HELP! I don't know binary, hexadecimal, octal, and bit-wise

Question

I haven't learned this in programming class before, but now I need to know it. What are some good resources for learning these numbers and how to convert them? I pretty much am going to memorise these like the times table.

Answer 1

In our everyday decimal system, the base number, or radix is 10. A number system's radix tells us how many different digits are in use. In decimal system we use digits 0 through 9.

The significance of a digit is radix ^ i, where i is digit's position counting from right, starting at zero.

Decimal number 6789 broken down:

 6  7  8  9              radix ^ i
 │  │  │  │             ──────────────
 │  │  │  └── ones       10 ^ 0 = 1
 │  │  └───── tens       10 ^ 1 = 10
 │  └──────── hundreds   10 ^ 2 = 100
 └─────────── thousands  10 ^ 3 = 1000

  ones      tens       hundreds    thousands
  ───────────────────────────────────────────────
  (9 * 1) + (8 * 10) + (7 * 100) + (6 * 1000)
= 9       + 80       + 700       + 6000
= 6789

This scheme will help us understand any number system in terms of decimal numbers.

---

Hexadecimal system's radix is 16, so we need to employ additional digits A...F to denote 10...15. Let's break down hexadecimal number CDEFh in a similar fashion:

 C  D  E  F              radix ^ i
 │  │  │  │             ──────────────
 │  │  │  └── ones       16 ^ 0 = 1
 │  │  └───── sixteens   16 ^ 1 = 16
 │  └──────── 256:s      16 ^ 2 = 256
 └─────────── 4096:s     16 ^ 3 = 4096

  ones       sixteens    256:s        4096:s
  ───────────────────────────────────────────────
  (Fh * 1) + (Eh * 16) + (Dh * 256) + (Ch * 4096)
= (15 * 1) + (14 * 16) + (13 * 256) + (12 * 4096)
= 15       + 224       + 3328       + 49152
= 52719

We have just converted the number CDEFh to decimal (i.e. switched base 16 to base 10).

---

In binary system, the radix is 2, so only digits 0 and 1 are used. Here is the conversion of binary number 1010b to decimal:

 1  0  1  0              radix ^ i
 │  │  │  │             ──────────────
 │  │  │  └── ones       2 ^ 0 = 1
 │  │  └───── twos       2 ^ 1 = 2
 │  └──────── fours      2 ^ 2 = 4
 └─────────── eights     2 ^ 3 = 8

  ones      twos      fours     eights
  ───────────────────────────────────────────────
  (0 * 1) + (1 * 2) + (0 * 4) + (1 * 8)
= 0       + 2       + 0       + 8
= 10

---

Octal system - same thing, radix is 8, digits 0...7 are in use. Converting octal 04567 to decimal:

 4  5  6  7              radix ^ i
 │  │  │  │             ──────────────
 │  │  │  └── ones       8 ^ 0 = 1
 │  │  └───── eights     8 ^ 1 = 8
 │  └──────── 64:s       8 ^ 2 = 64
 └─────────── 512:s      8 ^ 3 = 512

  ones      eights    64:s       512:s
  ───────────────────────────────────────────────
  (7 * 1) + (6 * 8) + (5 * 64) + (4 * 512)
= 7       + 48      + 320      + 2048
= 2423

So, to do a conversion between number systems is to simply change the radix.

To learn about bitwise operators, see http://www.eskimo.com/~scs/cclass/int/sx4ab.html.

1: http://en.wikipedia.org/wiki/RadixIn our everyday decimal system, the base number, or radix is 10. A number system's radix tells us how many different digits are in use. In decimal system we use digits 0 through 9.

The significance of a digit is radix ^ i, where i is digit's position counting from right, starting at zero.

Decimal number 6789 broken down:

 6  7  8  9              radix ^ i
 │  │  │  │             ──────────────
 │  │  │  └── ones       10 ^ 0 = 1
 │  │  └───── tens       10 ^ 1 = 10
 │  └──────── hundreds   10 ^ 2 = 100
 └─────────── thousands  10 ^ 3 = 1000

  ones      tens       hundreds    thousands
  ───────────────────────────────────────────────
  (9 * 1) + (8 * 10) + (7 * 100) + (6 * 1000)
= 9       + 80       + 700       + 6000
= 6789

This scheme will help us understand any number system in terms of decimal numbers.

---

Hexadecimal system's radix is 16, so we need to employ additional digits A...F to denote 10...15. Let's break down hexadecimal number CDEFh in a similar fashion:

 C  D  E  F              radix ^ i
 │  │  │  │             ──────────────
 │  │  │  └── ones       16 ^ 0 = 1
 │  │  └───── sixteens   16 ^ 1 = 16
 │  └──────── 256:s      16 ^ 2 = 256
 └─────────── 4096:s     16 ^ 3 = 4096

  ones       sixteens    256:s        4096:s
  ───────────────────────────────────────────────
  (Fh * 1) + (Eh * 16) + (Dh * 256) + (Ch * 4096)
= (15 * 1) + (14 * 16) + (13 * 256) + (12 * 4096)
= 15       + 224       + 3328       + 49152
= 52719

We have just converted the number CDEFh to decimal (i.e. switched base 16 to base 10).

---

In binary system, the radix is 2, so only digits 0 and 1 are used. Here is the conversion of binary number 1010b to decimal:

 1  0  1  0              radix ^ i
 │  │  │  │             ──────────────
 │  │  │  └── ones       2 ^ 0 = 1
 │  │  └───── twos       2 ^ 1 = 2
 │  └──────── fours      2 ^ 2 = 4
 └─────────── eights     2 ^ 3 = 8

  ones      twos      fours     eights
  ───────────────────────────────────────────────
  (0 * 1) + (1 * 2) + (0 * 4) + (1 * 8)
= 0       + 2       + 0       + 8
= 10

---

Octal system - same thing, radix is 8, digits 0...7 are in use. Converting octal 04567 to decimal:

 4  5  6  7              radix ^ i
 │  │  │  │             ──────────────
 │  │  │  └── ones       8 ^ 0 = 1
 │  │  └───── eights     8 ^ 1 = 8
 │  └──────── 64:s       8 ^ 2 = 64
 └─────────── 512:s      8 ^ 3 = 512

  ones      eights    64:s       512:s
  ───────────────────────────────────────────────
  (7 * 1) + (6 * 8) + (5 * 64) + (4 * 512)
= 7       + 48      + 320      + 2048
= 2423

So, to do a conversion between number systems is to simply change the radix.

To learn about bitwise operators, see http://www.eskimo.com/~scs/cclass/int/sx4ab.html.