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Card #1
1 3 5 7
9 11 13 15
17 19 21 23
25 27 29 31
Card #2
2 3 6 7
10 11 14 15
18 19 22 23
26 27 30 31Card #3
4 5 6 7
12 13 14 15
20 21 22 23
28 29 30 31Car
d #48 9 10 11
12 13 14 15
24 25 26 27
28 29 30 31
Card #5
16 17 18 19
20 21 22 23
24 25 26 27
28 29 30 31
What to expect…
• Learn some new things about our number system.
• Learn some stuff about other number systems.
• Learn some cool short-cuts that work for our number system.
• Learn how the Birthday cards work.
Let’s look at what we know:
• How many digits are there?
• How many numbers are there?
• Do we have to use 1,2,3… or can we use something else?
• Do we know any other number systems?
• When is 8 + 5 = 1?
10 digits
∞ (infinity)
Any symbol will work.
Yes!
On a Clock!
Back in the day…
• Different groups used different symbols.
• Symbols could be a single value or different values (depending on where they were).
• Here’s some examples:
In Mayan Math
Thanks to: http://www.michielb.nl/maya/math.html
This is 1 This is 2
But this is 21
The Mayans had up and down place value!
So….
• If this is one:
• And this is two:
• Then the sum is:
O O O O xx O O O xx O
O O O x xx x
(1)
(10)
(11)
Lights, Lights, Lights!
Binary Number Light 5 Light 4 Light 3 Light 2 Light 1 (1’s and 0’s)
1. __ O O O O x ____1_______2. __ O O O x O ____10______3. __ O O O x x ____11______4. __ O O x O O ____100_____5. __ O O x O x ____101_____6. __ O O x x O ____110_____
So let’s double some numbers
101 11 111 100 1010
1010 110 1110 1000 10100
Is there a pattern?
Why does it work for doubling?
Is it similar to a pattern we use in our system?
So to double over and over…
• Add a zero each time you double
• So in our number system we would write 1 x 2 x 2 x 2 if we wanted to double the number 1 three times.
• The shortcut for that would be 1 x 23
• In binary that number would be…
• 1000 (a zero for each double!) Exponent
Try writing these answers in binary --
3 x 24
4 x 23
7 x 25
13 x 23
= 11
= 100
= 111
= 1101
3 is 11 so with four zeroes it would be…
0000
00000000
000
So back to the Birthday Cards
• What is so special about the numbers on card #1?
• Look at your lights, lights, lights sheet and tell me if the numbers have something in common in binary.
• What about card #2? #3? #4? And #5?
Card #1
1 3 5 7
9 11 13 15
17 19 21 23
25 27 29 31
Birthday Cards
Card #1
1 3 5 7
9 11 13 15
17 19 21 23
25 27 29 31
Card #2
2 3 6 7
10 11 14 15
18 19 22 23
26 27 30 31Card #3
4 5 6 7
12 13 14 15
20 21 22 23
28 29 30 31Card #4
8 9 10 11
12 13 14 15
24 25 26 27
28 29 30 31
Card #5
16 17 18 19
20 21 22 23
24 25 26 27
28 29 30 31
So our base 10 system has shortcuts too…
• If binary had a shortcut for doubling ( x 2) then our system has one for…
• x 10
• So if I want to multiply a number by ten all I have to do is _______ ?
• And if I want to multiply by ten twice or three times?
For Example
• 34 x 10 =• 723 x 104 =
• 9 x 107 =• 4,571 x 102 =• 500 x 103 =
• This is TOO easy!
340
7,230,000
90,000,000
457,100
500,000
How is Xmania like our decimal system?
• Has a digit for zero.
• Uses place value (except they add digits to the left instead of the right).
• Has shortcuts for multiplying.
• _________________
• _________________
Your system should have:
• A name
• A digit for “zero”
• 3 or 4 digits total
• Place value• Multiplication shortcut (with explanation)
Let’s sum up!
• How are place valued number systems alike?
• What are the major differences?
• What are the shortcuts to our number system?
• Do the number shortcuts work with other number systems (like Xmania)?
Let’s sum up!
650,000
784,000
400,000
930
• Here’s a few for you to review:
• 65 x 104 =
• 784 x 103 =
• 4 x 105 =
• 93 x 10 =