BME303_lecture4_chap2 - BME303 Intro. to Computing Chapter...

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1 BME303 Intro. to Computing Chapter 2 – cont’d BME303 Intro. to Computing 2 Hexadecimal? 0001001010101011 0x 12AB – a convenient way to represent binary strings 0001 0010 1010 1011 12 A B
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2 BME303 Intro. to Computing 3 Hexadecimal? 0001 0010 1010 1011 12 A B Decimal value = ? http://easycalculation.com/hex-converter.php BME303 Intro. to Computing 4 D6 b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 1 1010110 Working with long strings of 1s and 0s is difficult We use hexadecimal (or hex) notation as a form of shorthand 1101 0110 = 0x D 6 = # ??? Hex is a 16-base number system How to convert to/from hex? hint: use binary as middle-man What about sign? Hexadecimal Notation
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3 BME303 Intro. to Computing 5 Hexadecimal Numbers Convert binary 0011011011010101 to hex 0x36D5 0011 0110 1101 0101 3 6 D 5 Binary Hex 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 A 1011 B 1100 C 1101 D 1110 E 1111 F 1001101100000111 0011101100000111 1101111000101111 1011111011101111 BME303 Intro. to Computing 6 Floating Point: Fractions Decimal Binary #3.5 11.1 b
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4 BME303 Intro. to Computing Floating Point: Fractions • Decimal: 2.010 · 10 3 = 2010 • Binary: 1.0011· 2 2 = 100.11 7 BME303 Intro. to Computing 8 10 2 10 1 10 0 10 -1 10 -2 10 -3 100 10 1 1/10 1/100 1/1000 2 2 2 1 2 0 2 -1 2 -2 2 -3 4 2 1 1/2 1/4 1/8 = #3.50 Floating Point: Fractions Decimal Binary #3.5, 4.75, 5.25 ??? = #4.75 = #5.25 = #3.50 = #4.75 = #5.25
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5 BME303 Intro. to Computing 9 Floating Point Computers represent real numbers using Floating Point notations Decimal: 2010 = 2.010 · 10 3 Binary: 100.11 = 1.0011· 2 2 IEEE Standard : ( 1 ) S ·1. fraction ·2 exponent 127 (1 exponent 254) S exponent (8-bit) fraction (23-bit) 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 5 2 1 0 32 bits total; exponent is an unsigned 8-bit integer BME303 Intro. to Computing 10 Floating Point Three step process: - convert the decimal number to a binary number - write binary number in “normalized” scientific notation - find the exponential term - store the number in the proper format ( 1 ) S ·1. fraction ·2 exponent 127 (1 exponent 254) S exponent (8-bit) fraction (23-bit) 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 5 2 1 0 IEEE Standard for Floating Point Arithmetic (“rules”)
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6 BME303 Intro. to Computing 11 Floating Point: Example 1 01111110 10000000000000000000000b – Sign is 1 , meaning the number is negative – Exponent field is 01111110 = 126 (decimal) – Fraction is . 100000000000 1. 1 ·2 ( 126 -127) b= 1. 1 -1 b= 0.11 b= 0.75 ( 1 ) S ·1. fraction ·2 exponent 127 (1 exponent 254) BME303 Intro. to Computing •0 10000000 00000000000000000000000 = +1 * 2**( -127) * 1.0 = 2 10000001 10100000000000000000000 = * 2**(129-127) * 1. = •1 10000001 10100000000000000000000 = 12 ( 1 ) S ·1. fraction ·2 exponent 127 (1 exponent 254)
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7 BME303 Intro. to Computing 13 •When exponent field is 00000000 and fraction is 0, the value is +/- 0 (depending on the sign bit) •When exponent field is 00000000 and fraction is not 0, use this formula instead: ( 1) S · 0 .fraction·2 -126 •When exponent field is 11111111 and fraction is 0, the value is +/- inf •When exponent field is 11111111 and fraction is not 0, the value is NaN (not a number) Question: What is the smallest non-negative, non-zero number that can be represented in IEEE 754 floating point format?
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BME303_lecture4_chap2 - BME303 Intro. to Computing Chapter...

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