Flashcards in Computer Arithmetic Deck (15):

1

## Decimal to Binary

### Divide the number by 2 and remainder (0 or 1), and stop when the result is 0. The answer is the remainders read from bottom to top.

2

## Binary to Decimal

###
Multiply the first non-zero bit by 2 and add the bit on its right.

Or from base 2

so an 8-bit number would look like

2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0

0 0 0 0 0 1 0 0 = 2^2 = 4

3

## Hexadecimal

###
Base 16

Notation: 3B_16 = 0x3B = $3B

0000_2 - 1001_2

0_16 - 9_16

1010_2 - 1111_2

A_16. - F_16

4

## Binary Subtraction in relation to Addition

### X - Y = X + (-Y)

5

## Sign and Magnitude Representation

###
First bit represents the sign.

So,

00001101_2 = + 13_10

10001101_2 = - 13_10

To negate, just flip the sign bit.

6

## Two's Complement Arithmetic

###
N represents -N using 2^n - N

Example:

N = 6_10 = 0110_2

-N = 2^4 - 6 = 10_10 = 1010_2

7

##
Decimal to Two's Complement:

1. If it's positive

2. If it's negative (How to complement N)

Ex.

Convert 6 to binary and convert -6 to binary

###
Convert to binary normally if positive.

Invert sign of decimal. Now convert it to binary. Then takes two complement (Invert bits and add 1).

Ex:

1. 6 -> 0110

2. -6 -> 6 -> 0110 -> 1001 -> 1010

or as before, -6 -> 2^4 - 6 -> 10 -> 1010

8

## Arithmetic Overflow

###
Overflow occurs in two's complement when addition of two positive number gives a negative result or when two negatives gives a positive.

How to detect it: Check the carry

01100

+ 01101

______

1 1001

1100

9

##
Fixed Point Arithmetic

How to convert 0.25 to binary

###
Base 2 representation of decimals

2^3 2^2 2^1 2^0 . 2^-1 2^-2 2^-3 2^-4

0 0 0 0 . 0 1 0 0

10

## The normalisation of IEE FP

###
1010.101 x 2^e

=

1.010101 x 2^(e+3)

11

##
Representing Biased forms

Convert 3 to Bias 3

What's the Bias for 8-bit

###
Bias: 2^(m-1) - 1

number - bias

3 - 3 = 0

Bias: 2^(7) - 1 = 127

12

## IEEE 754 floating point format

###
X = (-1)^S x 2^(E-B) x 1.F

B = 127

13

## Decimal to IEEE FP

###
Convert to binary.

Normalise.

Add Bias B then convert the exponent to binary.

S Biased Exponent Fractional Mantissa

1 bit 8 bits 23 bits

14

## IEEE FP to Decimal

### (-1)^S x (1+F) x 2^(E-B)

15