How many Pennies Are a Million?

One Penny 1 value  1¢, (one cent) width  0.75 inches, (3/4 of an inch) height  0.75 inches, (3/4 of an inch) thickness  0.0625 inches, (1/16 of an inch) weight  0.1 ounces, (1/10 of an ounce) area   0.5625 square inches

One Thousand Pennies 1,000 [5 pennies wide x 5 pennies high x 40 pennies tall]   A thousand pennies is only $10.00 worth of pennies, yet it weighs over six pounds.
value	$10.00, (Ten dollars and no cents)
width	3.75 inches
height	3.75 inches
thickness	2.5 inches
weight	100 ounces, (6.25 pounds)
height stacked	62.4 inches, (5.2 feet)
area (laid flat)	562.5 square inches (3.9 square feet)

Hundred Thousand Pennies

98,304 Ninety-eight thousand three hundred and four Pennies [ Two cubic feet ]   value  $983.04 (Nine hundred eighty-three dollars and four cents) width  24 inches, (two feet) height  12 inches, (one foot) thickness  12 inches, (one foot) weight  614.4 pounds height stacked  512 feet area (laid flat)  384 square feet

One Million Pennies 1,003,776 One million, three thousand, seven hundred and seventy-six Pennies [ A wall five by four by one feet thick with a 9-inch cube stepstool ]
value	$10,037.76 (Ten thousand, thirty-seven dollars and seventy-six cents)
width	Four feet
height	Five feet
thickness	12 inches, (one foot)
weight	6273.6 pounds (3.14 tons)
height stacked	5,228 feet ( 0.99 Miles )
area (laid flat)	3,921 square feet

One Billion Pennies 1,000,018,176 One billion, eighteen thousand, one hundred and seventy-six Pennies [ Five school buses. ]   Each of these blocks represents one 9x11x41 foot school bus - as seen below. If you were to stack all these pennies in a single pile, one atop the other, the stack would reach nearly one thousand miles high. For comparison, note that the Space Shuttle typically orbits only 225 miles above the Earth's surface. Only in North America and the general scientific community is this number (1,000,000,000) called a "billion". Most European countries call this number either "one thousand million" or, in some cases, a "milliard".
value	$10,000,181.76 (Ten million, one hundred eighty-one dollars and seventy-six cents)
width	45 feet
height	11 feet
thickness	41 feet
total weight	3,125 tons
height stacked	987 Miles
area (laid flat)	3,906,321 square feet (89.7 acres)

One Trillion Pennies 1,000,000,016,640 One trillion, sixteen thousand six hundred and forty Pennies [ One cube measuring 273 x 273 x 273 feet ]   The same football field, set beside our new cube for scale.
value	$10,000,000,166.40 (Ten billion, one hundred and sixty-six dollars and forty cents)
width	273 feet
height	273 feet
thickness	273 feet
total weight	3,125,000 tons
height stacked	986,426 Miles
area (laid flat)	89,675.2 acres

 

Prime Numbers Are Used To Buy Things

Whenever someone uses a credit card on the internet, prime numbers spring into action.

Before the card number is sent over the internet, it must be encrypted (put into code) for security and once the code is received by the store, it must be decrypted (decoded).

The code that is used the most is called RSA and it is based on prime numbers. It uses a "public key", information that is available to anyone and a "private key" information that only the store has.

The "public key" is a large number that is the product of two large primes and the "private key" is the two large primes themselves.

It is very difficult to factor a given large number into primes, which is what you would have to do to break the code. For example, it took researchers two years to factor a 232-digit number, even using hundreds of parallel computers.

So when a credit card is used, only the "public key" is sent from the store, the message is coded using the key, and sent to the store. The store already has the answer to the key and can easily uncode the message.

Prime Numbers to 400

Ok, if you've read the first blog on prime numbers, you know some rules that help you reduce the number of numbers you have to check to see if a number is prime.

One more rule that may help is the rule that:

All prime numbers are either +1 or -1 from a multiple of 6. Not all +1's and -1's are primes, but all primes are +1 or -1.

What this rule does is move you one away from all the numbers that can be divided by 2 or 3.

Here's another chart of what numbers you have to check as divisors for all primes up to 400.

This chart doesn't prove it, but it gives you hint of a basic rule of all numbers:

Every number is either prime or can be factored (divided) into primes.

This is true just from the way we define prime numbers

Prime Numbers

OK, here's the definition of a prime number:

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number.

In other words, a prime number is a counting number that can't be divided by another number.

So what does this mean?

1. First of all, except for 2, a prime number cannot be an even number. If it was even, 2 would divide into it.
2. Secondly, except for 5, any number that ends in a 5 is not prime. If it ends in 5, 5 would divide into it.
3. Thirdly, if the sum of the digits of a number is 3, 6, or 9, the number is not prime. If the sum of the digits is 3, 6, or 9, 3 divides into it.

So it isn't necessary to check for all the possible divisors to tell if a number is prime or not, just the primes less than the number and up to the square root of the number you're checking. (don't worry about the square root, you'll meet it in a year or two.) So this is the way it works for primes up to 100:

So we can now write down all the prime numbers up to 100. Write down your list and compare it to the answer below:

Answer: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 93, 97
Did you get caught with 91? 7 X 13 = 91

McNugget Numbers

With only 2 pence and 5 pence coins, one cannot make 3 pence,
 but one can make any higher amount.

McNugget Numbers are a special case of what is known as the Frobenius problem, after the mathematician Ferdinand Frobenius). The problem that asks for the largest monetary amount that cannot be obtained using only coins of specified denominations. For example, the largest amount that cannot be obtained using only coins of 3 and 5 units is 7 units. The solution to this problem for a given set of coin denominations is called the Frobenius number.
                                                                                                                  "Various British Pennys". Licensed under CC BY-SA 3.0 via Wikipedia

                                                                                            By Fritz Saalfeld (Own work) [CC BY-SA 2.5

The McNuggets version of the coin problem was introduced by Henri Picciotto, who included it in his algebra textbook. Picciotto thought of the application in the 1980s while dining with his son at McDonald's, working the problem out on a napkin. A McNugget number is the total number of McDonald's Chicken McNuggets in any number of boxes. The original boxes (prior to the introduction of the Happy Meal-sized nugget boxes) were of 6, 9, and 20 nuggets.

If you look at the numbers you can reach with boxes of 6, 9, and 20 nuggets, you notice that you can only reach multiples of 3 and multiples of 3 plus multiples of 20. So starting with 1, you can come up the the following list of numbers you cannot reach (non-McNugget numbers):

1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43

At 44, you realize that the next six numbers can be reached by:

44 = 6 + 9 + 9 + 20
45 = 9 + 9 + 9 + 9 + 9
46 = 6 + 20 + 20
47 = 9 + 9 + 9 + 20
48 = 6 + 6 + 9 + 9 + 9 + 9
49 = 9 + 20 + 20

Now you know that all the numbers above these can be reached. You can add 6 to each of these six numbers and get the next six numbers. You can do this to infinity, so 43 is the largest non-McNugget Number. Anytime you can get the smallest number that number times in a row, you've got the largest non-McNugget number.

Bonus Round: Since the introduction of the 4 piece Happy Meal, what is the largest non-McNugget number. Use 4, 6, 9 and 20. Answer Below.










Highlight to read answer
The answer is 11, did you get it?

 

Any larger integer can be obtained by adding some number of 6s to the appropriate partition above.                                                       (http://creativecommons.org/licenses/by-sa/2.5)], via Wikimedia Commons

Writing Roman Numerals

How did Roman Numerals begin?

Many people believe Roman Numerals began as a tally or marking system used by shepherds to keep track of how many sheep they had.
Each sheep was counted with a single notch cut into a stick with a knife. Every fifth sheep was recorded with two notches to form a V and then each tenth sheep was denoted by an X.
This method of record keeping was still being used by Italian shepherds in the nineteenth century.

The Basics

I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1000

The mnemonic 'I Value Xylophones Like Cows Do Milk' may really help you remember (Thanks to Adrian Bruce).

Put It Together

To write larger numbers in Roman numerals, Romans simply "added" all the symbols together. For instance:

LXXIII = 50 + 10 + 10 + 1 + 1 + 1 = 73
MDXII = 1000 + 500 + 10 + 1 + 1 = 1512

Hints:
- When writing roman numerals never use more than 3 of any symbol.
- When you have numerals on the right hand side of bigger ones, ADD.

Try these problems out (answers on the bottom)

1. Convert XXVIII to our number system
2. Convert XXXVI to our number system
3. What is LXII in our number system?
4. DCXXII =
5. MMXVI = 

When you have to Subtract

Sometimes you cannot follow the hint of not using more that 3 of any symbol, for instance in writing 9 in Roman numerals. Nine would be written VIIII, but that would break the rule of not using more than three I's. In these cases you need subtract the value of the smaller number from the larger.

IX = 10 - 1 = 9

Hint:
-When smaller numerals are on the left of larger ones... subtract.

Try these (answers on the bottom again):

1. IV =
2. VL =
3. CM =
4. XC =
5. CD =

Notable Roman Numeral Quote

'Class, please! If you don't learn roman numerals, you'll never know the date certain motion pictures were copyrighted.'

Edna Krabappel (from The Simpsons)

(Highlight the yellow answers to see clearly)Answers: 5. 2016 4. 622 3. 62 2. 36 1. 28
(Highlight again) 5. 400 4. 90 3.900 2. 45 1. 4

You can see a very good explanation of writing Roman numerals written and illustrated by  Adrian Bruce if you  Click Here .

Facts About Pi

The record for calculating pi, as of 2010, is to 5 trillion digits.

If you were to print 1 billion decimal values of pi in ordinary font it would stretch from New York City to Kansas.

3.14 backwards looks like PIE. Look through the back of the paper!

The first million decimal places of pi consist of 99,959 zeros, 99,758 ones, 100,026 twos, 100,229 threes, 100,230 fours, 100,359 fives, 99,548 sixes, 99,800 sevens, 99,985 eights and 100,106 nines. It cannot be proven, but it is pretty definite that the digits of pi are random.

Pi's evolution

Around 2000 B.C., Babylonians established the constant circle ratio as 3 1/8 or 3.125.

The ancient Egyptians arrived at a slightly different value of 3 1/7 or 3.143.

One of the earliest known records of pi was written by an Egyptian scribe named Ahmes (c. 1650 B.C.) on what is now known as the Rhind Papyrus. He was off by less than 1% of the modern approximation of pi (3.141592).

Plato (427-348 B.C.) supposedly obtained for his day a fairly accurate value for pi: √2 + √3 = 3.146.

The father of calculus (meaning "pebble used in counting," from calx or "limestone"), Isaac Newton, calculated pi to at least 16 decimal places.

William Jones (1675-1749) introduced the symbol "π" in the 1706, and it was later popularized by Leonhard Euler (1707-1783) in 1737.

Just How Many Stars are There?

People have been asking this question for a long time. Back before telescopes were invented by a Dutch eyeglass maker, Hans Lippershey in 1608, the number of stars was what we could see with our naked (or unaided) eyes. This number is estimated as no more than 10,000 stars. This number varies according to how you count them, as only 1/2 the stars are visible from a single place on Earth and then only 1/2 the time. You can't see stars through the Earth and during the daytime!

After the telescope was invented, the number of stars we could see increased as we could see more of the fainter stars we couldn't see before. This number is probably about 1,000,000,000 (1 billion), but no one could actually count them all. Instead, Jérôme Lalande published the Histoire Céleste Française in 1801, which contained an extensive star catalog. The observations made were made from the Paris Observatory and so it describes mostly Northern stars. This catalog contained the positions and magnitudes of 47,390 stars, out to magnitude 9 (very, very faint), and was the most complete catalog up to that time. This would give only about 100,000 stars of magnitude 9 or better if the Southern stars could be counted.

 

What we call the Milky Way was known long before telescopes, but it was not known that is was made up of stars. The name Milky Way is from a Greek fairy tale involving a goddess and her milk. After telescopes became commonplace, following the efforts of Galileo, it was soon realized that the Milky Way is an unusually dense collection of stars, all lined up.

 

When people first started looking through a telescope, they noticed fuzzy spiral objects which were grouped with other such "nebulae" and with star clusters. These were at first considered interesting curiosities in the sky, but not to be confused with really interesting things such as comets.

 

Edwin Powell Hubble (1889-1953) demonstrated that most "nebulae" are objects outside our galaxy, using the 100-inch telescope at Mt. Wilson (in 1924). With this powerful instrument, he was able to measure the distance to the Andromeda nebula itself, observing stars within that neighboring galaxy. Thus, the enormous distance to this neighbor became clear: Hubble gave it almost a million light years. As it turned out, this was still quite a bit short of the truth. The distance is greater than 1.5 million light years. With this greater distance, the information coming in from the Andromeda could be interpreted correctly, and it turned out to be a sister galaxy, quite similar to our own.

 Andromeda Galaxy

We now know that there are about 300,000,000,000 (300 billion) stars in the Milky Way Galaxy alone. The Milky Way is larger than average, an average galaxy has about 100 billion stars.

In 2012, Hubble scientists produced the Hubble Extreme Deep Field image. The area searched was an area of clear sky that was less than 1/100th the area of the full moon. In that little rectangle, scientists were able to count over 10,000 galaxies. If this area is typical, there are 100 billion galaxies in the universe. An average galaxy has 100 billion stars, so there are 10,000,000,000,000,000,000,000 stars in the universe (or 10²² in scientific notation).



Large Numbers With Names

The following table lists names of large numbers which are found in many English dictionaries and thus have a special claim to being "real words". The "Traditional British" values shown are not used in American English and are becoming very rare in British English, but their other-language variants are dominant in many non-English-speaking areas, including continental Europe and Spanish speaking countries in Latin America.

          English also has many words, such as "zillion", used informally to mean large but unspecified amounts.

Standard dictionary numbers

Name	Short scale (U.S., Canada and modern British)	Long scale (continental Europe, older British)
Million	106	106
Milliard		109
Billion	109	1012
Trillion	1012	1018
Quadrillion	1015	1024
Quintillion	1018	1030
Sextillion	1021	1036
Septillion	1024	1042
Octillion	1027	1048
Nonillion	1030	1054
Decillion	1033	1060
Undecillion	1036	1066
Duodecillion	1039	1072
Tredecillion	1042	1078
Quattuordecillion	1045	1084
Quindecillion	1048	1090
Sexdecillion (Sedecillion)	1051	1096
Septendecillion	1054	10102
Novemdecillion (Novendecillion)	1060	10114
Vigintillion	1063	10120
Centillion	10303	10600

In the list, scientific notation is used. For instance, a trillion is listed as  10^12. This is spoken as "ten to the twelth" and written as 1,000,000,000,000. Notice that when we write big numbers, we put a comma between every third digit.

The googol Family

The names googol and googolplex were introduced in Kasner and Newman's 1940 book, Mathematics and the Imagination, in the following passage:

The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would actually happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it.

Value	Name	Authority
10100	Googol	Kasner and Newman, dictionaries
10googol =	Googolplex	Kasner and Newman, dictionaries

If you wish to see a googolplex written out, here's a link you can click on, http://www.googolplexwrittenout.com/ . Of course it comes in multiple volumes, each containing 1 million digits (mostly zeros) of the written out number. A googolplex is so big that if you could read 1 million volumes of 1 million digits in only a 1/1000 of a second, it would still take you MUCH longer than the age of the universe to finish.

Citations:http://en.wikipedia.org/w/index.php?title=Names_of_large_numbers&oldid=665322020