numeration systems numerals did not start out as 1, 2, 3, … the first numeration systems were...

Numeration Systems• Numerals did not start out as 1, 2, 3, …

• The first numeration systems were often 1, 2, many. 0 wasn’t needed. More complex number systems arose out of need.

Numeration Systems • Tally systems: |||| at a time.• Egyptian--probably earliest known • Babylonian--only two symbols• Roman--most widespread, all over

Europe from the Roman Empire• Mayan--only three symbols• Hindu-Arabic--what we use today

A brief look at Egyptian Numerals

• Used at about the same time as Babylonian Numbers--many similarities

• More symbols:•

1 10 100 1000 10,000More symbols for 100,000 and 1,000,000

Egyptian Numerals• 300 + 10 = 310 (Notice no ones)

• 20,000 + 300 + 4 = 20304

Egyptian Numerals• 300 + 10 = 310 (Notice no ones)

• 20,000 + 300 + 4 = 20304

A brief look at Babylonian Numerals

• Initially, no zero. Later developed: • Two symbols only:

= 1; = 10.• Additive when written from greatest

to least: = 10+10+1+1+1• Use 60 as a base--there is a break after 60.

means 60+60+60+10+1

Babylonian Numerals =1 • ____ + 11 • 60 + 21 =

1 • 60 • 60 + 11 • 60 + 21 =

3600 + 660 + 21 = 4281

A brief look at Mayan Numerals

• Used the concept of zero, but only for place holders

• Used three symbols:• ---

1 5 0• Wrote their numbers vertically:

••• is 3 + 5 = 8, --- is 5 + 5 = 10

Mayan NumeralsNew place value… left a vertical gap.

• is one 20, and 0 ones = 20.

•• • is two ___ + 6

A brief look at Roman Numerals

• No zero.• Symbols:

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

• If the Roman Numerals are in order from greatest to least, then add:VII = 5 + 1 + 1 = 7; XVI = 10 + 5 + 1 = 16

Roman Numerals• I = 1, V = 5, X = 10, L = 50, C = 100,

D = 500, and M = 1000.

• If the Roman Numerals are NOT in order from greatest to least, then subtract where the order is wrong.IV = 5 - 1 = 4; IX = 10 - 1; XCII = 100 - 10 + 1 + 1 = 92

Exploration 2.8• Exploration 2.8--We will not do this

entire thing--it is too long! But we will do parts. Today, we will begin with manipulatives.

Alphabitia• Read the introduction.• Use the “artifacts”.

UnitLong

Flat

Your job:• See if you can create a numeration

system, like the ones we have just seen. Make notes, and be ready to explain it to others who are not part of this tribe. It should be logical, and be able to be continued past Z.

• Keep the materials for Monday.

Alphabitia Numeration System Proposals

• What did you come up with in your group?

• What are the pros and cons of your group’s system and the other groups’ systems?

What makes an efficient numeration system?

Alphabitia• A numbering system is only powerful if

it can be reliably continued.

• Ex: 7, 8, 9, … what comes next?

• Ex: 38, 39, … what comes next?

• Ex: 1488, 1489, … what comes next?

The Numeration System we use today:

The Hindu-Arabic System • Zero is used to represent nothing and as a place

holder.• Base 10 Why?• Any number can be represented using only 10

symbols.• Easy to determine what number comes next or what

number came before.• Operations are relatively easy to carry out.

In Base 10…

• Digits used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9• We can put the digit 9 in the units

place. Can we put the next number (ten) in the units place?

• Only one digit per place• Placement of digits is important!• 341 ≠ 143. Can you explain why not?

Exploration 2.9

• Different Bases

In another base…• We need a 0, and some other digits

• So, in base 10, we had 0 plus 9 digits

• What will the digits be in base 9?

• What will the digits be in base 3?

• Which base was involved in alphabitia?

So, let’s count in base 6• Digits allowed: 0, 1, 2, 3, 4, 5

• There is no such thing as 6

• When we read a number such as 2136, we don’t typically say “two hundred thirteen.” We say instead “two, one, three, base 6.”

Count! In base 6• 1, 2, 3, 4, 5, …• 10, 11, 12, 13, 14, 15, …• 20, 21, 22, 23, 24, 25, …• 30, 31, 32, 33, 34, 35, …• 40, 41, 42, 43, 44, 45, …• 100, …• 100, 101, 102, 103, 104, 105, … 110

Compare base 6 to base 10

• Digits 0,1,2,3,4,5,6,7,8,9

• New place value after 9 in a given place

• Each place is 10 times as valuable as the one to the right

• 243 = 2 • (10 • 10) + 4 • 10 + 3 • 1

• Digits 0 - 5• New place value after 5

in a given place• Each place is 6 times

as valuable as the one to the right.

• 243base 6 = 2 • (6 • 6) + 4 • 6 + 3 • 1or 99 in base 10

Compare Base 6 to Base 10

• 312 =

3 • 100 +

1 • 10 + 2 • 1

• 312base 6 =

3 • 36 +

1 • 6 +

2 • 1 =

116 in base 10

How to change from Base 10 to Base 6?

• Suppose your number is 325 in base 10.

• We need to know what our place values will look like.

• _____ _____ _____ _____6•6•6 6•6 6 1

Now, 6•6•6 = 216. 216 = 1000 in base 6.

Base 10 to Base 6

• ___1__ _____ _____ _____6•6•6 6•6 6 1

• Now, 325 - 216 = 109. Since 109 is less than 216, we move to the next smaller place value: 6 • 6 = 36.

• 109 - 36 = 73. Since 73 is greater than 36, we stay with the same place value.

Base 10 to Base 6

• __1___ ___3__ _____ _____

6•6•6 6•6 6 1• We had 109: 109 - 36 - 36 - 36 = 1. We

subtracted 36 three times, so 3 goes in the 36ths place.

• We have 1 left. 1 is less than 6, so there are no 6s. Just a 1 in the units place.

Base 10 to Base 6

• __1___ ___3__ __0___ __1___

6•6•6 6•6 6 1

• Check: 1 • 216 + 3 • 36 + 1 • 1 = 325

• So 325 = 13016

Count! In base 16 1, 2, 3, 4, 5, 6, 7, 8, 9, a,

b, c, d, e, f,

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 1e, 1f,

20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2a, 2b, 2c, 2d, 2e, 2f

Homework for Friday

• Exploration 2.9: Part 1: for Base 6, 2, and 16, do #2; Part 3: #2, 3, Part 4: #1, 2, 4. For the base 16 section, change all the base 12 to base 16 (typo)

• Read Textbook pp. 109-118• Do Textbook Problems pp. 120-121:

15b,c, 16b,d, 17a,i, 18b,f, 19, 29