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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

Floating Point Arithmetic
How to add

1.110100 x 2^5
+ 1.01001 x 2^3

change the exponent so they could be the same.