Number Systems

The number forty-two can be written in a variety of ways:

"forty-two" (English)
"quarantadue" (Italian)
"zweiundvierzig" (German)
"42" (base 10 - decimal)
"52" (base 8 - octal)
"101010" (base 2 - binary)
"2A" (base 16 - hexadecimal)
"132" (base 5)
"1S" (base 23)
"10" (base 42)

Notation System	Symbols Used
Base 10 (decimal)	0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Base 8 (octal)	0, 1, 2, 3, 4, 5, 6, 7
Base 2 (binary)	0, 1
Base 16 (hexadecimal)	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Counting in Hexadecimal

Dec.	Hex.	Dec.	Hex.	Dec.	Hex.	Dec.	Hex.
0	0	11	B	22	16	33	21
1	1	12	C	23	17	34	22
2	2	13	D	24	18	35	23
3	3	14	E	25	19	36	24
4	4	15	F	26	1A	37	25
5	5	16	10	27	1B	38	26
6	6	17	11	28	1C	39	27
7	7	18	12	29	1D	40	28
8	8	19	13	30	1E	41	29
9	9	20	14	31	1F	42	2A
10	A	21	15	32	20	43	2B

Converting From Other Bases to Base 10

Base 16 to Base 10:

1AC = 1 × 16² + A × 16¹ + C × 16⁰

= 1 × 16² + 10 × 16¹ + 12 × 16⁰

= 256 + 160 + 12

= 428

FF2E05 = F × 16⁵ + F × 16⁴ + 2 × 16³ + E × 16² + 0 × 16¹ + 5 × 16⁰

= 15 × 16⁵ + 15 × 16⁴ + 2 × 16³ + 14 × 16² + 0 × 16¹ + 5 × 16⁰

= 15,728,640 + 983,040 + 8,192 + 3,584 + 0 + 5

= 16,723,461

Base 5 to Base 10:

12034 = 1 × 5⁴ + 2 × 5³ + 0 × 5² + 3 × 5¹ + 4 × 5⁰

= 625 + 250 + 0 + 15 + 4

= 894

Base 2 to Base 10:

110101 = 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰

= 32 + 16 + 0 + 4 + 0 + 1

= 53

Converting From Base 10 to Other Bases

Base 10 to Base 16:

     1 ÷ 16 = 0 remainder 1     >>>   1         = 1AC
   26 ÷ 16 = 1 remainder 10   >>>   A
428 ÷ 16 = 26 remainder 12 >>>   C

Base 10 to Base 5:

3 ÷ 5 = 0 remainder 3 = 3203
17 ÷ 5 = 3 remainder 2
85 ÷ 5 = 17 remainder 0
428 ÷ 5 = 85 remainder 3

Base 10 to Base 2:

        1 ÷ 2 = 0 remainder 1                        = 110011000
        3 ÷ 2 = 1 remainder 1
        6 ÷ 2 = 3 remainder 0
      12 ÷ 2 = 6 remainder 0
    25 ÷ 2 = 12 remainder 1
    51 ÷ 2 = 25 remainder 1
102 ÷ 2 = 51 remainder 0
214 ÷ 2 = 102 remainder 0
428 ÷ 2 = 214 remainder 0

Dec.	Hex.	Dec.	Hex.	Dec.	Hex.	Dec.	Hex.
0	0	11	B	22	16	33	21
1	1	12	C	23	17	34	22
2	2	13	D	24	18	35	23
3	3	14	E	25	19	36	24
4	4	15	F	26	1A	37	25
5	5	16	10	27	1B	38	26
6	6	17	11	28	1C	39	27
7	7	18	12	29	1D	40	28
8	8	19	13	30	1E	41	29
9	9	20	14	31	1F	42	2A
10	A	21	15	32	20	43	2B

Dec.	Hex.	Dec.	Hex.	Dec.	Hex.	Dec.	Hex.
0	0	11	B	22	16	33	21
1	1	12	C	23	17	34	22
2	2	13	D	24	18	35	23
3	3	14	E	25	19	36	24
4	4	15	F	26	1A	37	25
5	5	16	10	27	1B	38	26
6	6	17	11	28	1C	39	27
7	7	18	12	29	1D	40	28
8	8	19	13	30	1E	41	29
9	9	20	14	31	1F	42	2A
10	A	21	15	32	20	43	2B

Dec.	Hex.	Dec.	Hex.	Dec.	Hex.	Dec.	Hex.
0	0	11	B	22	16	33	21
1	1	12	C	23	17	34	22
2	2	13	D	24	18	35	23
3	3	14	E	25	19	36	24
4	4	15	F	26	1A	37	25
5	5	16	10	27	1B	38	26
6	6	17	11	28	1C	39	27
7	7	18	12	29	1D	40	28
8	8	19	13	30	1E	41	29
9	9	20	14	31	1F	42	2A
10	A	21	15	32	20	43	2B