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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
Idrissa Reply
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Sherica
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Sherica
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hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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I'm interested in nanotube
Uday
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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