Binary Representation

Binary Representation for Numbers

Assume 4-bit numbers5 as an integer

0101-5 as an integer

How?

5.0 as a real number How? What about 5.5?

Sign Bit

Reserve the most-significant bit to indicate sign

Consider integers in 4 bits Most-significant bit is sign: 0 is positive, 1 is negative The 3 remaining bits is magnitude 0010 = 2 1010 = -2

How many possible combinations for 4 bits?How many unique integers using this

scheme?

Two’s Complement

Advantages # of combinations of bits = # of unique integers Addition is “natural”

Convert to two’s complement (and vice versa)1. invert the bits2. add one3. ignore the extra carry bit if present

Consider 4-bit numbers 0010 [2] -> 1101 -> 1110 [-2]

Addition

0010 [2] + 1110 [-2] 0000 [ignoring the final carry—extra bit]

0011 [3] + 1110 [-2] 0001 [1]

1110 [-2] + 1101 [-3] 1011 [-5]

Range of Two’s Complement

4-bit numbers Largest positive: 0111 (binary) => 7 (decimal) Smallest negative: 1000 (binary) => -8 (decimal) # of unique integers = # of bit combinations = 16

n bits ?

Binary Real Numbers

5.5 101.1

5.25 101.01

5.125 101.001

5.75 101.11

… 23 22 21 20 . 2-1 …

8 bits only

5.5 101.1 -> 000101 10

5.25 101.01 -> 000101 01

5.125 101.001 -> ??

With only 2 places after the point, the precision is .25 What if the point is allowed to move around?

25 24 23 22 21 20 2-1 2-2

Floating-point Numbers

Decimal 54.3 5.43 x 101 [scientific notation]

Binary 101.001 10.1001 x 21 [more correctly: 10.1001 x 101] 1.01001 x 22 [more correctly: 1.01001 x 1010]

What can we say about the most significant bit?

Floating-point Numbers

General form: sign 1.mantissa x 2exponent the most significant digit is right before the dot

Always 1 [no need to represent it] Exponent in Two’s complement

1.01001 x 22

Sign: 0 (positive) Mantissa: 0100 Exponent: 010 (decimal 2)

Java Floating-point Numbers

Sign: 1 bit [0 is positive]

Mantissa: 23 bits in float 52 bits in double

Exponent: 8 bits in float 11 bits in double

sign exponent mantissa

Imprecision in Floating-Point Numbers

Floating-point numbers often are only approximations since they are stored with a finite number of bits.

Hence 1.0/3.0 is slightly less than 1/3.1.0/3.0 + 1.0/3.0 + 1.0/3.0 could be less

than 1.

www.cs.fit.edu/~pkc/classes/iComputing/FloatEquality.java

Abstraction Levels

Binary Data

Numbers (unsigned, signed [Two’s complement], floating point) Text (ASCII, Unicode)

• HTML Color

• Image (JPEG)Video (MPEG)

Sound (MP3) Instructions

Machine language (CPU-dependent) Text (ASCII)

Assembly language (CPU-dependent)• High-level language (CPU -independent: Java, C++, FORTRAN)