Math1003 1.16 - Real Numbers

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