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Numerical Methods
IS089IU
Floating Point Representation and Errors – Chapter 2
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Scientific Notation
Format Short
Format Long
Format short e
Format long e
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Floating Point Puzzles
 For each of the following expressions, do you understand what they are?
int x = …;
float f = …;
double d = …;
Problem with a "regular number" (fixed-point): A fixed-point system allocates a fixed
number of bits to the integer part and a fixed number to the fractional part.
• To store the small number, like 0.0000000001 mm, you need to allocate many bits for
the fractional part, leaving very few for the integer part.
• To store the large number, like 94,607,304,725,808 km, you need to allocate many
bits for the integer part, leaving no room for the fractional part to represent small
items.
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IEEE Floating Point
 IEEE Standard 754
 Established in 1985 as uniform standard for floating point arithmetic
 Before that, many idiosyncratic formats
 Supported by all major CPUs
 Driven by Numerical Concerns
 Nice standards for rounding, overflow, underflow
 Hard to make go fast
 Numerical analysts predominated over hardware types in defining standard
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Binary Number
 Base 2 number
 Using Matlab/Octave
 Decimal to Binary
 10(base 10)=1010 (base 2) =
1*23+0*22+1*21+0*20=8+0+2+0=10(base 10)
 10/2=5R0, 5/2=2R1,2/2=1R0, 1/2=0R1, Reverse the
remainders: 1010
 Binary to Decimal
 1001(base 2)=1*23+0*22+0*21+1*20=8+0+0+1=9(base 10)
 Decimal points number
 10.5(base 10)=1010.1(base 2)
 10.25(base 10)=1010.01(base 2)
 10.125(base 10)=1010.001(base 2)
Write a matlab
code to convert
decimal to binary
Use dec3bin
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Fractional Binary Number Examples
 Value
5-3/4
2-7/8
63/64
Representation
101.112
10.1112
0.1111112
 Observation
 Divide by 2 by shifting right
 Numbers of form 0.111111…2 just below 1.0
 Use notation 1.0 – 
 Limitation
 Can only exactly represent numbers of the form x/2k
 Other numbers have repeating bit representations
 Value
1/3
1/5
1/10
Representation
0.0101010101[01]…2
0.001100110011[0011]…2
0.0001100110011[0011]…2
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Floating Point Representation
 Numerical Form
 –1s M 2E
 Sign bit s determines whether number is negative or positive
 Significand M normally a fractional value in range [1.0,2.0).
 Exponent E weights value by power of two
 Encoding
s
exp
 MSB is sign bit
 exp field encodes E
 frac field encodes M
 Sizes
 Single precision: 8 exp bits, 23 frac bits
 32 bits total
 Double precision: 11 exp bits, 52 frac bits
 64 bits total
frac
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Step-by-Step Example using 32-bit IEEE 754 Format
 Let's convert the decimal number -10.625 to a 32-bit IEEE 754 format
 Step 1. Convert to Binary:
 Integer Part (10): 10 in decimal is 1010 in binary.
 Fractional Part (0.625):
 0.625 * 2 = 1.25 (write down 1)
 0.25 * 2 = 0.5 (write down 0)
 0.5 * 2 = 1.0 (write down 1)
 The binary fractional part is .101.
 Combined, -10.625 in binary is -1010.101.
Write Matlab code to
convert fractional
decimal to binary
(for .5, .25, .125, .75)
 2. Convert to Binary Scientific Notation:
 Move the binary point to create a normalized number, where there's a "1" to the
left of the point.
 -1010.101 becomes -1.010101.
 We moved the point 3 places to the left, so the exponent is 3.
 The number in scientific notation is -1.010101 x 2³.
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Step-by-Step Example using 32-bit IEEE 754 Format
 3. Determine the Sign Bit:
 Since the original number is negative, the sign bit is 1.
 4. Calculate the Biased Exponent:
 The exponent is 3.
 For 32-bit single-precision, the bias is 127.
 Biased Exponent = 3 + 127 = 130.
 Convert 130 to binary:
 130 / 2 = 65 remainder 0
 65 / 2 = 32 remainder 1
 32 / 2 = 16 remainder 0
 16 / 2 = 8 remainder 0
 8 / 2 = 4 remainder 0
 4 / 2 = 2 remainder 0
 2 / 2 = 1 remainder 0
 1 / 2 = 0 remainder 1
 Reading the remainders in reverse, 130 in binary is 10000010.
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Step-by-Step Example using 32-bit IEEE 754 Format
 5. Determine the Mantissa:
 The mantissa is the fractional part of the binary scientific notation: 010101.
 Since we have a limited number of bits for the mantissa in the 32-bit format (23
bits), we pad the end with zeros: 01010100000000000000000.
 6. The 32-bit IEEE 754 is: 11000001001010100000000000000000.
Check online for the conversion
https://www.h-schmidt.net/FloatConverter/IEEE754.html
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HW (for advance student only)
Write a Matlab code to convert a Decimal number to
IEEE 754 32-bit floating point number. Test with the
following numbers:
1- 10.12556
2- -125.47523
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HW week 3:
• Read chapter 1 and 2 of
Matlab book:
• Perform example 1.2
Provide the plot just as
figure 1.4 in Matlab