What do we now know about number systems? Number systems are predicable (we can figure out which symbol [number] comes next), they have an order/arrangement (permutation), and represent real quantities.

Let’s review from yesterday We created a new number system using 3 unique ‘symbols’ [Circle, Square, Triangle] How could we use these three shapes to create a predictive system [be able to know what comes next using a pattern or rule]? We would need some sort of “order” in our system – decimal, we know it begins with 0, then 1, then 2…all the way to 9. We know that 3 never comes after 8 or some other number and we can predict which number (symbol) will come next.

Step 1 The slide states “1 place”. Think to how the decimal system works. We have a ones place, a tens place, a hundreds place, et cetera. In this example, all ‘permutations’ or arrangements of the shapes are shown.

Step 2 Now we add a new level. This is akin to adding the 10’s place in our decimal system. However, in this system, there are only 3 shapes. We take the order: Circle, Triangle, Square and repeat that pattern adding a “tens” place in front. In this example, this would be similar to showing 10, 11, 12 then 20, 21, 22, then 30, 31, 32 – remember this system only uses 3 shapes, so ’32’ would be as high as it would go without needing another place value ~ a ‘hundreds’ place. Just to be clear, I am not saying that these shapes directly mean “10, 11, 12” “20, 21, 22” and “30, 31, 32” I just wanted to show how it works using the place value.

Step 3

Step 4

Binary Numbers Place has it’s value

Binary number system Proper number system Uses only two ‘shapes’ or ‘symbols’ Predictive set of rules for using 1s and 0s to represent values Allows for counting and arithmetic operations In Computer Science, binary numbers are represented as 0 and 1. These are the most simple (simple, because there are only two symbols) number system and the most efficient way for us to transmit information using computers. This is why we use binary in programming.

What did the Rich computer scientist say when she walked into the pun shop? Warning: bad joke on the next slide

I’m sorry, but I did warn you.  “I’m binary thing!”

Odometer Think of a vehicle’s odometer… How does it function? (yes, it records mileage, but think of the mechanics… what does a rolling odometer look like?)

Patterns Any binary number with a single 1 is a power of 2 Just like any in our number system, any number ending in 0 (10, 20, 30, 40… etc.) is a power of 10 In this slide, we see that in order to represent the number 1, we only need one bit (binary digit). In order to represent the number 2, we need two digits, or bits. In order to represent the number 4, we will need 3 bits (binary digits) and in order to represent the number 8, we will need 4 digits, or places. In decimal with those same amount of digits – we can represent up to 1000.

The Decimal system In the “One’s place”, there are 10 (decimal – base 10) possibilities: 10 x 1 = 10 In the “Ten’s place”, there are 100 possibilities: 10 x 10 = 100 In the “Hundred’s place”, there are 1000 possibilities: 100 x 100 = 1,000 Etc.

This may seem basic. It is This may seem basic. It is. But it is important to remember what each number actually means. For decimal, it is easy, the number 4,017 represents 4,017 ones as depicted above. This won’t be the case with binary (base 2), or other base systems as what is represented means something else. For example (we will see this in later slides): 100 in binary does not mean one hundred ones, it actually represents the number 4 (decimal). Essentially, we are finding out how many ‘ones’ are in this number by determining how many ‘ones’ are in each place value

Circle Triangle Square ___ in the 3’s place ___ in the 9’s place ___ in the 27’s place

With Binary… ___ in the 2’s place ___ in the 4’s place

Constructing a binary number means figuring out which powers of 2 add up to the number you want 128 64 32 16 8 4 2 1 0 0 1 0 1 0 0 1 In this slide, we can see that the binary 00101001 is 41. With a 1 in the 32 place, and a 1 in the 8’s place and a 1 in the 1’s place, we add that all up and get 41. 11111111 would mean 128 +64+32+16+8+4+2+1 = 255 + 1 = 41 + 8 32

Let’s make a “Flippy do” Cut along the dotted lines Fold the bottom row up. On the top row, write all the powers of 2, from right to left 27 26 25 24 23 22 21 20 On the bottom row, write a 0 in every cell. Flip the bottom row up and write a 1 in every cell. Calculate the powers of 2 in the 2nd row

We now have a decimal to binary converter! The largest value our “flippy do” will create is 255 in decimal. With more place values, we could create a “decimal / binary converter” with a larger value possibility.

There are 10 types of people in the world, Those who understand binary and those who don’t. Why is this a joke? Because it appears to read “There are ten…” when it is really stating there are 2 (one, zero = 2 in decimal) types… If you “understood binary” then the joke would make sense to you.