Binary Numbers#

Decimal number#

The number that we see daily.

$$N_{10} = \Sigma a_i\times10^i, 0\le a_i\lt 10$$

e.g. : $15_{10} = 1\times10^1+5\times10^0$

Hex (Base-16) number#

$$N_{16} = \Sigma a_i\times16^i, 0\le a_i\lt F$$ $$a_i = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E\}$$

e.g. : $B65F_{16} = 11\times16^3+6\times16^2+5\times16^1+15\times16^0=46687_{10}$

Binary number#

$$N_{2} = \Sigma a_i\times2^i, 0\le a_i\lt 2$$

e.g. : $15_{10} = 1\times2^3+1\times2^2+1\times2^1+1\times2^0 = 1111_{2}$

This method is most often used in computer,there are few terminology as shown below.

Sign Magnitude#

Info:

$$N'=\pm N$$

It can represent the number in $[-2^{n-1}+1, 2^{n-1}-1]$.

Use the first bit to record the magnitude, 1 means negative, 0 is positive.

Example#

We want to use 4 digits to represent $-3$.

$$-3_{10}=1\space 011_2$$

1’s complement#

Info:

$$N' = (2^n - 1) - N$$

It can represent the number in $[-2^{n-1}+1,2^{n-1}-1]$.

Just simply invert every bit.

Example#

We want to use 4 digits to represent $-3$.

$$-3 = (16-1) - 3 = 12$$

Thus, we have the representation of $-3$ :

$$-3_{10} = 1\space 100_2$$

2’s complement#

Info:

$$N' = 2^n - N$$

It can represent the number in $[-2^{n-1},2^{n-1}-1]$.

Invert every bit and add 1.

Example#

We want to use 4 digits to represent $-3$.

$$-3 = 16 - 3 = 13$$

Thus, we have the representation of $-3$ :

$$-3_{10} = 1\space 101_2$$

Special Power of 2#

Binary Coded Decimal (BCD)#

A common coding method to code each digit of a decimal number in terms of binary bits. (Since the number is between 0 to 9, so we need 4 digits)

Sometimes, BCD will have a carryout, which means the higer number is 1 or 0.

Example#

• $3_{10} = 0011_{BCD}$
• $31_{10} = 0011\space 0001_{BCD}$
• $19_{10} = 1\space1001_{BCD}$ ($C_{out} = 1$)
• $8_{10} = 0\space 1000_{BCD}$ ($C_{out} = 0$)

ASCII#

ASCII (American Standard Code for Information Interchange) is the standard binary code for the set of alphanumeric characters, the character set includes :

• numerals
• alphabets
• special printable characters
• control characters

tags: Logic Design EECS1010#