Lecture 1: Number Representation

Big Idea: Everything is Bits

All data in your computer is stored as bits, sequences of 0s and 1s.

High voltage wire = 1. Low voltage wire = 0.

Numbers in Different Bases

• Binary System (Base 2)
• Decimal System (Base 10)
• Octal System (Base 8)
• Hexadecimal System (Base 16)

We use letters if we need more digits in Hex

The base-16 digits: 0 1 2 3 4 5 6 7 8 9 A(10) B(11) C(12) D(13) E(14) F(15)

Converting from Base-2/8/16 to Base-10

To convert bases(-2/8/16...) to base-10: Write out the powers of that base (e.g. powers of 16).

Converting from Base-10 to Base-2/8/16

Use the "leftover algorithm".

Leftover Algorithm:

• If you use a box, you have to fill it up.
• Use as few boxes as possible.

Numbers in Different Bases

Commonly Used Bases

• Base 10 (decimal):
  • Digits: 0 1 2 3 4 5 6 7 8 9 ○ Understandable by humans.
• Base 2 (binary):
  • Digits: 0 1 ○ Converting numbers to base 2 lets us represent numbers as bits!
• Base 16 (hexadecimal):
  • Digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F ○ A convenient shorthand for writing long sequences of bits.

Commonly Used Bases – Notation

Writing numbers is ambiguous when we have different bases.

1011 Is this base 2? Base 10? Base 16? Base 7?

Converting Between Commonly-Used Bases

How to padding with zeros?

Groups of bits have special names

• 1 byte = 8 bits. 2 hex digits.
• 1 nibble = 4 bits. 1 hex digit.

Representing Integers (Including Negatives)

Unsigned Integers: Number Line View

Overflow occurs when we exceed the largest value and wrap back around to 0.

• Example in 8-bit binary: 11111111 + 00000001 = 00000000.

Negative overflow occurs when we try to go below 0 and wrap around to large values.

• Example in 8-bit binary: 00000001 – 00000010 = 11111111.
• Don't call it underflow. That's a different thing.

underflow

在计算机科学中，underflow 指的是在执行浮点数计算时，结果的绝对值小于计算机能够表示的最小正数，从而被近似为零的情况。这与整数的negative overflow不同。

Underflow 发生在浮点数计算中，具体来说，当浮点数计算的结果绝对值非常小，以至于计算机无法表示（因为它小于浮点数格式能表示的最小正数），结果就会被舍入为零。这是一种精度丢失现象。

简单来说，negative overflow 是在整数运算中出现的现象，而 underflow 则是浮点数运算中特有的情况。

Different Representations

Sign-Magnitude (Not Commonly Used)

idea

Use the left-most bit to indicate if the number is positive (0) or negative (1).

sign-magnitude representation

0b1000...000 and 0b0000...000 both represent zero!

One's Complement (Not Commonly Used)

idea

If the number is negative, flip(翻转) the bits.

• There's an unavoidable overflow, just like in unsigned.
• Most positive number + 1 = Most negative number.
• 0b1111...111 and 0b0000...000 both represent zero!

Two's Complement (What we use today)

idea

If the number is negative, flip the bits, and add one (Because we shifted to avoid double-zero).

Two's Complement: Conversion Algorithm

From Binary to Decimal

To convert two's complement to a signed decimal number (base 10):

• If left-most digit is 0: Positive number.
  • Just read it as unsigned.
• If left-most digit is 1: Negative number.
  • Flip the bits, and add 1.
  • Convert to base-10, and stick a negative sign in front.

From Decimal to Binary

To convert a signed decimal number to two's complement:

• If number is positive:
  • Just convert it to base-2.
• If number is negative:
  • Pretend it's unsigned, and convert to base-2.
  • Flip the bits, and add 1.

Two's Complement: Deep Connection to Modular Arithmetic

Bias Notation

idea

Just like unsigned, but shifted on the number line

\(Standard Bias\) of –(\(2^{N–1}\)– 1)

Convert biased notation to decimal

1. Read as unsigned decimal.
2. Add the bias.

Convert decimal to biased notation

1. Subtract the bias.
2. Convert to unsigned binary.