A presentation on how numerical values are stored in a computer. It is an essential topic to understand when one wants to study on how mathematical calculations are done in a computer. It helps understand what to expect after a certain calculation is completed by a computer.

1. NumericalMethods
Lecture 1
Representation of Numerical Values in the Computer

2. Overview
– Number Systems
– Number Conversion
– Representation of Numbers
– Computer Arithmetic
– Errors in Arithmetic
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3. NumberSystems
Let us discuss mainly about “Positional Number
Systems”
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4. Number Systems
– Non-positional Number Systems
– Positional Number Systems
– Decimal
– Binary
– Hexadecimal
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5. NumberConversion
Converting numbers represented in one format, to
other
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6. Number Conversion
– Decimal to Non-decimal Number
– Non-decimal to Decimal Number
– Binary to Hexadecimal Number
– Hexadecimal to Binary Number
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7. Representationof
Numbers
How modern computers store and represent numbers
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8. Representation of Numbers
– Dependent on words
– Words are a number of bits
– Word size varies from machine to machine
– Computers process using binary
– Humans easily understand decimal
– Hexadecimal expresses more in less digits
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9. Representation of Numbers
– Integer Representation
– For word size n, maximum possible number to store is 2n – 1
– Negative numbers are stored as 2’s complement
– The use of sign bit makes the highest possible number 2n-1 – 1
– First toggle all the bits of the number
– Add 1 to the whole number
– Use the sign bit as it is
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10. Representation of Numbers
– Floating Point Representation
– Exponential form, 𝑥𝑥 = 𝑓𝑓 × 10𝐸𝐸
– 𝑓𝑓 is mantissa and 𝐸𝐸 is exponent
– The entire memory location is divided to three parts
– Sign, mantissa and exponent
– Typically 24 bits for mantissa and 7 bits for exponent are used on a
32 bit word representation system
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11. Representation of
Numbers
A sample floating point representation from
Numerical Methods textbook (Figure 3.1)
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SIGN EXPONENT MANTISSA
1 bit 7 bits 24 bits
0
24 23
31 30

12. Representation of Numbers
– Floating Point Representation
– Shifting of decimal pints  Normalisation
– Numbers in normalized form  Normalised Floating Point Numbers
– Two possible notations: 0.596 x 10-2 or .596E-2
– Conditions to be satisfied by mantissa, 𝑓𝑓
– For positive numbers: 0.1 ≤ 𝑓𝑓 < 1.0
– For negative numbers: −1 < 𝑓𝑓 ≤ −0.1
– In general: 0.1 ≤ |𝑓𝑓| < 1.0
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13. ComputerArithmetic
How computers process arithmetic operations
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14. Computer Arithmetic
– Integer Arithmetic
– Integer arithmetic always results in integer
– Exception:
– Cannot represent infinite numbers, bounded above and below
– Integer division provides two results: quotient and remainder
– Sample integer arithmetic:
– 25 + 12 = 37, 25 - 12 = 13, 12 – 25 = -13
– 25 x 12 = 300, 25 ÷ 12 = 2, 12 ÷ 25 = 0
– Show the following is incorrect for integer arithmetic
–
𝑎𝑎+𝑏𝑏
𝑐𝑐
=
𝑎𝑎
𝑐𝑐
+
𝑏𝑏
𝑐𝑐
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15. Computer Arithmetic
– Floating Point Arithmetic: Addition
– Let 𝑥𝑥 and 𝑦𝑦 be added to result in 𝑧𝑧; where 𝑥𝑥 ≥ 𝑦𝑦
– The fractional parts: f𝑥𝑥, f𝑦𝑦, f𝑧𝑧
– The exponent parts: E𝑥𝑥, E𝑦𝑦, E𝑧𝑧; where E𝑥𝑥 ≥ 𝐸𝐸𝑦𝑦
– Algorithm:
– Set E𝑧𝑧 = max(E𝑥𝑥, E𝑦𝑦)  E𝑧𝑧 = E𝑥𝑥
– Shift f𝑦𝑦 to the right by E𝑥𝑥 − E𝑦𝑦 places
– Set f𝑧𝑧 = f𝑥𝑥 + f𝑦𝑦
– Normalize if necessary
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16. Computer Arithmetic
– Floating Point Arithmetic: Subtraction
– Addition with different sign bits
– May result in mantissa overflow
– Left shift f𝑧𝑧 and decrease E𝑧𝑧 accordingly
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17. Computer Arithmetic
– Floating Point Arithmetic: Multiplication
– Multiply mantissa, f𝑧𝑧 = 𝑓𝑓𝑥𝑥 × 𝑓𝑓𝑦𝑦
– Add the exponents, E𝑧𝑧 = 𝐸𝐸𝑥𝑥 + 𝐸𝐸𝑦𝑦
– Normalise if necessary
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18. Computer Arithmetic
– Floating Point Arithmetic: Division
– Divide mantissa, f𝑧𝑧 = 𝑓𝑓𝑥𝑥 ÷ 𝑓𝑓𝑦𝑦
– Subtract the exponents, E𝑧𝑧 = 𝐸𝐸𝑥𝑥 − 𝐸𝐸𝑦𝑦
– Normalise if necessary
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19. Computer Arithmetic
– Floating Point Arithmetic: Practice
– Add 0.964572 E2 and 0.586351 E5
– Add 0.735816 E4 and 0.635742 E4
– Subtract 0.994576 E-3 from 0.999658 E-3
– Multiply 0.20 E4 and 0.40 E-2
– Divide 0.876543 by 0.200000 E-3
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20. Computer Arithmetic
– Floating Point Arithmetic: Practice Answers
– Add 0.964572 E2 and 0.586351 E5  0.587315 E5
– Add 0.735816 E4 and 0.635742 E4  0.1371558 E5
– Subtract 0.994576 E-3 from 0.999658 E-3  0.508200 E-5
– Multiply 0.20 E4 and 0.40 E-2  0.80 E1
– Divide 0.876543 by 0.200000 E-3  0.438271 E-1
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21. ErrorsinArithmetic
Errors that are seen in arithmetic operations
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22. Errors in Arithmetic
– Floating Point Arithmetic
– Inexact binary representation of decimal number: 0.1 to
0.0001100110011…
– Errors due to rounding-off a number: Add the numbers
0.500000 E1 and 0.100000 E-7
– Subtractive cancellation: Subtract 0.499998 from 0.500000
– Overflow or underflow of numbers
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23. Errors in Arithmetic
– Laws of Arithmetic
– Errors in arithmetic induce error in associative and distributive law
errors
– 𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧 ≠ 𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧 and 𝑥𝑥 × 𝑦𝑦 × 𝑧𝑧 ≠ 𝑥𝑥 × 𝑦𝑦 × 𝑧𝑧
– 𝑥𝑥 × 𝑦𝑦 + 𝑧𝑧 ≠ 𝑥𝑥 × 𝑦𝑦 + 𝑥𝑥 × 𝑧𝑧
– Provide an example where associative law of addition fails due to
an arithmetic error of floating point number
– 𝑥𝑥 = 0.456732 × 10−2
, 𝑦𝑦 = 0.243451, 𝑧𝑧 = −0.248000
– 𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧 ≠ 𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧
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24. Thank You
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