
Power CardsApplication  Binary Numbers 
Let's use this mathemagic trick to demonstrate a nifty way to write numbers as binary or in Base 2. Read this excellent description from the Math Forum of how computers or calculators use binary numbers.
We ordinarily write numbers in base 10. That means there are 10 digits, 09. For example, consider 209 in ordinary base 10. You know that is 2 hundreds, plus 0 tens, plus 9 ones. Hundred, ten, and one are all powers of 10, our base. So we could think of it this way.
Base 10  

100  10  1 
10^{2}  10^{1}  10^{0} 
2  0  9 
Note: If you need an explanation of 0 as an exponent, here's a good one from the Math Forum.
Now we can easily use this same method to write numbers in base 2. Instead of powers of 10, we use powers of 2, and remember we can only use 2 digits  0 and 1. Let's convert 28 in base 10 to base 2. Remember from our Power Cards that 28 = 16 + 8 + 4.
Base 2  

16  8  4  2  1 
2^{4}  2^{3}  2^{2}  2^{1}  2^{0} 
1  1  1  0  0 
So 28 in Base 10 = 11100 in Base 2.
Now see how the Power Cards relate to binary numbers? See the chart below. If the number is in the card, it gets a 1 in that slot; if not, it gets a 0.
Base 2  

Card 5  Card 4  Card 3  Card 2  Card 1 
16  8  4  2  1 
2^{4}  2^{3}  2^{2}  2^{1}  2^{0} 
1  1  1  0  0 

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