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MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1.16Real Numbers
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Goal
To be able to represent real numbers in IEEE standard form.
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Real Numbers are stored by a totally different method from integers.
35.0
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Real Numbers are stored by a totally different method from integers.
Although many schemes are possible, we will explore a scheme that is widely used and accepted by the IEEE (Institute for Electrical and Electronic Engineering).
35.0
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Real numbers are stored as 32 bits (usually referred to as the type ‘float’) or64 bits (usually referred to as the type ‘double’).
35.0
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Real numbers are stored as 32 bits (usually referred to as the type ‘float’) or64 bits (usually referred to as the type ‘double’).
For our discussions, we’ll use 32 bitsto represent a real number.
35.0
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0The 32 bits are divided into 3 sections.
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0The 32 bits are divided into 3 sections.
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0signbit
The 32 bits are divided into 3 sections.
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0signbit
The 32 bits are divided into 3 sections.
exponent
8 bits
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0signbit
The 32 bits are divided into 3 sections.
exponent
8 bitsnumber
23 bits
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0signbit
The 32 bits are divided into 3 sections.This is the 1-8-23 pattern.
exponent
8 bitsnumber
23 bits
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
35.0signbit
The 32 bits are divided into 3 sections.This is the 1-8-23 pattern.
sign bit : 0 for a positive number, 1 for negative
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Steps to represent a number in IEEE standard form
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Steps to represent a number in IEEE standard form1. calculate the binary form of the number
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary)
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary)5. store the normalized binary form without the first 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1. 25.010 = 11001.02
Real Numbers
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1. 25.010 = 11001.02
2. normalize 11001.02
Real Numbers
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1. 25.010 = 11001.02
2. normalize 11001.02
Real Numbers
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1.1001 x 24
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1. 25.010 = 11001.02
2. normalize 11001.02
Real Numbers
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1.1001 x 24
this is similar to scientific notation except that we use
powers of 2
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1. 25.010 = 11001.02
2. normalize 11001.02
Real Numbers
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1.1001 x 24
we moved the point 4 positions to
the left, so our exponent is 4
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1. 25.010 = 11001.02
2. normalize 11001.02
3. set the sign bit
Real Numbers
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1.1001 x 24
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 25.010 in IEEE standard1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
25.010
is a positive number
1.1001 x 24
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit 25.010
is a positive number
1.1001 x 24
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section
1.1001 x 24
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section
1.1001 x 24we want to store both negative and positive exponents
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section
1.1001 x 24we’ll store it so that it corresponds to the
following table
00000000 -12700000001 -12600000010 -125
… …01111111 010000000 110000001 210000010 310000011 4
… …11111111 128
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section
1.1001 x 24we’ll store it so that it corresponds to the
following table
00000000 -12700000001 -12600000010 -125
… …01111111 010000000 110000001 210000010 310000011 4
… …11111111 128
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section
1.1001 x 24
00000000 -12700000001 -12600000010 -125
… …01111111 010000000 110000001 210000010 310000011 4
… …11111111 128
to correspond to the table, we’ll add 127
to the exponent
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section
1.1001 x 24
00000000 -12700000001 -12600000010 -125
… …01111111 010000000 110000001 210000010 310000011 4
… …11111111 128
4 + 127 = 131131 = 100000112
to correspond to the table, we’ll add 127
to the exponent
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 1 1
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section
1.1001 x 24
4 + 127 = 131131 = 100000112
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 1 1
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section5. store the normalized binary form
1.1001 x 24
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 1 1
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section5. store the normalized binary form
1.1001 x 24
1.1001
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 1 1
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section5. store the normalized binary form
1.1001 x 24
1.1001
the normalized form will always have a 1 to the left
of the point, so let’s ignore the 1 (but when we use this number in a
calculation, we’ll put it “back”)
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 1 1
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section5. store the normalized binary form
1.1001 x 24
1.1001
the normalized form will always have a 1 to the left
of the point, so let’s ignore the 1 (but when we use this number in a
calculation, we’ll put it “back”)
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 1 1 1 0 0 1
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section5. store the normalized binary form
1.1001 x 24
1.1001
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 1 1 1 0 0 1
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section5. store the normalized binary form
1.1001 x 24
1.1001
we’ll fill in the rest
with 0s
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Represent 25.010 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 25.010 = 11001.02
2. normalize 11001.02 = 1.1001 x 24
3. set the sign bit4. store 4 in the exponent section5. store the normalized binary form
1.1001 x 24
1.1001
we’ll fill in the rest
with 0s
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112 3410 = 1000102
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112 3410 = 1000102
.2 x 2 = 0.4
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112 3410 = 1000102
.2 x 2 = 0.4
.4 x 2 = 0.8
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112 3410 = 1000102
.2 x 2 = 0.4
.4 x 2 = 0.8
.8 x 2 = 1.6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112 3410 = 1000102
.2 x 2 = 0.4
.4 x 2 = 0.8
.8 x 2 = 1.6
.6 x 2 = 1.2
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112 3410 = 1000102
.2 x 2 = 0.4
.4 x 2 = 0.8
.8 x 2 = 1.6
.6 x 2 = 1.2
.2 x 2 = 0.4
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112 3410 = 1000102
.2 x 2 = 0.4
.4 x 2 = 0.8
.8 x 2 = 1.6
.6 x 2 = 1.2
.2 x 2 = 0.4
34.210 = 100010.00112
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
3. set the sign bit
1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers1 1
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
3. set the sign bit
1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers1 1
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
3. set the sign bit4. store 5 in the exponent section as (5 + 127 = 132) 100001002
1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers1 1 0 0 0 0 1 0 0 0
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
3. set the sign bit4. store 5 in the exponent section as (5 + 127 = 132) 100001002
1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers1 1 0 0 0 0 1 0 0 0
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
3. set the sign bit4. store 5 in the exponent section as (5 + 127 = 132) 100001002
5. store the normalized binary form
1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
3. set the sign bit4. store 5 in the exponent section as (5 + 127 = 132) 100001002
5. store the normalized binary form
1.000100011 x 25
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
3. set the sign bit4. store 5 in the exponent section as (5 + 127 = 132) 100001002
5. store the normalized binary form
1.000100011 x 25
we’ll fill in the rest with the repeating
pattern
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
Real Numbers1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
Represent -34.210 in IEEE standard1. 34.210 = 100010.00112
2. normalized as 1.000100011 x 25
3. set the sign bit4. store 5 in the exponent section as (5 + 127 = 132) 100001002
5. store the normalized binary form
1.000100011 x 25
we’ll fill in the rest with the repeating
pattern
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05.05 x 2 = 0.1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05.05 x 2 = 0.1.1 x 2 = 0.2
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05.05 x 2 = 0.1.1 x 2 = 0.2.2 x 2 = 0.4
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05.05 x 2 = 0.1.1 x 2 = 0.2.2 x 2 = 0.4.4 x 2 = 0.8
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05.05 x 2 = 0.1.1 x 2 = 0.2.2 x 2 = 0.4.4 x 2 = 0.8.8 x 2 = 1.6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05.05 x 2 = 0.1.1 x 2 = 0.2.2 x 2 = 0.4.4 x 2 = 0.8.8 x 2 = 1.6.6 x 2 = 1.2
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05.05 x 2 = 0.1.1 x 2 = 0.2.2 x 2 = 0.4.4 x 2 = 0.8.8 x 2 = 1.6.6 x 2 = 1.2.2 x 2 = 0.4
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112 .025 x 2 = 0.05.05 x 2 = 0.1.1 x 2 = 0.2.2 x 2 = 0.4.4 x 2 = 0.8.8 x 2 = 1.6.6 x 2 = 1.2.2 x 2 = 0.4
0.02510 = 0.00000112
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.000001120.000001100110011
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
0.000001100110011
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
1.100110011001
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
1.1001 x 2-6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
3. set the sign bit
1.1001 x 2-6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 0
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
3. set the sign bit
1.1001 x 2-6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 0
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
3. set the sign bit4. store -6 in the exponent section as (-6 + 127 = 121) 011110012
1.1001 x 2-6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 0 1 1 1 1 0 0 1 1
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
3. set the sign bit4. store -6 in the exponent section as (-6 + 127 = 121) 011110012
1.1001 x 2-6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 0 1 1 1 1 0 0 1 1
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
3. set the sign bit4. store -6 in the exponent section as (-6 + 127 = 121) 011110012
5. store the normalized binary form
1.1001 x 2-6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
Represent 0.02510 in IEEE standard
Steps to represent a number in IEEE standard form1. calculate the binary form of the number2. calculate the normalized binary form3. set the sign bit4. store the exponent (+127, store as an 8-bit binary) 5. store the normalized binary form without the first 1
1. 0.02510 = 0.00000112
2. normalized as 1.1001 x 2-6
3. set the sign bit4. store -6 in the exponent section as (-6 + 127 = 121) 011110012
5. store the normalized binary form
1.1001 x 2-6
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10001010 (= 138)
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10001010 (= 138)3. the decimal exponent is 138 - 127 = 11
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10001010 (= 138)3. the decimal exponent is 138 - 127 = 114. from the number section, we have 1.112
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10001010 (= 138)3. the decimal exponent is 138 - 127 = 114. from the number section, we have 1.112
5. therefore the number is 1.112 x 211
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10001010 (= 138)3. the decimal exponent is 138 - 127 = 114. from the number section, we have 1.112
5. therefore the number is 1.112 x 211
6. 1.112 = 1.75 and 211 = 2048
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10001010 (= 138)3. the decimal exponent is 138 - 127 = 114. from the number section, we have 1.112
5. therefore the number is 1.112 x 211
6. 1.112 = 1.75 and 211 = 20487. 1.75 x 2048 = 3584
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10001010 (= 138)3. the decimal exponent is 138 - 127 = 114. from the number section, we have 1.112
5. therefore the number is 1.112 x 211
6. 1.112 = 1.75 and 211 = 20487. 1.75 x 2048 = 3584
3584
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10000000 (= 128)
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10000000 (= 128)3. the decimal exponent is 128 - 127 = 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10000000 (= 128)3. the decimal exponent is 128 - 127 = 14. from the number section, we have 1.11001
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10000000 (= 128)3. the decimal exponent is 128 - 127 = 14. from the number section, we have 1.110015. therefore the number is 1.11001 x 21
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10000000 (= 128)3. the decimal exponent is 128 - 127 = 14. from the number section, we have 1.110015. therefore the number is 1.11001 x 21
6. 1.11001 = 1.78125 and 21 = 2
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10000000 (= 128)3. the decimal exponent is 128 - 127 = 14. from the number section, we have 1.110015. therefore the number is 1.11001 x 21
6. 1.11001 = 1.78125 and 21 = 27. 1.78125 x 2 = 3.5625
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 0, we know this is a positive number2. the exponent section is 10000000 (= 128)3. the decimal exponent is 128 - 127 = 14. from the number section, we have 1.110015. therefore the number is 1.11001 x 21
6. 1.11001 = 1.78125 and 21 = 27. 1.78125 x 2 = 3.5625
3.5625
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 1, we know this is a negative number
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 1, we know this is a negative number2. the exponent section is 01111000 (= 120)
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 1, we know this is a negative number2. the exponent section is 01111000 (= 120)3. the decimal exponent is 120 - 127 = -7
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 1, we know this is a negative number2. the exponent section is 01111000 (= 120)3. the decimal exponent is 120 - 127 = -74. from the number section, we have 1.001001
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 1, we know this is a negative number2. the exponent section is 01111000 (= 120)3. the decimal exponent is 120 - 127 = -74. from the number section, we have 1.0010015. therefore the number is 1.001001 x 2-7
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 1, we know this is a negative number2. the exponent section is 01111000 (= 120)3. the decimal exponent is 120 - 127 = -74. from the number section, we have 1.0010015. therefore the number is 1.001001 x 2-7
6. 1.001001 = 1.140625 and 2-7 = 0.0078125
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 1, we know this is a negative number2. the exponent section is 01111000 (= 120)3. the decimal exponent is 120 - 127 = -74. from the number section, we have 1.0010015. therefore the number is 1.001001 x 2-7
6. 1.001001 = 1.140625 and 2-7 = 0.00781257. 1.140625 x 0.0078125 = -0.008911132813
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Real Numbers1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Determine the real number1. since the sign bit is 1, we know this is a negative number2. the exponent section is 01111000 (= 120)3. the decimal exponent is 120 - 127 = -74. from the number section, we have 1.0010015. therefore the number is 1.001001 x 2-7
6. 1.001001 = 1.140625 and 2-7 = 0.00781257. 1.140625 x 0.0078125 = -0.008911132813
-0.008911132813
