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Questions and Answers
What is the result of adding 1111 and 10001 in binary?
What is the result of adding 1111 and 10001 in binary?
Which type of complement can be used to simplify the subtraction of n-digit binary numbers?
Which type of complement can be used to simplify the subtraction of n-digit binary numbers?
How is the 1's complement of a binary number obtained?
How is the 1's complement of a binary number obtained?
What is the definition of the 2's complement of a binary number?
What is the definition of the 2's complement of a binary number?
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What is the result of adding 11001 and 111 in binary?
What is the result of adding 11001 and 111 in binary?
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Which type of complement is referred to as the (r - 1)'s complement?
Which type of complement is referred to as the (r - 1)'s complement?
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What is the value of the 1's complement of the binary number 10011011?
What is the value of the 1's complement of the binary number 10011011?
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When is the 2's complement of a binary number 0?
When is the 2's complement of a binary number 0?
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Study Notes
Number Systems Overview
- Number systems represent and process numerical data in computing, using electric signals with two states (on and off).
- Four primary number systems are: decimal, binary, octal, and hexadecimal.
Decimal System
- Base 10 system using digits from 0 to 9.
- Value of a digit depends on its position, represented as coefficients of powers of 10.
- Example: 748 in decimal = 710² + 410¹ + 8*10⁰ = 700 + 40 + 8.
Binary System
- Base 2 system using only digits 0 and 1.
- Compactly represents data; effective for digital systems.
- Example: 10001 in binary = 12⁴ + 02³ + 02² + 02¹ + 1*2⁰ = 16 + 1 = 17 in decimal.
Octal System
- Base 8 system with eight digits: 0-7.
- Compact representation of binary; octal digits correspond to groups of three binary digits.
- Example: 135 in octal = 18² + 38¹ + 5*8⁰ = 64 + 24 + 5 = 93 in decimal.
Hexadecimal System
- Base 16 system using digits 0-9 and letters A-F (A=10, B=11, C=12, D=13, E=14, F=15).
- Efficient representation of binary; groups of four binary digits convert to one hex digit.
- Example: 27FB in hexadecimal = 216³ + 716² + 1516¹ + 1116⁰ = 8192 + 1792 + 240 + 11 = 10235 in decimal.
Conversion between Number Systems
- To convert from decimal to another base, divide by the new base and collect remainders.
- For fractional decimal conversions, multiply the fraction by the new base and collect integer parts.
- Example conversions:
- 320 in decimal to binary = 101000000.
- 153 in decimal to octal = 231.
- 46687 in decimal to hexadecimal = B65F.
Changing Ranges and Representations
- The range of numbers in a base relates to the number of bits used.
- For n bits in unsigned binary, the range is 0 to (2^n - 1).
- Unsigned fractions range from 0 to (2^n - 1)/2^n.
Arithmetic Operations in Different Bases
- Operations follow the same principles as in base 10.
- Pay attention to allowable digits in each base and carry operations when sums exceed the base's digit limit.
Important Formulas and Findings
- For converting radix, for example, to find r in (211)r = (152)8 leads to the equation involving r and numerical equivalences between bases.
- Quadratic formula can be used to solve complex relationships in base conversions.
Summary of Course Content
- Topics include computer basics, number systems, HTML, the internet, JavaScript, networking fundamentals, and system security.
Studying That Suits You
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Description
Learn how to convert numbers from the decimal system to other bases by following a series of steps. This quiz covers the identification of the base, division by the base, accumulation of remainders, and more.
