Numbers in binary system code (base two and computer representation). Numerical conversions.

Introduction into the binary system

Binary numbers. Binary numeral system

In mathematics and digital electronics, a binary number is a number expressed in the binary numeral system, which is a base 2 numeral system. In this system, numeric values are represented using only two different symbols: typically 0 (zero) and 1 (one). The base 2 system is a positional numeric notation with a radix of 2. Binary is the way a computer holds information, the 1's and 0's.

Examples of binary numbers: 01, 10, 001, 010, 011, 100, 101, 110, 111, etc.

The decimal number (denary) system we're all familiar with is a base-ten system, meaning it uses ten distinct digits - 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Examples of numbers in the decimal system: 1, 2, 3, 10, 101, 304, 579, 2746, 54206, etc.

Counting in binary system

After 0 and 1 comes 10. In fact, whenever a number made up of entirely 1's is reached, an extra digit is added. This is exactly the same thing that happens in the decimal system when a number made up of entirely 9's is reached.

Here are the first 32 (33) numbers expressed in binary and their decimal equivalent:

Decimal numberBinary number
0(10)0(2)
1(10)1(2)
2(10)10(2)
3(10)11(2)
4(10)100(2)
5(10)101(2)
6(10)110(2)
7(10)111(2)
8(10)1000(2)
9(10)1001(2)
10(10)1010(2)
11(10)1011(2)
12(10)1100(2)
13(10)1101(2)
14(10)1110(2)
15(10)1111(2)
16(10)1 0000(2)
17(10)1 0001(2)
18(10)1 0010(2)
19(10)1 0011(2)
20(10)1 0100(2)
21(10)1 0101(2)
22(10)1 0110(2)
23(10)1 0111(2)
24(10)1 1000(2)
25(10)1 1001(2)
26(10)1 1010(2)
27(10)1 1011(2)
28(10)1 1100(2)
29(10)1 1101(2)
30(10)1 1110(2)
31(10)1 1111(2)
32(10)10 0000(2)

As you can see, there are 32 distinct numbers that can be represented on 5 digits or less (1 through 31, as well as 0). This can be also calculated, because 32 = 25. The total quantity of distinct numbers that can be represented in 8 digits is 28 = 256. 1 through 255 as well as 0. So 255 in binary is 11111111. Because binary uses base two as opposed to the decimal base ten, the numbers get larger much more quickly, but they still obey the same principles.

Binary Letters

There are multiple methods of representing letters and symbols in binary code. These methods are called encodings. For example, the ASCII encoding assigns unique binary bytes to 128 different characters. This makes it possible to encode any string of text. Below you can find the alphabet letters in binary.

Alphabet in binary, capital letters & lower case

Capital letterBinary codeLower caseBinary code
A0100 0001a0110 0001
B0100 0010 b011 00010
C0100 0011 c011 00011
D0100 0100 d011 00100
E0100 0101 e0110 0101
F0100 0110 f0110 0110
G0100 0111 g0110 0111
H0100 1000 h0110 1000
I0100 1001 i0110 1001
J0100 1010 j0110 1010
K0100 1011 k0110 1011
L0100 1100 l0110 1100
M0100 1101 m0110 1101
N0100 1110 n0110 1110
O0100 1111 o0110 1111
P0101 0000 p0111 0000
Q0101 0001 q0111 0001
R0101 0010 r0111 0010
S0101 0011 s0111 0011
T0101 0100 t0111 0100
U0101 0101 u0111 0101
V0101 0110 v0111 0110
W0101 0111 w0111 0111
X0101 1000 x0111 1000
Y0101 1001 y0111 1001
Z0101 1010 z0111 1010
* Spaces were used inside binary codes, to group digits by four, in order to make them more readable

How computer works

At a physical level, the 0's and 1's are stored in the central processing unit (CPU) of a computer system using logic gates or transistors. Transistors are microscopic switches that control the flow of electricity. If a current passes through the transistor (switch closed), this represents a 1. If a current doesn't pass through (switch open), this represents a 0. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices. Each digit is referred to as a bit. The term also refers to any digital encoding/decoding system in which there are exactly two possible states. In digital data memory, storage, processing, and communications, the 0 and 1 values are sometimes called "low" and "high," respectively. Binary information is also transmitted using magnetic properties; the two different types of polarities are used to represent zeros and ones. An optical disk, such as a CD-ROM or DVD, also stores binary information in the form of pits and lands (the area between the pits).

Computer software translates between binary information and the information you actually work with on a computer such as decimal numbers, text, photos, sound, and video. Binary information is sometimes also referred to as machine language since it represents the most fundamental level of information stored in a computer system.

Bits and bytes

Bits can be grouped together to make them easier to work with. A group of 8 bits is called a byte. Other groupings include:

GroupingEquivalent
Nibble4 bits (half a byte)
Byte8 bits
Kilobyte (KB)1024 bytes (or 1024 x 8 bits)
Megabyte (MB)1024 kilobytes (or 1024 bytes x 1024 bytes = 1048576 bytes)
Gigabyte (GB)1024 Megabytes
Terabyte (TB)1024 Gigabytes

Most computers can process millions of bits every second. A hard drive's storage capacity is measured in Gigabytes or Terabytes. RAM is often measured in Megabytes or Gigabytes (as of 2016...).

Below you can find all the numeral conversions operations that can be run on this website: conversions between binary and decimal positional numeral systems for integers (unsigned, signed, one's complement, two's complement) and decimals (float or double on 32/64 bit, single or double precision, IEEE 754 floating point standard)