Section 2 2 Whole Numbers and Numeration Three

Section 2. 2 Whole Numbers and Numeration

Three common uses of numbers • Cardinal number - tells the size of a set - example: I have five brothers. • Ordinal number - tells the position of an object in a list (or in other words, how far away the object is from the beginning of the list). - example: you are the 17 th on the waiting list. • Identification number - gives an object a name or identity - example: The zip code here is 92020.

Three common uses of numbers • Cardinal number - tells the size of a set - example: I have five brothers. • Ordinal number - tells the position of an object in a list (or in other words, how far away the object is from the beginning of the list). - example: you are the 17 th on the waiting list. • Identification number - gives an object a name or identity - example: The zip code here is 92020.

Concept of a Cardinal number. Numbers are abstract concepts rather than physical objects. It is therefore more difficult to teach numbers than to teach the names of tangible objects such as fruits and vegetables. The best way (to define cardinal numbers) is to use the concept of one-to-one correspondence. For example, 3 is the abstract attribute common to all sets that match the set {a, b, c} (or the set {s, t, u}, or the set { , , } etc. ), 4 is the abstract attribute common to all sets that match the set {a, b, c, d} (or the set { , , , } etc. ), 5 would be the abstract attribute common to all sets that match the set {a, b, c, d, e} (or the set { , , , , } etc. ).

Draw 1 -to-1 correspondences to identify the groups that match the example. Those groups will then be defined to have 9 elements.

Numbers and Numerals A number is an abstract attribute. A numeral is a symbol or a string of symbols (representing a number). 3 8 Which one represents a larger number? The on the right Which one is a larger numeral? The on the left

Digits and Numerals A digit is a single character numeral in a numeration system. In our Hindu-Arabic system, there are exactly ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A number (or more precisely, a numeral) is either a single digit or a string of several digits, such as 327, which has three digits. A digit is analogous to a letter in our alphabet, and a number (or numeral) is analogous to a word. A word can have many letters.

Numeration systems • A numeration system is a systematic way to assign symbols to numbers. Tally system • converts each object into a tally Even though the tally system is easy to learn, it is difficult to read big numbers such as the following.

Improve clarity by grouping The base of a system is the size of each basic group. The base of the tally system is five and • five tallies = one bundle • five bundles = one superbundle (a super bundle) • five superbundles = one super-superbundle • etc. What are the short comings of the Tally system?

Egyptian System • a base ten system • with a special symbol for each power of ten one ten hundred thousand tenhundredthousand million • an additive system = One thousand one hundred twenty one. • a non-positional system has the same value as

The US currency system basically is the Egyptian numeration system. The present denominations of our currency in production are $1, $2, $5, $10, $20, $50 and $100. The denominations of our currency in production includes each power of ten: $1, $100, (before 1969, there were even $1, 000, $10, 000, and $100, 000 bills. However, $1, 000 bills were never produced by the treasury department. ) The $2, $5, $20, $50 bills are added for convenience. This system is clearly additive and non-positional, just like the Egyptian system. =

Million Dollar Display This venue closed shortly after the death of Ted Binion. Rumor has it that his sister removed it from the casino. You used to be able to get your picture taken next to Binion's Million! It was located downtown in Binion's Horseshoe Hotel & Casino. Before 1997, you could find this cool display of 100, discontinued, yet entirely legal tender $10, 000 bills (a genuine US$1, 000 in all!) and you could walk right up and stand next to them. For many of us it was the closest we'll ever get to that much money.

The Horseshoe Casino in downtown L. V. , on Fremont Street Experience.

From 1966 to 1999 one of the legendary icons of Las Vegas was the Million Dollar Display at Binion’s Horseshoe Casino. This collection of one hundred 1934 $10, 000 bills, encased in Plexiglas and framed by an inverted golden horseshoe, was the backdrop for over 5 million photographs. Tourists, celebrities, even royalty came to have their pictures taken next to the million dollars in cash. When the casino decommissioned the display in the late 1990’s, Jay Parrino’s The Mint was there and bought all 100 notes. $10, 000 bills are rare in their own right, being the largest denomination note issued for general circulation ($100, 000 notes were printed, but used only for transfers within the Federal Reserve Bank system – private ownership is prohibited). This particular note has the dual distinction of being not only the best of the 100 note hoard, but the finest known example in the world.

The Egyptian System Advantages of this system • simplicity in structure, easy to learn • intuitive approach Drawbacks of this system • new symbols must be invented whenever larger powers of ten are encountered. • lots of symbols may be needed for a relatively small number. • the length of an expression is not proportional to the size of the number it represents; means thirty two means one thousand - this leads to confusion and makes calculations very cumbersome.

The Roman System I V X L C one five ten fifty hundred • a base ten system • positional eg. VI is different from IV • additive and subtractive eg. VI = 5 + 1 and IV = 5 – 1 • multiplicative as well eg. XII = 12 and XII = 12, 000 D five hundred M thousand 2020 MIAMI

This is the New York Public library in NYC. In the front wall, you can see what years it was constructed. (see next slide for enlargement)

From what year to what year was it built?

The Roman System Almost no advantage. Disadvantages • new symbols are needed for higher powers of ten • subtractive system is very unnatural to use • pattern is too complex to be practical • length of expression is not proportional to the number being represented

Babylonian System • a base sixty system – such as the time measurement system • requires only three symbols one • is positional: ten place-holder is different from • has a very sophisticated feature – place value, i. e. the value of a symbol depends on its position in the whole expression.

The currently used Hindu-Arabic system also has place value. In fact, it copies the idea from the Babylonian system. 3532 This is 3 thousands. This is 3 tens. The concept of place value also exists in music scores. The key of each note depends on its position on the staff, while the shape tells you the duration of that note.

Babylonian System Examples: one two three four five six seven eight nine ten … … fifty nine eleven sixty twelve sixty one thirteen sixty two fourteen sixty three

Babylonian System Examples: seventy one … eighty ninety one hundred ten one hundred twenty one

The numerals in the Babylonian system is very similar to what we have in the face of a digital clock. The only important difference is that the Babylonian numerals will not have the : symbol to separate the groups.

Examples means 60 + 42 = 102 means 12× 60 + 21 = 741 means 602 + 0× 60 + 14 = 3614

Babylonian System Advantage of the system • a very small set of symbols is used, and this set will never need to be expanded. Disadvantages of the system • the base is too large • ambiguity exists • place value is difficult to learn • length of the whole expression is still not proportional to the number it represents.