Ch3a- 2EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Crunching Numbers

• Topics we need to explore

• Representing numbers on a computer• Negatives, too

Hardware Alert!Hardware Alert!

• Building hardware to work with Floating Point numbers

• Building hardware to do logic and math• And, Or• Addition, Subtraction• Comparisons• Multiplication and Division

Ch3a- 3EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Representation

• All data on a computer is represented in binary

• 32 bits of data may be:• 32-bit unsigned integer• 4 ASCII characters• Single-precision IEEE floating point number• Who knows...

data: 1000 1001 0100 0110 0000 0101 0010 1000

As 32-bit unsigned integer: 2,303,067,432

As 32-bit 2’s complement integer: -1,991,899,864

As 4 ASCII characters: ‘??’, ‘F’, ENQ, ‘(‘

Note: Limited ASCII chart on p. 142

Ch3a- 4EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

ASCII Representation of Numbers

• Terminal I/O (keyboard, display) only deals with ASCII characters

• Typically, strings of characters

• “We’re #1” --> 87,101,44,114,101,32,35,49,0

NULL Termination

• Note that the number ‘1’ is represented by 49

• Numbers in I/O consist of their ASCII representations

• To output 103, use ASCII values 49, 48, 51 (3 bytes)• Outputting 103 (one byte) won’t work

‘1’, ‘0’, ‘3’ code for ‘g’

Ch3a- 5EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Number Systems- Negative Numbers

• What about negative numbers?

• We need to represent numbers less than zero as well as zero or higher

• In n bits, we get 2n combinations• Make half positive, half negative...

Sign bit: 0-->positive, 1-->negativeSign bit: 0-->positive, 1-->negative 31 remaining bits for magnitude31 remaining bits for magnitude

First method: use an extra bit for the sign0 000 0000 0000 0000 0000 0000 0000 01011 000 0000 0000 0000 0000 0000 0000 0101

+5

-5

Ch3a- 6EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Sign and Magnitude Representation

• Two different representations for 0!

0000

0111

0011

1011

11111110

1101

1100

1010

1001

1000

0110

0101

0100

0010

0001

+0+1

+2

+3

+4

+5

+6

+7-0

-1

-2

-3

-4

-5

-6

-7

Note: Example is shown for 4-bit numbers

Note: Example is shown for 4-bit numbers

Inner numbers:Binary

representationSeven Positive Numbers and “Positive” Zero

Seven Negative Numbers and

“Negative” Zero

• Number range for n bits = +/- 2n-1 -1

• Three low order bits represent the magnitude: 0 (000) through 7 (111)

• Two discontinuities

• High order bit is sign: 0 = positive (or zero), 1 = negative

Ch3a- 7EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Two’s Complement Representation

• Only one discontinuity now

0000

0111

0011

1011

11111110

1101

1100

1010

1001

1000

0110

0101

0100

0010

0001

+0+1

+2

+3

+4

+5

+6

+7-8

-7

-6

-5

-4

-3

-2

-1

Note: Example is shown for 4-bit numbers

Note: Example is shown for 4-bit numbers

Inner numbers:Binary

representationEight Positive

Numbers

Re-order Negative

Numbers to Eliminate

Discontinuities

• Only one zero

• One extra negative number

Note: Negative numbersstill have 1 for MSB

Note: Negative numbersstill have 1 for MSB

Ch3a- 8EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

2’s Complement Negation Method #1To calculate the negative of a 2’s complement number:

1. Complement the entire number

2. Add one

Examples:

n = 0110= 6

complement n = 01000100 = 68n = 10010000= -1121001

add 1

-n =1010 = -6

complement

10111011add 1

-n =10111100 = -68

complement

01101111add 1

-n =01110000 = 112

WARNING: This is for calculating the negative of a number. There is no such thing as “taking the 2’s complement of a number”.

Ch3a- 9EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

2’s Complement Negation Method #2To calculate the negative of a 2’s complement number:

1. Starting at LSB, search to the left for the first one

2. Copy (unchanged) all of the bits to the right of the first one and the first one itself

Examples:

n = 0110= 6

-n = 1010 = -6

copycomplement

n = 01000100 = 68

-n = 10010111 = -68

copycomplement

n = 10010000= -112

-n = 10000011 = 112

copycomplement

3. Complement the remaining bits

Ch3a- 10

EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Adding Two’s Complement Numbers

4

+ 3

7

0100

0011

0111

-4

+ 3

-1

1100

0011

1111

-4

+ (-3)

-7

1100

1101

11001

4

- 3

1

0100

1101

10001

Just add the completenumbers together.Just add the completenumbers together.

Sign taken care of automatically.Sign taken care of automatically.

Ignore carry-out (for now)Ignore carry-out (for now)

A carry out from sign bit does not necessarily mean overflow!A carry out from sign bit does not necessarily mean overflow!

Ch3a- 11

EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

OverflowAdd two positive numbers to get a negative numberor two negative numbers to get a positive numberAdd two positive numbers to get a negative numberor two negative numbers to get a positive number

5 + 3 = -85 + 3 = -8

-7 - 2 = +7-7 - 2 = +7

Overflow cannot occur when adding a positive and negative number together

0000

0111

0011

1011

11111110

1101

1100

1010

1001

1000

0110

0101

0100

0010

0001

+0+1

+2

+3

+4

+5

+6

+7-8

-7

-6

-5

-4

-3

-2

-1

Overflow occurs when crossing discontinuity

Not a discontinuity - No Overflow

A carryout from the MSB could mean crossing at either of these places – One is OK, one is Overflow

Ch3a- 12

EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Detecting OverflowOverflow occurs when:

We add two positive numbers and obtain a negative

Looking at the sign bit (MSB):

0+ 0

0Cin

0Cout

+

++

0+ 0

1Cin

0Cout

+

+-

No overflow Overflow

We add two negative numbers and obtain a positive

1+ 1

0Cin

1Cout

-

-+

1+ 1

1Cin

1Cout

-

--

Overflow No Overflow

Overflow when carry in to sign bit does not equal carry outOverflow when carry in to sign bit does not equal carry out

Cin

Cout

Overflow

0 110

1+ 0

0Cin

0Cout

-

+-

No Overflow

1

1+ 0

1Cin

1Cout

-

++

No Overflow

0

Ch3a- 13

EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Signed and Unsigned operationsConsider the following:

$t0 = 0000 0000 0000 0000 0000 0000 0000 0101

$t1 = 1111 1111 1111 1111 1111 1111 1111 1001

execute: slt $s0, $t0, $t1

What’s the result?

If we mean for $t0 to be 5 and $t1 to be 4,294,967,289 (treatas unsigned integers) then $s0 should get 1

If we mean for $t0 to be 5 and $t1 to be -7 (treat as signedintegers) then $s0 should get 0

The default is to treat as signed integers

Use sltu for unsigned integers

Ch3a- 14

EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Using more bits

What’s different between 4-bit 2’s complement and 32-bit?

MSB has moved from bit 3 to bit 31!

Copy MSB of original number into all remaining bits

0111

1010

710710

(32-bit 2’s comp.)

-610-610 (32-bit 2’s comp.)

Sign ExtensionSign Extension

0000 0000 0000 0000 0000 0000 0000 0111

1111 1111 1111 1111 1111 1111 1111 1010

4-bit2’s comp.

To convert from 4-bit 2’s complement to 32-bit: Copy all 4 bits to 4 Least significant bits of the 32-bit number.

Ch3a- 15

EE/CS/CPE 3760 - Computer OrganizationSeattle Pacific University

Loading a single byte from memory

We can read a single 8-bit byte from memory location 3000 by using:

lb $t0, 3000($0) # read byte at mem[3000]

assuming mem[3000] = 0xF3, we get...

$t0: 0xFFFFFFF3 (sign-extension for other 3 bytes)

If we only want the byte at 3000 (without extension), used an unsigned load:

lbu $t0, 3000($0) # read byte a mem[3000]

$t0: 0x000000F3 (no sign-extension)

0x prefix means Hex