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So: -(-128) = -128.

Because 128 is out of the range of signed 8bits numbers.

3.2 addition and subtraction:

Addition and Subtraction is done using following steps:

  • Normal binary addition
  • Monitor sign bit for overflow
  • Take twos compliment of subtrahend and add to minuend ,i.e. a - b = a + (-b)

Hardware for addition and subtraction:

3.3 multiplying positive numbers:

The multiplying is done using following steps:

  • Work out partial product for each digit
  • Take care with place value (column)
  • Add partial products

Hardware implementation of unsigned binary multiplication:

Execution of example:

Flowchart for unsigned binary multiplication:

3.4 multiplying negative numbers

Solution 1:

  • Convert to positive if required
  • Multiply as above
  • If signs were different, negate answer

Solution 2:

  • Booth’s algorithm:

Example of Booth’s Algorithm:

3.5 division:

  • More complex than multiplication
  • Negative numbers are really bad!
  • Based on long division
  • (for more detail, reference to Computer Organization and Architecture, William Stalling)

4. floating-point representation

4.1 principles

We can represent a real number in the form

± S × B ± E size 12{ +- S times B rSup { size 8{ +- E} } } {}

This number can be stored in a binary word with three fields:

  • Sign: plus or minus
  • Significant: S
  • Exponent: E.

(A fixed value, called the bias, is subtracted from the biased exponent field to get the true exponent value (E). Typically, the bias equal 2 k 1 1 size 12{2 rSup { size 8{k - 1} } - 1} {} , where k is the number of bits in the binary exponent)

  • The base B is implicit and need not be stored because it is the same for all numbers.

4.2 ieee standard for binary floating-point representation

The most important floating-point representation is defined in IEEE Standard 754 [EEE8]. This standard was developed to facilitate the portability of programs from one processor to another and to encourage the development of sophisticated, numerically oriented programs. The standard has been widely adopted and is used on virtually all contemporary processors and arithmetic coprocessors.

The IEEE standard defines both a 32-bit (Single-precision) and a 64-bit (Double-precision) double format with 8-bit and 11-bit exponents, respectively. Binary floating-point numbers are stored in a form where the MSB is the sign bit, exponent is the biased exponent, and "fraction" is the significand. The implied base (B) is 2.

Not all bit patterns in the IEEE formats are interpreted in die usual way; instead, some bit patterns are used to represent special values. Three special cases arise:

  1. if exponent is 0 and fraction is 0, the number is ±0 (depending on the sign bit)
  2. if exponent = 2 e size 12{2 rSup { size 8{e} } } {} -1 and fraction is 0, the number is ±infinity (again depending on the sign bit), and
  3. if exponent = 2 e size 12{2 rSup { size 8{e} } } {} -1 and fraction is not 0, the number being represented is not a number (NaN).

This can be summarized as:

Single-precision 32 bit

A single-precision binary floating-point number is stored in 32 bits.

The number has value v:

v = s × 2 e size 12{2 rSup { size 8{e} } } {} × m

Where

s = +1 (positive numbers) when the sign bit is 0

s = −1 (negative numbers) when the sign bit is 1

e = Exp − 127 (in other words the exponent is stored with 127 added to it, also called "biased with 127")

m = 1.fraction in binary (that is, the significand is the binary number 1 followed by the radix point followed by the binary bits of the fraction). Therefore, 1 ≤ m<2.

In the example shown above:

S=1

E= 011111100(2) -127 = -3

M=1.01 (in binary, which is 1.25 in decimal).

The represented number is: +1.25 × 2−3 = +0.15625.

5. floating-point arithmetic

The basic operations for floating-point X1 = M1 R E1 size 12{X1=M1*R rSup { size 8{E1} } } {} and X2 = M2 R E2 size 12{X2=M2*R rSup { size 8{E2} } } {}

  • X1 ± X2 = ( M1 R E1 E2 ) R E2 size 12{X1 +- X2= \( M1*R rSup { size 8{E1 - E2} } \) R rSup { size 8{E2} } } {} (assume E1 size 12{<= {}} {} E2)
  • X1 X2 = ( M1 M2 ) R E1 + E2 size 12{X1*X2= \( M1*M2 \) R rSup { size 8{E1+E2} } } {}
  • X1 / X2 = ( M1 / M2 ) R E1 E2 size 12{X1/X2= \( M1/M2 \) R rSup { size 8{E1 - E2} } } {}

For addi­tion and subtraction, it is necessary lo ensure that both operands have the same exponent value. I his may require shifting the radix point on one of the operands to achieve alignment. Multiplication and division are more straightforward.

A floating-point operation may produce one of these conditions:

  • Exponent overflow: A positive exponent exceeds the maximum possible expo­nent value. In some systems, this may be designated as
  • Exponent underflow: A negative exponent is less than the minimum possible exponent value (e.g.. -200 is less than -127). This means that the number is too small to be represented, and it may be reported as 0.
  • Significand underflow: In the process of aligning significands, digits may flow off the right end of the significand. Some form of rounding is required.
  • Significand overflow: The addition of two significands of the same sign may result in a carry out of the most significant bit. This can be fixed by realign­ment.

Questions & Answers

what is application of humanistic theory of classroom
Afreen Reply
The humanistic learning theory developed further and harnesses the idea that if students are upset, sad, or distressed, they're less likely to be able to focus on learning. This encourages teachers to create a classroom
Dhwani
state the brief history of psychology
James
The history of psychology as a scholarly study of the mind and behavior dates back to the Ancient Greeks. ... Psychology as a self-conscious field of experimental study began in 1879, in Leipzig Germany, when Wilhelm Wundt founded the first laboratory dedicated exclusively to psychological research
Dhwani
can anyone explain sleep and it's stages?
Shachi Reply
yes
Dhwani
please let me know
Shachi
sleep works in three hour stages. falling asleep rem, waking back up. do you have a spicific question?
Nathaniel
When thinking about getting the sleep you need, it’s normal to focus on how many hours of sleep you get. While sleep duration is undoubtedly important, it’s not the only part of the equation. It’s also critical to think about sleep quality and whether the time spent sleeping is actually restorative
Dhwani
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Nathaniel
I mean stages of sleep
Shachi
and it's explaination
Shachi
How brain power can be used to control physical things?
Ali Reply
can anyone explain sleep and it's stages?
Shachi
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wazir Reply
what is the role of psychology in the development of therapies for disorders of circadian rhythm?
wazir
To study Abnormal Psychology means, to study mainly the nature of
Baddo Reply
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la
abnormal, this branch of psychology typically deals with behavior in a clinical context.
pakki
yo
Baddo
To study Abnormal Psychology means, to study mainly the nature of 1). Conscious Mind 2). Unconscious Mind 3). Subconscious Mind 4). Normal Mind
pakki
yo
Kylee
abnormal psychology- is the scientific study of Psychological disorders and their treatments
utkarsh
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Baddo
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Baddo
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pakki
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Anye
Yo
Monica
What is the role of psychology in the development of therapies for circadian rhythm disorders?
wazir
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Anye
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Stepphun Reply
examine any 5 perceptual principals of organization
kundai Reply
Is it possible for a child to be schizophrenic?
Esther Reply
yes. early onset of schizophrenia starts at 13 to 18. very rare but possible.
Kristel
Good question
Mulder
It's uncommon for children to be diagnosed with schizophrenia.
pakki
what is the role of psychology in the development of therapies for circadian rhythm disorders?
wazir
is there a difference between age and romantic attachment style?
Alysha Reply
I don't think so. Well, not always. Certain things should be timeless.
Know
I'm 48 & he's 26 ...so we shall see?
Tanya
in my opinion if you both are comfortable with each other then definitely age is nothing but a number but then also there will be little difference in the cognition due to age
utkarsh
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Shailaja
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utkarsh
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Ashley
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Sujeet
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Smitha
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krish
can empirical research be performed on people being diagnosed or people who have borderline personality disorder?
Stepphun Reply
yes it can
Angela
***pubmed.ncbi.nlm.nih.gov/3711874/
Angela
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Stepphun
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Angela
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Stepphun
discuss the three models of positional coding in working memory citing examples.Please help me with points to this question
jane Reply
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Mimi Reply
What are the vocational choices for women in pakistan also discuss its aims and objectives.
Malik Reply
I want to research on Egocentrism and Autistic thinking on myself.
RAVI Reply
I want to research on Egocentrism and Autistic thinking on myself. Have you any suggestion for me?
RAVI
I want to research on Egocentrism and Autistic thinking on myself. Have you any suggestion for me?
RAVI
why do you think they correlate?
James
what is the role of psychology in the development of therapies for circadian rhythm?
wazir
what is Egocentrism
Ishfaq
External goal that have the capacity to motivate behaviour
Haider Reply
Reinforcements and Punishments
Khyzar
indeed! but positive punishment if I may add
Angela
hi. i am a student of B.A. Psychology (major), i wanted to enquire as to which textbook would be most informative on 'Biopsychology' ?
Bhargavi
Hey, I'm currently a BS Psychology major and I'd like to recommend the 10th Edition Biological Psychology book by Kalat
Ayessa
thanks, it really means a lot😀
Bhargavi
You're welcome.
Ayessa
what is logical atomism
Koshivani
When we have data register and address registers in user mode (visible registers) then why we need MAR and MBR
Hassan Reply
with the given example MOV BX,AX describe the sequence that will be followed using instruction state diagram
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Source:  OpenStax, Computer architecture. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10761/1.1
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