1. CS 101 – Sept. 4 Number representation Integer Unsigned √ Signed √
Sign-magnitude
BCD
Real
Convert from decimal to binary
Binary scientific notation

2. Scheme III: Sign-magnitude
Leftmost bit works like a +/- sign.
Remaining bits behave like unsigned.
Example: what is ?
Leftmost bit “1” means number is negative
Remaining bits are 1001 = 9, so entire number is –9
What would –20 look like in 8 bits?
+20 looks like or
To make it negative, change first bit to 1.

3. Scheme IV: BCD “Binary Coded Decimal” Ex. 278 = 0010 0111 1000
Only good for positives.
Each decimal digit corresponds to 4 bits.
Easy conversion for binary  decimal
Ex =
If we have 8 bit BCD, what is the range?
Waste
Purpose – easy conversion to/from base 10

4. Having enough bits
Question – Are 6 bits sufficient to represent the number 50?
Answer – Depends on the rep’n scheme. Let’s look at each one.
Avoiding overflow

5. Integer vs. Real
Integer arithmetic on computer is quick and exact, but limited range
Real arithmetic needs wide range, reasonable degree of precision
Preferred for numerical computation
14 significant digits is usually enough!

6. Binary examples Look at this sequence: 111 = 7 1110 = 14 11100 = 28
= 56
= 112
Let’s go the other way
111. = 7
11.1 = 3.5 or 7/2
1.11 = 1.75 or 7/4
.111 = 7/8
.0111 = 7/16

7. Another example 10100 = 20 1010 = 10 101 = 5 10.1 = 2.5 or 5/2
.101 = 5/8
Each digit corresponds to (+/–) power of 2.
For example, the binary equals what in decimal?

8. Convert to binary
Separate number (e.g. 5.7) into integer and fractional parts.
Integer part? Binary store.
Fractional part:
Keep multiplying fractional part by 2 until it becomes zero, or until you have a repeating pattern.

9. Example Convert 9.625 to binary. “9” becomes 1001.
Fractional part is 0.625:
.625 * 2 = 1.25
.25 * 2 = 0.5
.5 * 2 = 1.0
Fractional part reached 0. So answer is

10. Example Let’s convert 4.6875 to binary The “4” becomes 100.
Fractional part is :
.6875 * 2 = 1.375
* 2 = 0.75
* 2 = 1.5
* 2 = 1.0
Fractional part reached 0. So answer is

11. Repeating pattern Let’s try converting 0.7 to binary. .7 * 2 = 1.4
.4 * 2 = 0.8
.8 * 2 = 1.6
.6 * 2 = 1.2
.2 * 2 = 0.4
And we’ve seen “.4” already ____
Answer is … or
Note the bar.