04 number representations

7/31/2019 04 Number Representations

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Number representations

We often want to do numerical computations in digital systems so we need to

understand how to represent and manipulate binary numbers (and, in general, in

number systems other than decimal).

ECE124 Digital Circuits and Systems Page 1


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Position number representation


Lets assume no decimal portions Any number can be written as:

Here, r is called the base or radix and the a_i are called the coefficients. The

coefficients are always between 0,1, (r-1).

Examples:


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Different bases

Some important bases:

Base-10 (decimal)we are familiar with this system. Digits are 0, 1, 2, , 8, 9.

Base-2 (binary) Digits are 0, 1.

Base-8 (octal)Digits are 0, 1, 2, , 7.

Base-16 (hexadecimal)Digits are 0, 1, 2, , 9, A, B, C, D, E, F.

In a binary system (base-2), the digits/coefficients are often called bits.



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Fixed point number representations

If we have fractional numbers (i.e., something after the decimal) we can do effectively

the same thing.

We can do this in any base or radix:


Example:


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Number conversions (base-r to base-10).

Conversion from base-r to base-10 is easy; Just expand the base-r representation and

the result is a base-10 result.



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Number conversions from base-10 to base-r (1)

We do the integer part and the fractional part separately.

Consider the integer part of a number in base-r and its division by the base-r:


Conversion procedure should be clear; we take a decimal number and performsuccessive division by r until the quotient becomes 0.

The remainders provide the coefficients in the new base.


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Convert (53)10 to base-2.


We read the digits from bottom to top, writing them left to right.

Result is that (53)10 = (110101)2

The left most bit is the MSB (most-significant bit) and the right most bit is the LSB

(least significant bit).


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Consider the fractional part of a number in base-r and its multiplication by the base-r:


Conversion should be clear; We can take a decimal number and perform successive

multiplication by r;

Continue until the fractional part of the multiplication becomes 0.

The integer parts provide the coefficients in the new base.


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Convert (0.625)10 to base-2.


We read the bits top to bottom, and write them after the radix point left-to-right.

Result is (0.625)10 = (0.101)2


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Conversions between binary, octal and hexadecimal numbers

Nice simple ways to convert between these three number systems, since all are a

power of 2.

Binary to Octal simply requires grouping bits into groups of 3-bits and converting.

Binary to Hexadecimal simply requires grouping bits into groups of 4-bits and

converting.

Going the other direction (Octal to Binary or Hexadecimal to Binary) should follow

Example (100111100101)2 = (4745)8 = (9E5)16



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Complements

Complements are used to simplify the subtraction operation when performing digital

computation.

Two types of complements for base-r:

Diminished radix complement also referred to as (r-1)s complement.

Radix complementalso referred to as rs complement.

So, for binary systems (base-2), we have 1s complements and 2s complements.

I am only going to talk about radix complements (2s complements in base-2) since it is

the most useful.

Note: Its important to understand that the complement of a number will return the

original number.


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Radix complements

Given a number N in base-r having n digits, the rs complement is defined as:

Examples:

The 10s complement of (546700)10 is (1000000)-546700 = (453300)10

The 2s complement of (1011000)2 is (10000000)-1011000 = (0101000)2

Note: In base-2, the 2s complement of a number is obtained by flipping the bits and

adding 1.


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Addition of unsigned binary numbers

Binary addition of unsigned numbers is done just like in decimal. We add digits and

generate carries.


Note: We can get a non-zero carry out; if we are using (and are limited to) n-bits, thisbecomes an overflowcondition for unsigned addition.


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Subtraction of unsigned binary numbers

Binary subtraction of unsigned numbers can be done just like in decimal. We subtract

numbers using borrows when necessary.


Note: It sort of doesnt make sense to subtract binary numbers if the minuend issmaller than the subtrahend; A negative result can often be considered anunderflow.


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Subtraction of unsigned binary numbers using complements

The way we perform subtraction by hand is a hassle when done by a digital circuit and

this is where complements help a

The subtraction of two numbers M-N in base-r can be done by: Adding the minuend

M to the rs complement of the subtrahend N. The result mathematically is:


Two possible results:

Case 1: IfM >= N we will get a carry out of 1 (rn) which can be discarded to givethe result.

Case 2: IfM < N we will not get a carry out. Since the result is rn (N-M), wesimply complement the sum and put a negative sign in front.


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Example of subtraction using rs complement in base-10

Subtract 72532-3250 assuming

these are base-10 numbers.

There is a carry out (equal to 1 in

this example) which we discard

and our result is 69282

Subtract 3250 72532 assuming

these are base-10 numbers.

There is no carry out (shown as an

extra 0) so we need to complement

the sum and add a negative sign.

10s complement of 30718 is 69282,

and therefore our answer is -69282


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Example of subtraction using rs complement in base-2

Subtract 101-011 (5-3) using

assuming base-2representations.

There is a carry out so we need to

discard the carry out.

Therefore our answer is (010)2

Subtract 011-101 (3-5) using assuming

base-2 representations.

There is no carry out (shown as extra

leading 0) so we need to complement

the result and add a negative sign.

2s complement of 110 is 010.

Therefore our answer is(010)2 which is

-2.


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Signed numbers

We need to consider methods to represent signed numbers in addition to unsigned

numbers.

In a computer, signed or unsigned numbers are both represented as strings of bits;

given a string of bits it is up to us to interpret the number.

If a number is signed, the left-most bit (the most-significant bit) tells us the sign:

0 means a +ve number.

1 means ave number.

E.g., 01001 means +9 in either its signed or unsigned representation given only 5

bits to represent a number.

E.g., 11001 means +25 in its unsigned representation; some other negative number

in its unsigned representation given only 5 bits to represent a number.


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Signed Number Representations (1)

Only 1 way to represent a +ve number, but 3 ways to represent its negative version

in a binary number system:

Signed magnitude.

Signed 1s-complement (we will not talk about this further).

Signed 2s-complement.

E.g., consider +9 represented with 8 bits: +9 is (00001001).

E.g., consider -9 represented with 8 bits:

Signed magnitude: -9 is (10001001).

Signed 1s-complement: -9 is (11110110).

Signed 2s-complement: -9 is (11110111).


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Signed Number Representations (2)

Signed magnitude representations are obtained by flipping the leading bit of the +ve

version.

Signed 2s complement is obtained by flipping all the bits and adding 1

i.e., taking the 2s complement of the +ve version.


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Different schemes to represent numbers assuming 4 available bits.

Note the multiple representations of 0 in signed-magnitude and 1s complement.

Note: we get an extra digit in 2s complement.


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Addition of signed binary numbers using 2s complement

Signed arithmetic is easy with 2s complement numbers. We just perform the addition

and ignore the carry out.


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Subtraction of signed binary numbers using 2s complement

Things are easy using 2s complement representation forve numbes. We take the 2s

complement of the subtrahend and add. Why does this work?

E.g., Consider (-6)-(-13) = +7.

Ignoring the carry out (as we do for addition) we get the correct result of +7.


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Illustration of 2s complement numbers and why they are good for signed binary arithmetic

Consider that we only have 4-bits in which to represent a signed number. We can

represent the numbers -8 to +7 (16 different values).

We can visualize the numbers on a ring.

Arithmetic operations are modulo.

When we perform

addition/subtraction, we are simply

moving around the ring

1100

0000

1011

10100110

0100

0011

0010

1000

0001

0101

01111001

1101

1110

1111

0+1

+6

+5

+4

+3

+2

-3

-4

-5

-7

-8+7

-6

-2

-1

With 2s complement numbers, we get only one representation of zero. Hence, interms if the circuit, 2s complement is nice since we dont need to worry about

jumping over and extra 0 when performing addition or subtraction.

04 number representations

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