We use the decimal numbering system (base 10) in our everyday lives. This can be seen below:

$$632 = (100 * 6) + (10 * 3) + (1*2)$$

This may be seen through the formula $$\sum^2_{i=0}d_i = dn * 10^n + dn_{-110^{n-1+...}}d_110^1+d_0*10^0$$

$d_n$ is the most significant digit and $d_0$ is the least significant digit.

Generalize - Numbering System with base X

In Base X, a non-negative intereger $d_nd_{n-1}...d_1d_0$

$$ \sum^2_{i=0}d_i * Xi = dn * Xn + dn_{-1}X^{n-1+...+}d_1X^1+d_0X^0 $$

• The same number can have many representations on many bases
• For example, consider the decimal number 23
• Computers use a binary system so 23 would be $10111_2$
• Sometimes we use a hexadecimal system to make reading binary representations 'easier'. $23_{10} = 0x17(17_{16})$

cout converts the binary storage of the number into the requested format The C language uses specifiers to convert within scanf() this can be seen in [[C1]]

Base 10	Base 2	Base 8	Base 16
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B

Number Representation and Binary Arithmetic

• To convert to binary repeatedly divide the decimal number by 2, until the quotient is 0.
• Collect the remainders as you go
• Write down the remainders from left to right

[[Computer Science Notes/CDA 3100/Index|Index]]