Basic Arithmetic (adding and subtracting)

Digital logic to show add/subtractBoolean algebra

abstraction of physical, analog circuit behavior

1

0

CPUcomponents – ALUlogic circuitslogic gatestransistors

Digital Logic

A B A * B0 0 00 1 01 0 01 1 1

A B A B0 0 00 1 11 0 11 1 1

∨

AB

out

outAB

and (* or ^)

or ( )∨

Digital Logic

A out

outAB

not (~, ⌐, −)

xor ( )

A ~A0 11 0

A B A B0 0 00 1 11 0 11 1 0

⊕

⊕

Digital Logic

AB

out

outAB

nand

nor

A B0 0 10 1 11 0 11 1 0

B*A

A B A nor B0 0 10 1 01 0 01 1 0

Digital Logic

Given the truth table:

F = + + +

Sum of products from the truth table. Often we can simplify.

A B F

0 0 10 1 11 0 11 1 1

BA∗BA∗BA∗BA∗

BA∗ BA∗ BA∗ BA∗

Unsigned Binary Arithmetic

The half adder is an example of a simple digital circuit built from two logic gates.

half adder logic - two one-bit inputs (a, b) and two one-bit outputs (carry_out, sum)

a 0 0 1 1+ b + 0 + 1 + 0 +1

carry_out sum 00 01 01 10

Unsigned Binary ArithmeticBinary Addition

A B cout sum

0 0 0 00 1 0 11 0 0 11 1 1 0

Input Output

A B Cout S

0 0 0 00 1 0 11 0 0 11 1 1 0

cout

sumAB

Half Adder

carry_out = a and b

sum = a xor b

Unsigned Binary Arithmetic

The problem with a half-adder is that it doesn't handle carries. Consider adding the following two numbers:

When we add the two numbers, we get

Look at the middle and leftmost columns. You add 3 bits. Half adders can only add two bits.

1 11 1 1

+ 0 1 1(1) 0 1 0

1 1 1

+ 0 1 1

Unsigned Binary Arithmetic

For adding multi-bit fields/words, e.g., 4 bits

a3 a2 a1 a0

+ b3 b2 b1 b0

---------------------------------------sum3 sum2 sum1 sum0

we also need to add a carry_in with ai and bi, where i > 0

Unsigned Binary Arithmetic

A full adder for a_i + b_i + carry_in is given in the figure below.

– three one-bit inputs (a, b, carry_in) and– two one-bit outputs (carry_out, sum)– cascade two half adders (sum output bit of first

attached to one input line of the other) and then ortogether the carry_outs

Input OutputCin A B Cout S0 0 0 0 01 0 0 0 10 1 0 0 11 1 0 1 00 0 1 0 11 0 1 1 00 1 1 1 01 1 1 1 1

Cin

AB

sum

Cout

Full Adder• An n-bit adder built by connecting n full adders• carries propagate from right to left (i.e., connect the

carry_out of an adder to the carry_in of the adder in the next leftmost bit position

• the initial, that is, rightmost, carry_in is zero)

• overflow occurs when a number is too large to represent. • for unsigned arithmetic, overflow occurs when a carry out

occurs from the most significant (i.e., leftmost) bit position

Full Adder

(there are faster forms of addition hardware where the carries do not have to propagate from one side to the other, e.g., carry-lookahead adder)

Signed Binary Addition/Subtraction

fundamental idea #1finite width arithmetic- modulus rn, where r is radix, n is number of digits wide- wraps around from biggest number to zero, ignoring overflow

e.g., 4-bit arithmetic => modulus is 24 = 160, 1, 2, ..., 15 then wrap around back to 0

thus an addition of rn to an n-digit finite width value has no effect on the n-digit value

Signed Binary Addition/Subtraction

fundamental idea #2subtraction is equivalent to adding the negative of number

e.g., a - b = a + (-b)

observationa - b == a - b + rn == a + (rn - b)

\______/ \_______/#1 #2

\______/this term is our representationfor (-b)

it turns out that we can more easily perform rn - b than a - b

Signed Binary Addition/Subtraction

digit complement for n digits == (rn - 1) - number in binary, this is called one's complement and equals a value of n ones minus the bits of the number

for binary, one's complement (2n - 1 - number) is equivalent to inverting each bit

in decimal, this is called nine's complement and equals a value of n nines minus the digits of the number

in hexadecimal, this is n f's (fifteens) minus the digits of the number

Signed Binary Addition/Subtraction

radix complement for n digits == (rn - 1) - number + 1

two's complement in binaryten's complement in decimal

for binary, two's complement (2n - 1 - number + 1) is equivalent to inverting each bit and adding one

Signed Binary Addition/SubtractionWe can easily make a full adder do subtraction by adding an inverter in front of each bi and setting carry into the rightmost adder to one

Signed Binary Addition/Subtractionrange for n-bit field: unsigned is [ 0, 2n - 1 ]2's compl. signed is [ -2n-1, 2n-1 - 1 ]

signed overflow occurs whenever the sign bits of the two operands agree, but the sign bit of the result differs (i.e., add two positives and result appears negative, or add two negatives and result appears nonnegative)range diagrams for three bits

unsigned

signed(2's compl)

000 001 010 011 100 101 110 111|------- |------- |------- |------- |------- |------- |-------| 0 1 2 3 4 5 6 7

100 101 110 111 000 001 010 011|------- |------- |------- |------- |------- |------- |-------|

-4 -3 -2 -1 0 +1 +2 +3

b2 b1 b0

sign b1 b0

Signed Binary Addition/Subtractionmodulo arithmetic (keep adding +1 and wrap around)

000 001 010 011 100 101 110 111

(unsigned) 0 1 2 3 4 5 6 7

(or 2'scompl)

0 +1 +2 +3 -4 -3 -2 -1^^--carry occurs on wrap around

Signed Binary Addition/Subtraction3-bit examples

bits unsigned signed111 = 7 = (–1)

+001 = +1 = +(+1)----- --- -------000 0 (0)(carry) OVF

^^^-- this is what the ALU computes for either unsigned or signed. but, while it is an unsigned overflow, it is CORRECT for signed

Signed Binary Addition/Subtraction3-bit examples

Example 2bits unsigned signed011 3 = (+3)

+001 = +1 = +(+1)----- -- -----100 4 (–4)

OVF

^^^-- this is what the ALU computes for either unsigned or signed, but, while it is correct for unsigned, it is SIGNED OVERFLOW!

Signed Binary Addition/Subtraction

16-bit signed (2's complement) examples

in 16-bit arithmetic, we can represent values as four hex digits;if the leading hex digit is between 0 and 7 (thus it has a leading bit of 0), it is a nonnegative value; if the leading hex digit is between 8 and f (thus it has a leading bit of 1), it is a negative value

signed overflow occurs ifa. (+) added with (+) gives a (-), orb. (-) added with (-) gives a (+)

Signed Binary Addition/Subtraction

hexadecimal hexadecimal decimal

0x7654 = 0x7654 = (+30292)+0xffed = +(-0x13) = +( –19)

0x7641 0x7641 (+30273)(carry)

carry occurs but there is no signed overflow (thus carry is ignored)

(+) added with (-) cancels out, so signed overflow is not possible

Signed Binary Addition/Subtraction

hex decimal0x7654 = (+30292)

+0x1abc = +( + 6844)---------- ------------0x9110 (-28400) should be 37136, but is > max

positive 32767

no carry occurs but there is signed overflow(+) added with (+) giving (-) => SIGNED OVERFLOW!

Signed Binary Addition/Subtraction

hex decimal0x7654 = (+30292)

+0x1abc = +( + 6844)

0x9110 (-28400) should be 37136, but is > max positive 32767

no carry occurs but there is signed overflow(+) added with (+) giving (-) => SIGNED OVERFLOW!

Signed Binary Addition/Subtraction

hexadecimal0x7654 change subtraction to addition by ffff-0xff8d taking two's complement of 0xff8d -ff8d

0072+ 10073

hexadecimal hexadecimal decimal

0x7654 = 0x7654 = (+30292)-0xff8d = +0x0073 = +( +115)

0x76c7 = (+30407)

no carry occurs and no signed overflow(+) added with (+) giving (+) => no signed overflow