Mathematische Sachaufgaben & Denkmodelle · Computer Science Principles (Binary & Bases)
Binärsystem & Balkenwaagen-Visualisierer
Zweiersystem & Gewichte-Teilung: Mechanische Kugelbahn und Gewichte 1, 2, 4, 8, 16g
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Experimentelle Steuerungsparameter
g
Basis
Echtzeit-Herleitung & Kennzahlen
Binärausgabe
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Aufgelegte Gewichte
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Hexadezimaldarstellung
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Benötigte Bits
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Echtzeit-Formelherleitung
Grundformel:
--Einsetzungsschritte:
--Ergebnis: --
Typische Beispielaufgaben & Prüfungsfallen
Worked Problem
Aufgabe: Scale with weights 1, 2, 4, 8, 16g on right pan only. 1) How many distinct weights can be measured? 2) How to weigh 25g?
Schrittweise Herleitung:
- 1) 2⁵ - 1 = 31 distinct weights from 1g to 31g. 2) 25 in binary is 11001_2 = 16 + 8 + 1, so use 16g, 8g, and 1g weights.
Worked Problem
Aufgabe: Convert decimal 45 into binary and hexadecimal.
Schrittweise Herleitung:
- Divide by 2 successively: remainders bottom-up yield (101101)₂. Split into 4-bit nibbles: 0010 1101 = 0x2D.
Arbeitsblatt für den Unterricht & Hausaufgaben
Q1. Variant 1: 1000 bottles of wine, 1 poisoned (24h onset). Minimum mice to identify the poisoned bottle in 24 hours?
Answer Key: 2¹⁰ = 1024 > 1000, so only 10 mice! Assign 10-bit binary labels to bottles; dead mice encode the binary digits of the poison.
Q2. Variant 2: Weights 1, 3, 9, 27g can be placed on EITHER pan. Max weighable weight? How to balance 11g?
Answer Key: Max weight = 1+3+9+27 = 40g. For 11g: 11 = 9 + 3 - 1. Left pan: object + 1g. Right pan: 9g + 3g (balanced at 12g).
Q3. Variant 3: Why do digital computers use binary (0 and 1) instead of familiar decimal?
Answer Key: Hardware reliability! Representing two physical states (on/off, high/low voltage) provides vast noise immunity compared to 10 voltage levels.
Häufige Fragen & typische Missverständnisse
What does "1 Byte = 8 bits" mean? Why is 1KB = 1024 Bytes rather than 1000?
1 bit is a 0/1 binary digit. 8 bits form 1 Byte. In base-2 systems, 2¹⁰ = 1024 is the natural metric multiplier closest to 1000.
Why are binary weight sums strictly unique without redundant combinations?
The Binary Representation Uniqueness Theorem proves every natural number has a strictly unique expansion into powers of 2.
Why is hexadecimal ubiquitous in computing (e.g. 0xFF)?
Hex is shorthand for binary! Exactly 4 bits condense into 1 hex digit (1111 = F). 0xFF compactly denotes 11111111 (decimal 255).
If 4 weights can balance any integer from 1 to 40g using both pans, what are their masses?
Powers of 3: 1g, 3g, 9g, 27g! Bachet's weighing puzzle proves balanced ternary (-1, 0, +1) maximizes encoding efficiency.