Difference between revisions of "Number Systems"
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== Customs == | == Customs == |
Revision as of 09:45, 24 November 2021
Curriculum[edit]
Coder Merlin™ Computer Science Curriculum Data | |
Unit: Numbers Experience Name: Number Systems (W1011) Next Experience: () Knowledge and skills:
Topic areas: Positional notation Classroom time (average): 60 minutes Study time (average): 180 minutes Successful completion requires knowledge: understand positional notation Successful completion requires skills: ability to use positional notation to represent numbers in the binary, octal, decimal, and hexadecimal systems; ability to convert between representations of numbers in the binary, octal, decimal, and hexadecimal systems |
Experience[edit]
This is a multi-page experience, consisting of the following pages. Be sure to visit them all in sequence. A navigation bar at the bottom of each page is available to assist you.
- Positional Notation
- Alternative Bases
- Formal Representation
- Conversions to Decimal
- Conversions from Decimal
- Customs
- Shortcut Conversions
Customs[edit]
- It is customary to separate groups of three decimal digits by commas. For example: 123,45610.
- It is customary to separate groups to two hexadecimal digits by spaces and prefix a leading zero to what would otherwise be a single digit. For example: 01 2F16.
- It is customary to separate groups of three octal digits by spaces and prefix leading zeros to pad to a three digit grouping. For example: 001 2378.
- It is customary to separate groups of four binary digits by spaces and prefix leading zeroes to pad to a four digit grouping. For example: 0001 1011 11002.
Shortcut Conversions[edit]
Because both the octal and hexadecimal bases contain and possible digits, respectively, and the binary base contains possible digits, we're able to utilize a shortcut when converting between these systems. Note that each octal digit is comprised of three binary digits and each hexadecimal digit is comprised of four binary digits. We can therefore establish (and memorize) the following table:
Decimal | Binary | Octal | Hexadecimal |
---|---|---|---|
00 | 0000 | 00 | 0 |
01 | 0001 | 01 | 1 |
02 | 0010 | 02 | 2 |
03 | 0011 | 03 | 3 |
04 | 0100 | 04 | 4 |
05 | 0101 | 05 | 5 |
06 | 0110 | 06 | 6 |
07 | 0111 | 07 | 7 |
08 | 1000 | 10 | 8 |
09 | 1001 | 11 | 9 |
10 | 1010 | 12 | A |
11 | 1011 | 13 | B |
12 | 1100 | 14 | C |
13 | 1101 | 15 | D |
14 | 1110 | 16 | E |
15 | 1111 | 17 | F |
The shortcut is very straight-forward yet a significant time saver and based on the above information. When converting between binary, octal, and hexadecimal, there's no need to use an intermediate, decimal step.
Binary to Hexadecimal[edit]
Group the binary number in sets of four digits, from right to left. Each binary set represents a single hexadecimal digit. (It's sometimes helpful to pad the final set with zeroes.) For example, given the number , regroup the binary digits in sets of four:
.
Then, use the memorized table to translate each set of four binary digits to a single hexadecimal digit:
1100 | 0111 | 0010 | 0101 |
C | 7 | 2 | 5 |
Hexadecimal to Binary[edit]
Simply follow the above procedure in reverse. Use the memorized table to convert each hexadecimal digit to a set of four binary digits.
Binary to Octal[edit]
Group the binary number in sets of three digits, from right to left. Each binary set represents a single octal digit. (It's sometimes helpful to pad the final set with zeroes.) For example, given the number , regroup the binary digits in sets of three:
.
Then, use the memorized table to translate each set of three binary digits to a single octal digit:
001 | 100 | 011 | 100 | 100 | 101 |
1 | 4 | 3 | 4 | 4 | 5 |
Octal to Binary[edit]
Simply follow the above procedure in reverse. Use the memorized table to convert each octal digit to a set of three binary digits.
Octal to Hexadecimal (and vice versa)[edit]
This requires slightly more work. Convert the number in the source base to binary, and then convert from binary to the target base.
Exercises[edit]
References[edit]
- Positional Notation (Wikipedia)
- Radix (Wikipedia)
Experience Metadata
Experience ID | W1011 |
---|---|
Next experience ID | |
Unit | Numbers |
Knowledge and skills | §10.311 |
Topic areas | Positional notation |
Classroom time | 60 minutes |
Study time | 3 hours180 minutes <br /> |
Acquired knowledge | understand positional notation |
Acquired skill | ability to use positional notation to represent numbers in the binary, octal, decimal, and hexadecimal systems ability to convert between representations of numbers in the binary, octal, decimal, and hexadecimal systems |
Additional categories |