## Math Circle Week of March 21

We continued to count in binary today.   We started out by reviewing how to count the basic numbers in binary.

From there we learned how to add in binary, and use addition as a short cut to figure out some larger binary numbers.

Since binary is a base 2 system, we talked about powers of 2 and exponents.

So for instance, if I am translating the first several numbers from decimal to binary we use the following table.  We can see that each time we get to a power of 2 that we need a new column.

0 = 0

1 = 1

2 = 10

3 = 11

4= 100

5 = 101

6 =110

7 = 111

8 = 1000

9 = 1001

10 = 1010

11 = 1011

12 = 1100

So just looking at the powers of 2 we can see the relationship between the number of columns in binary and how many factors of two we have.

2 = 2 = 10

4 = 2 ×2  = 100

8= 2 ×2 ×2 = 1000

16 =2 ×2  ×2 ×2 =10,000

32 =2 ×2 ×2 ×2 ×2 = 100,000

64= 2×2×2×2×2×2 = 1,000,000

128 = 2×2×2×2×2×2×2 = 10,000,000

So using this and simple addition we can quickly find harder numbers without counting on our fingerless hands.

So for instance in decimal, 13 = 8 + 4 + 1

We could then write this in binary as 1000 + 100 + 1 = 1101

To find the number 19 = 16 + 2 + 1 = 10,000 + 10 + 1 = 10,011

Some of the kids at the end wanted to dive into multiplication and we touched on subtraction.  We’ll see if we can handle these two topics as we go forward,  They might be a bit too advanced.