Binary Representation
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GH Feedback from Josh (email address in Issue #221) 9/2/15:
- No way that binary numerals will take 15-20 minutes as suggested in the teacher's manual. It's not a long lecture for me, but I could imagine it consuming more than one class period, depending on the teacher and the students.
- This page feels out of place and rushed. The explanation is fine for someone who is inclined to do mathematics, but several of my students find it awkward. Of all the labs so far, this one seems to require the most intervention.
- It would be helpful if there were some small binary numbers to translate. Instead of writing numbers with lead zeroes (we don't do that with decimal numbers, do we?), just write the binary number. (A 2 subscript for binary numbers might not be bad.) 101 base 2 is 5. That would be a nice, small starting problem. Instead, the first part of #7 is 10000111 base 2, and that could be a load for a lot of students.
- In #8, I love how you are communicating the meaning of "bit" but I think it will be lost on a lot of people unless supplemented with something like, "Wonder where the word "bit" comes from?"
Here's a collection of powers of two:
1 |
2 |
4 |
8 |
16 |
32 |
64 |
128 |
256 |
512 |
1024 |
2048 |
4096 |
8192 |
16384 |
32768 |
- Find three different powers of two that add up to 21.
- Can you find any other ways to get 21?
- Find some different powers of two that add up to 66.
- Can you find any other ways to get 66?
- Find some different powers of two that add up to 15.
- Are there any whole numbers (positive integers) that aren't a sum of two different powers of two?
In decimal notation, each place value represents a power of ten: the units place (100 = 1), the tens place (101 = 10), the hundreds place (102 = 100), the thousands place (103 = 1000), etc. So, for example:
The symbol "XVIII" in Roman numerals represents the same number as the decimal representation "18." In binary, we write "10010" for "18," but it's still the same number. So there's no such thing as a "Binary number," just binary representation.
9827 = 9 × 103 + 8 × 102 + 2 × 101 + 7 × 100
Binary uses the same idea but with powers of two instead of powers of ten. So, for example:
10010 (in binary) = 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20 = 16 + 2 = 18 (in decimal)
Decimal notation is called base 10. It's the usual way of representing numbers with the digits "0-9." Binary is base 2. It only uses two digits: "0" and "1."
In both systems, place value always starts from the units place and goes up right to left (just as you learned in elementary school):
| base 10: |
|
|
0 |
9 |
8 |
2 |
7 |
| (higher places) |
... |
ten-thousands place |
thousands place |
hundreds place |
tens place |
units place |
| base 2: |
|
|
1 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
| (higher places) |
... |
128s place |
64s place |
32s place |
sixteens place |
eights place |
fours place |
twos place |
units place |
Computers store information in binary using bits and bytes. A bit is a "0" or "1". A byte is 8 bits grouped together like "10001001", which is binary for the number "137".
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Represent these bytes in decimal notation:
- 10000111
- 00100000
- 00011111
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Represent these decimal numerals in binary notation:
- 27
- 28
- 239
What's the rightmost digit in the binary representation of 15551212?- What's the rightmost bit ("binary digit") of 123456789?
- What numbers can be represented in a single bit?
- What numbers can be represented in a single byte?
- What numbers can be represented in a kilobyte?
Play the Cisco Binary Game.