the sum of the powers of two

You know base ten cards? They are little cards with pictures of cubes on them. Some have just one cube, others ten, others 100, and others 1000. Using the cards helped my oldest son learn adding and subtracting from four digit numbers.

The other day I made binary cards. The first one had one square, the second two, the third 4, the fourth 8 etc, etc. Then I could show my children that just as they could use base ten cards to make up any number, they could use binary cards. We talked about how we only need one set of each number to be able to make up any other number, but that having two sets would allow us to add and subtract numbers. Using the cards gives us a physical way of seeing that every number can be represented by the sum of powers of two.

Using binary cards gives an easy way to start writing numbers in binary. Lay out all the cards with the one with the most squares on  the left and then with numbers going progressively lower. Choose any number under 63 (or you have to make more cards than the ones pictured above). Pick the cards needed to make up that number and flip the rest over. Now think of the flipped up cards as representing 1s and the flipped down cards as representing 0s.

The ancient Egyptians used this fact in how they did multiplication. A video about it is found at: http://bestofyoutube.com/egyptian-maths and there is a description of the method in The secret life of math : discover how (and why) numbers have survived from the cave dwellers to us! by Ann McCallum. A hands-on way to do this is with sets of binary cards. Try printing out the cards illustrated. Take the first set of six cards and use them to multiply by three. Try 3 x 9. Thinking in terms of binary and powers of two, nine equals one group of eight plus a one, so take the card with the eight squares and the card with the single square, add the numbers on them together and you get: 3 x 9 = 27. Try it with a higher number. What is 3 x 19? 19 = 16 + 2 + 1, so 3 x 19 = 48 + 6 + 3 or 57.

Although no knowledge of binary is necessary for the trick it can be thought of as having the person identify in which column the number 1 would be and in which a 0 would be. So the number 14, written as 1110 in binary, would require the card starting with 8, the card starting with 4 and the card starting with 2. Using Ms. Colgan’s cards, from the book, gives a chance to explore binary in a different way.