY about everything you learn, no matter how trivial it may seem, will allow you to find some really clever applications!
Presentation Suggestions:
If you practice this a LOT beforehand, you can start off by asking students to name any 2-digit number and you will do it in your head quickly. Then tell them the trick. But only do this with a LOT OF PRACTICE!
The Math Behind the Fact:
If you look closely, we are using the identity:
a2 = (a-b)(a+b) + b2.
Roman and "Arabic" Numerals
The use of Roman numerals has been mathematically obsolete for more than 1100 years. Nonetheless, the Roman symbols for numbers continue to be used in a variety of ways, most of them rather stereotyped: to mark the hours on clock faces, to number pages in the prefaces of books, to express copyright dates, and to count items in a series (such as the Super Bowls of U.S. professional football).
The form of Roman numeration used today was established during the Middle Ages in Western Europe. It is derived from the systems actually used in Roman times, but with certain improvements. The basic Roman numerals as used today are:
I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000
The symbols are repeated to form larger numbers, and when different symbols are combined, the larger unit precedes the smaller. Thus VIII represents 8, CLXXX is 180, and MMDCCXXV is 2725.
The Romans usually wrote IIII for 4 and XXXX for 40. The number 949 was DCCCCXXXXVIIII. To shorten the length of such numbers a "subtraction rule" was sometimes used in Roman times and was commonly used in medieval times. The "subtraction rule" allows the use of six compound symbols in which a smaller unit precedes the larger:
IV = 4 IX = 9 XL = 40 XC = 90 CD = 400 CM = 900
Using these symbols, 949 is written more compactly as CMXLIX. (Other "subtracted" symbols are not allowed. Thus 99 must written XCIX, not IC.) The use of subtracted symbols was never mandatory, so IIII and IV can be used interchangeably for 4.
Actually, the symbols D (500) and M (1000) were originally written using a vertical stroke with surrounding arcs; these arcs can only be approximated on this page by using parentheses. D appeared as I ) and M as ( I ). This system allowed powers of ten larger than 1000 to be written by increasing the number of arcs: 10 000 was written (( I )) and 100 000 was written ((( I ))). The Romans had no word for 1 000 000 and rarely considered numbers of that size or larger. In late Roman and medieval times, after D and M were adopted as the symbols for 500 and 1000, a custom arose of writing a bar over a number to multiply that number by 1000. Thus 10 000 became X with a bar over it and 100 000 became C with a bar over it. These "overbarred" symbols are almost never seen today.
In Roman times, only the capital letters were used for number symbols. Later, after lower case letters came into use, Roman numbers were often written in lower case. Thus "vi" means 6 and "cxxii" means 122. Sometimes cases were even mixed, as in "Mcxl" for 1140. Furthermore, the lower case letter "j" was sometimes used in place of "i". A common custom was to write "j" for the last in a series of one's, as in "xiij" for 13.
Roman numerals continued in use in Europe after the fall of the Roman Empire, and they remained in general use for centuries after our modern number system became available. As we see, their use in certain applications continues even today.
The modern system of numeration is based on place value, with the same symbol, such as 4, taking on different meaning (4, 40, 400, etc.) depending on its location within the representation of the number. Place value notation was used long ago in Babylonian cuneiform numerals, but our modern decimal place value system was invented by Hindu mathematicians in India, probably by the sixth century and perhaps even earlier. The modern numerals 1, 2, 3, ..., are sometimes called "Arabic" numerals in the West because they were introduced to Europeans by Arab scholars. The key figure was the great Arab mathematician Mohammed ibn-Musa al-Khowarizmi, who taught at Baghdad sometime between 800 and 850. He wrote a book on the Hindu number system known today only in a later Latin translation as De numero indorum, "On the Hindu numbers." Subsequently he wrote a longer and very influential work, Al-jabr w'al muqabalah, known in Europe as Algebra, which included all the techniques of arithmetic still taught in schools today. The author's name, Latinized as "Algorismus," is the root of the English word "algorithm".
The Hindu-Arabic numeration system was known in Europe by 1000, but at first it didn't make much of a dent in the use of Roman numerals. During the 1100's the "Arabic" numerals were a topic of great interest among European scholars, and several translations of the Algebra appeared. In 1202, Leonardo of Pisa (ca. 1180-1250) published a famous book Liber abaci explaining and popularizing the Hindu-Arabic system, the use of the zero, the horizontal fraction bar, and the various algorithms of the Algebra. (Leonardo is better known today by his patronymic Fibonacci, "son of Bonaccio.") Thereafter modern numerals and the standard operations of arithmetic were commonly used by scholars, but Roman numerals continued to be used for many purposes, including finance and bookkeeping, for many centuries to come.
Incidentally, the numerals 0123456789 are more properly known as European digits. The numerals actually used in Arabic script, the true Arabic numerals, are of different forms;
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