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Further Reading for Teachers
Binary Expansions
Any number 0, 1, 2, 3, 4, 5 and so on, can be written as a sum of powers of
2: 20 = 1; 21 = 2; 22 = 4; 23 = 8;
24 = 16, 25 = 32 and so on. For example: 5 = 4 + 1; 11 = 8 + 2 + 1; 8 = 8 (a power of 2 itself). This is called the binary expansion (the so-called "secret codes" of the lesson). Each number has its very own binary expansion.
Binary expansions have important applications in math and science.
Dictionary for Numbers and Their
Binary Expansions
Let's choose a range of numbers to fix ideas. The best way is to fix a
range of powers of 2. For example, let's use 1, 2, 4, 8, 16, as in the
lesson. The biggest number will be
31 = 16 + 8 + 4 + 2 + 1
In other
words, we are working with numbers from 0 to 31.
-
Pick any number between 0 and 31, say 13. Succesive divisions by 2
will lead to its binary expansion. Start by dividing 13 by 2.
| | 13 = |
2 × 6 + 1 |
| Now divide 6 (the quotient) by 2 |
| | 13 = |
2 × (2 × 3) + 1 |
| and simplify the right-hand side. |
| |
13 = |
22 × 3 + 1 |
| Next divide 3 (the quotient) by 2 |
|
13 = |
22 × (2 × 1 + 1) + 1 |
| and simplify the right-hand side. |
| |
13 = |
23 + 22 + 1 |
| We find: |
| |
13 = |
8 + 4 + 1 |
So 13 = 8 + 4 + 1 is the binary expansion. In the
language of the lesson, the secret code for 13 is 8, 4, 1 .
- Now pick a binary expansion (or a secret code). For example, 16 + 4 +
1 (that is, pick the secret code 16, 4, 1). The corresponding number
is 21, since
21 = 16 + 4 + 1
Comments
- There are several other ways to find the binary expansion of a
number. One is to use the Human Counting Machine; another technique
with beans is explained in the lesson "Guess the Number" Magic Trick.
- Consider the number 7105. Its decimal expansion is
7 × 1000 + 100 + 5 = 7 × 1000 + 1 × 100 + 0
× 10 + 5
This decimal expansion is sometimes written
[7105]10.
This means that the digits 7, 1, 0, and 5 multiply the powers of 10:
1000, 100, 10, and 1 respectively.
Consider now the number 50. Its binary expansion is
32 + 16 + 2 = 1 × 32 + 1 × 16 +
0 × 8 + 0
× 4 + 1 × 2 + 0 × 1
This binary expansion is sometimes written [110010]2.
This means that the digits 1, 1, 0, 0, 1 and 0 multiply the powers of 2:
32, 16, 8, 4, 2, and 1 respectively.
 
Send questions, comments to asofer@math.utexas.edu. |
