sc lecture 5

Chapter 3: Representation of numbers

Key Concepts

Positional numeral systems

• A numeral system is a writing system for expressing numbers, that is, a notation for expressing numbers of a given set, using digits or other symbols.
• Computers use the binary numeral system.
• Humans use the decimal (Hindu-Arabic) numeral system.
• They both are positional numeral systems, which represent any number by a sequence of juxtaposed digits. This means that the value of each digit depends on it’s position in the number.
  • Juxtaposed digits refer to placing digits side-by-side without an explicit operator, commonly representing multi-digit numbers (e.g., 2 and 5 becoming 25). In the Arabic numeral system, digits are placed together to represent tens, hundreds, etc. (e.g., the number “254” is a juxtaposition of 2, 5, and 4.
  • $4682=4\ast 1000+6\ast 100+8\ast 10+2\ast 1$
• Positional numeral systems can use any integer $r\ge 2$ for the base, and the digit in position $i$ has weight $r^i$
  • $256_6=5\ast 6^0+3\ast 6^1+2\ast 6^2$
• Digital devices adopt the binary base ($r=2$), which uses two possible digits (0 and 1).
  • The general forma of a number B in such cases is
    • $B=b_{n-1}{b}_{n-2}\dots b_1{b}_0$
  • The value of this natural number is:
    • $B={\sum }_{i=0}^{n-1}{b}_i\ast 2^i$
• The range of these numbers is defined as the difference between the largest and the smallest representable number in a given numeral system.
  • A representation with $n$ bits ranges from 0 to $2^n-1$. With eight bits, this range goes from 0 to +255.
• The integer part of a number is separated from its fractional part by a radix point
  • In the decimal system, the radix is called decimal point.
    • $468.51=4\ast 1000+6\ast 100+8\ast 10+2\ast 1+5\ast 0.1+1\ast 0.01$
  • Binary fractions have the binary point. But if the computer just sees 1’s and 0’s how does it know if the number has a binary point? That information has to be in the program that its been written that maybe is the reason for which in the programming language C you have to declared that a variable is a float.

Before you go # 5

What is the most appropriate numeral system to represent money —plata—(Euros) in a computer?

For some reason I did not catch the answer is integer lol

Related topics

• Essentials-of-computing-systems, page 36
• SC-slidesEN-chap3-RepresentationNumbers, page 1