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In the early days of computing, designers made computers express numbers using unsigned binar

To include negative numbers, designers came up with sign magnitude.

Then designers created ones complement.

Finally, designers developed twos complement.

But the computer had to count backwardsfor the negative numbers.

Plus, this introduced positive and negative zero.

Hey guys!do you ne

number

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Comparing Integer RepresentationsNegation and Zeroes

Hi! And welcome to the Best IntegerRepresentation competition!

Here, well choose who gets to be theworlds standard for computer integers!But first, lets introduce our contestants:

Round 1 - Negation

Round 2 - Zeroes

In this competition, well use 8-bit numbers. Now lets get started!

Unsigned SignMagnitude

OnesComplement

TwosComplement

Bias

Flip sign bit Flip all bitsFlipallbi

ts

andadd1

Subtractfrom12710

Oh, dear! It looks like Unsigned cant negate. But this competition has onlystarted, so Unsigned still has a chance of catching up to the others.

0000 00001000 0000

000000001111 1111

0111111100000000

Now things are getting interesting! Unsigned and Twos Complement get two points eachfor having one zero and being able to represent zero with all zero bits. Biass zero isnt allzero bits, but it gets a point for having only one zero. And though they have two zeroes,

Sign Magnitude and Ones Complement get a point for having a zero of all zero bits.

Round 1 is easy. Just tell mehow you negate a number!

Now for Round 2! Show meall the ways you representzero!

Um...

00000000

-1271

-1271

-127

1

0 1 1 1 1

2 2 2 3 2

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255

0

-255

128

-128

00000000 01111111 11111111

1

-1

127

-127

10000000

255

0

-255

128

-128

00000000 01111111 11111111

1

-1

127

-127

10000000

255

0

-255

128

-128

00000000 01111111 11111111

1

-1

127

-127

10000000

255

0

-255

128

-128

00000000 01111111 11111111

1

-1

127

-127

10000000

255

0

-255

128

-128

00000000 01111111 11111111

1

-1

127

-127

10000000

Comparing Integer RepresentationsIncrements and Monotonicity

Round 3 - Incrementing

255

0

-255

128

-128

00000000 01111111 11111111

1

-1

127

-127

10000000

Very nice! Unsigned has a graph that is continuousand has a unit slope. This means we can use an

unsigned comparator to compare integers!Well give Unsigned two points for that.

Whats this? Sign Magnitude has a very unusualincrement indeed. It has a unit slope for positiveintegers, but the slope becomes -1 for negatives.

Sorry, but no points for Sign Magnitude this round.

Another monotonically increasing graph with unitslope! You can use an unsigned comparator to

compare integers here, too. Bias gets two points!

Just like Ones Complement, Twos Complementhas a discontinuous graph and unit slope. So

well give Twos Complement a point.

Theres a discontinuity in the graph forOnes Complement, but we do like how

it has a consistent unit slope.Thats one point for Ones Complement.

4

2 3

4 4

-12

On to Round 3! Using this board, graph how your value changes when you

increment your bit pattern from 00000000 to 11111111! Well give each playera point for having a continuous graph and a point for a consistent unit slope.

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Unsigned

Sign Magnitude

Ones Complement

Twos Complement

Bias

Comparing Integer RepresentationsThe Thrilling Conclusion!

One Zero?Negation? Zero =0000 0000

MonotonicaIncreasing

Continuous?Weve finally arrived at theend of our competition. Letssee that scoreboard!

Well, well! It appears we have a three-way tie among Unsigned, Twos Complement,and Bias! We can certainly give each of our winners a prize, though!

Unsigned, youll be the representationfor data whenever users call upon theunsignedmodifier in C! Ive heard thatother languages use it, too, so youllwork for them as well.

Bias, youll represent the exponent inIEEE-754 floating-point numbers! Thefact that we can compare exponentswith an unsigned comparator will come inhandy!

And you, Twos Complement, becauseyou can negate and have one zero thatis expressed as all zero bits, you will bethe representation of integers forbinary computers all around the world!

-12710

unsignedchar foo = 24;

000110002

signedchar bar = -24;

111010002

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3.141592

Floating-Point Numbers!

An IEEE754 floating point number consists of three parts:

(Also known as the Significand)

the Exponent,the Sign, and the Mantissa.

The Sign, as its name suggests,determines the sign of the number.

The Exponent plays a vital role in determininghow big (or small) the number is. However, itsencoded so that unsigned comparison can be

used to check floating-point numbers.

To see the true magnitude of theExponent, youd need to subtract theBias, a special number determined bythe length of the Exponent.

And last but not least, the Mantissaholds the significant digits of thefloating point number.

+

100000002

0100000000000

00000000002

-12710

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Floating-Point Numbers:All Together Now!

0100000000000

00000000002

+

Once all the parts of the floating-point number are obtained, convertingit to decimal is just a matter of applying the following formula:

Notice that the Mantissa actually represents a fraction, instead of an integer!In addition to representing real numbers, the IEEE 754 representation can also indicate...

positive or negative infinity,

and even when something is not anumber! This is called NaN.

+ 111111112 000000000

000000000000002

-

- 111111112+

the set of numbers known as denormalizednumbers (including zero),

+-

-12610

-12710

Example:

=02

NaNs arent

comparable, butthey can bedifferent!

If this is all zeroes,the float is zero!

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Floating-Point Numbers:The Great Number Line

Due to the format of the IEEE-754 standard, the floating-point numbers can be plotted on a number line.In fact, the floating-point numbers are arranged so that they can be incremented like a binary odometer!

NaN

+ FloatingPoint

Number

+ DenormalizedNumber

+ 0-

-

-

+-