Difference between revisions of "Number Systems"

Latest revision as of 17:58, 8 January 2022

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Curriculum[edit]

Coder Merlin™ Computer Science Curriculum Data

Unit: Numbers

Experience Name: Number Systems (W1011)

Next Experience: ()

Knowledge and skills:

§10.311 Demonstrate proficiency in the use of positional notation to represent and convert between numbers in the binary, octal, decimal, and hexadecimal systems

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]

Start One page

Exercises[edit]

Exercises

M1011-10 Complete Merlin Mission Manager Mission M1011-10.

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

@@ Line 1: / Line 1: @@
-= Number Systems =
+== Curriculum ==
-== Positional Notation ==
+{{MerlinCurriculumData|{{ROOTPAGENAME}}}}
-'''Positional notation''' (sometimes called '''place-value notation''') is a method of encoding numbers. It differs from other notations (such as Roman numerals) in that it uses the same symbol for different orders of magnitude depending on its position. For example, consider the number 23. The “3” indicates 3 ''ones'', because it is in the ''ones'' position. The “2”, however, indicates 2 ''tens'', because it is in the ''tens'' position. We know the value of a position by its location within the number. As we move ''left'' in a number, each position is valued at ten times the prior position. It might help if we label each position using power notation. Consider the number 123:
+== Experience ==
+{{MerlinMultipageExperienceSubpages
+  |Pages=Positional Notation;Alternative Bases;Formal Representation;Conversions to Decimal;Conversions from Decimal;Customs;Shortcut Conversions
+}}
-[[File:Positional Notation.png]]
+== Exercises ==
+{{Exercises|
-The “3” is the right-most, and therefore the lowest-valued position, representing “ones” with a position multiplier of 100 (that is, 10 raised to the zero power, or 1). The “2” is located one position to the left, so we multiply by 10 again giving us a position multiplier of 101 (that is, 10 raised to the first power, or 10). Finally, the “1” is located one position to the left, so we again multiply by 10 giving us a position multiplier of 102 (that is, 10 raised to the second power, or 100). By multiplying each digit by its corresponding position multiplier, we can obtain the value of the entire number:
+* {{MMMAssignment|M1011-10}}
+}}
-* 10<sup>2</sup> + 2 * 10<sup>1</sup> + 3 * 10<sup>0</sup> =<br />
+== References ==
-+ 20 + 3 =<br />
+* [https://en.wikipedia.org/wiki/Positional_notation Positional Notation] (Wikipedia)
-<br />
+* [https://en.wikipedia.org/wiki/Radix Radix] (Wikipedia)
+{{Experience
-While we generally understand the mechanics of this process in the decimal system, thinking about how it actually works will enable us to consider other systems.
+|experienceID=W1011
+|experienceUnit=Numbers
-== Number Base ==
+|knowledgeAndSkills=§10.311
-The '''radix''' or '''base''' is the number of unique digits, ''including zero'', used to represent numbers in a positional numeral system. The base is normally written as a subscript to the right of the number. For example, the decimal number 123 would formally be written as (123)<sub>10</sub><br />
+|topicAreas=Positional notation
-Note that the parentheses are sometimes not written: 123<sub>10</sub><br />
+|classroomTime=60 minutes
-In the case of decimal (base 10) numbers, the subscripted 10 is often assumed and not written.
+|studyTime=3 hours
+|acquiredKnowledge=understand positional notation;
-== Decimal System ==
+|acquiredSkill=ability to use positional notation to represent numbers in the binary, octal, decimal, and hexadecimal systems;
-The decimal system is the system with which we are most familiar. It is a decimal system because it contains ten unique digits:<br />
+ability to convert between representations of numbers in the binary, octal, decimal, and hexadecimal systems;
-<pre>
+}}
-1 2 3 4 5 6 7 8 9
-</pre>
-For any (integer) value larger than 9, we’re required to use positional notation. Please keep in mind that “10” is not a digit. Rather, it’s a number consisting of two digits, a “1” in the tens (10<sup>1</sup>) position and a “0” in the ones position (10<sup>0</sup>).
-== Octal System ==
-The octal system uses eight unique digits to represent a number:<br />
-<pre>
-1 2 3 4 5 6 7
-</pre>
-Let’s consider the value of an octal number: 4735<sub>8</sub>
-What is the ''decimal'' value of this number? We use exactly the same method that we use for knowing the value in any system:
-{| class="wikitable" style="text-align:right;"
- | Digit at position:  || 4 || 7 || 3 || 5
- |-
- | Position multiplier: || 8<sup>3</sup> || 8<sup>2</sup> || 8<sup>1</sup> || 8<sup>0</sup>
- |-
- | Position value: || 4 &bull; 8<sup>3</sup> || 7 &bull; 8<sup>2</sup> || 3 &bull; 8<sup>1</sup> || 5 &bull; 8<sup>0</sup>
- |-
- |                 || 2048 || 448 || 24 || 5
- |}