Base 10, Base 2 & Base 5
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9.
In any standard positional numeral system, a number is conventionally written as (x)y with x as the string of digits and y as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)10 is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.
Contents
In numeral systems
In the system with radix 13, for example, a string of digits such as 398 denotes the (decimal) number 3 × 132 + 9 × 131 + 8 × 130 = 632.
More generally, in a system with radix b (b > 1), a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, where 0 ≤ di < b. In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix b would have a ones' place, then a b1s' place, a b2s' place, etc.
Commonly used numeral systems include:
| Base/radix | Name | Description |
|---|---|---|
| 2 | Binary numeral system | Used internally by nearly all computers, is base 2. The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electric counters. |
| 8 | Octal system | Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits (23). |
| 10 | Decimal system | Used by humans in the vast majority of cultures. Its ten digits are "0"–"9". Used in most mechanical counters. |
| 12 | Duodecimal (dozenal) system | Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses. |
| 16 | Hexadecimal system | Often used in computing as a more compact representation of binary (1 hex digit per 4 bits). The sixteen digits are "0"–"9" followed by "A"–"F" or "a"–"f". |
| 20 | Vigesimal system | Traditional numeral system in several cultures, still used by some for counting. Historically also known as the score system in English, now most famous in the phrase "four score and seven years ago" in the Gettysburg Address. |
| 60 | Sexagesimal system | Originated in ancient Sumer and passed to the Babylonians. Used today as the basis of modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes, and seconds) by analogy to the rotation of the Earth. |
Base 2
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).
The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language.
Representation
Any number can be represented by a sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of the following rows of symbols can be interpreted as the binary numeric value of 667:
| 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| | | ― | | | ― | ― | | | | | ― | | | | |
| ☒ | ☐ | ☒ | ☐ | ☐ | ☒ | ☒ | ☐ | ☒ | ☒ |
| y | n | y | n | n | y | y | n | y | y |
The numeric value represented in each case is dependent upon the value assigned to each symbol. In the earlier days of computing, switches, punched holes and punched paper tapes were used to represent binary values. In a modern computer, the numeric values may be represented by two different voltages; on a magnetic disk, magnetic polarities may be used. A "positive", "yes", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.
In keeping with customary representation of numerals using Arabic numerals, binary numbers are commonly written using the symbols 0 and 1. When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations are equivalent:
- 100101 binary (explicit statement of format)
- 100101b (a suffix indicating binary format; also known as Intel convention)
- 100101B (a suffix indicating binary format)
- bin 100101 (a prefix indicating binary format)
- 1001012 (a subscript indicating base-2 (binary) notation)
- %100101 (a prefix indicating binary format; also known as Motorola convention)
- 0b100101 (a prefix indicating binary format, common in programming languages)
- 6b100101 (a prefix indicating number of bits in binary format, common in programming languages)
- #b100101 (a prefix indicating binary format, common in Lisp programming languages)
When spoken, binary numerals are usually read digit-by-digit, in order to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced one zero zero, rather than one hundred, to make its binary nature explicit, and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as one hundred (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct value), but this does not make its binary nature explicit.
Binary arithmetic
Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.
Addition
The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:
- 0 + 0 → 0
- 0 + 1 → 1
- 1 + 0 → 1
- 1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21) )
Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:
- 5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101) )
- 7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101) )
This is known as carrying. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:
1 1 1 1 1 (carried digits)
0 1 1 0 1
+ 1 0 1 1 1
-------------
= 1 0 0 1 0 0 = 36
In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).
When computers must add two numbers, the rule that: x xor y = (x + y) mod 2 for any two bits x and y allows for very fast calculation, as well.
Long carry method
A simplification for many binary addition problems is the Long Carry Method or Brookhouse Method of Binary Addition. This method is generally useful in any binary addition in which one of the numbers contains a long "string" of ones. It is based on the simple premise that under the binary system, when given a "string" of digits composed entirely of n ones (where n is any integer length), adding 1 will result in the number 1 followed by a string of n zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of n 9s will result in the number 1 followed by a string of n 0s:
Binary Decimal
1 1 1 1 1 likewise 9 9 9 9 9
+ 1 + 1
——————————— ———————————
1 0 0 0 0 0 1 0 0 0 0 0
Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations. In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 02 (95810) and 1 0 1 0 1 1 0 0 1 12 (69110), using the traditional carry method on the left, and the long carry method on the right:
Traditional Carry Method Long Carry Method
vs.
1 1 1 1 1 1 1 1 (carried digits) 1 ← 1 ← carry the 1 until it is one digit past the "string" below
1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 cross out the "string",
+ 1 0 1 0 1 1 0 0 1 1 + 1 0 1 0 1 1 0 0 1 1 and cross out the digit that was added to it
——————————————————————— ——————————————————————
= 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1
The top row shows the carry bits used. Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series. The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique. Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 12 (164910). In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort.
Addition table
| 0 | 1 | |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 10 |
The binary addition table is similar, but not the same, as the truth table of the logical disjunction operation . The difference is that , while .
Subtraction
Subtraction works in much the same way:
- 0 − 0 → 0
- 0 − 1 → 1, borrow 1
- 1 − 0 → 1
- 1 − 1 → 0
Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as borrowing. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value.
* * * * (starred columns are borrowed from) 1 1 0 1 1 1 0 − 1 0 1 1 1 ---------------- = 1 0 1 0 1 1 1
* (starred columns are borrowed from) 1 0 1 1 1 1 1 - 1 0 1 0 1 1 ---------------- = 0 1 1 0 1 0 0
Subtracting a positive number is equivalent to adding a negative number of equal absolute value. Computers use signed number representations to handle negative numbers—most commonly the two's complement notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula:
- A − B = A + not B + 1
Multiplication
Multiplication in binary is similar to its decimal counterpart. Two numbers A and B can be multiplied by partial products: for each digit in B, the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used. The sum of all these partial products gives the final result.
Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:
- If the digit in B is 0, the partial product is also 0
- If the digit in B is 1, the partial product is equal to A
For example, the binary numbers 1011 and 1010 are multiplied as follows:
1 0 1 1 (A)
× 1 0 1 0 (B)
---------
0 0 0 0 ← Corresponds to the rightmost 'zero' in B
+ 1 0 1 1 ← Corresponds to the next 'one' in B
+ 0 0 0 0
+ 1 0 1 1
---------------
= 1 1 0 1 1 1 0
Binary numbers can also be multiplied with bits after a binary point:
1 0 1 . 1 0 1 A (5.625 in decimal)
× 1 1 0 . 0 1 B (6.25 in decimal)
-------------------
1 . 0 1 1 0 1 ← Corresponds to a 'one' in B
+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in B
+ 0 0 0 . 0 0 0
+ 1 0 1 1 . 0 1
+ 1 0 1 1 0 . 1
---------------------------
= 1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal)
See also Booth's multiplication algorithm.
Multiplication table
| 0 | 1 | |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
The binary multiplication table is the same as the truth table of the logical conjunction operation .
Division
Long division in binary is again similar to its decimal counterpart.
In the example below, the divisor is 1012, or 5 in decimal, while the dividend is 110112, or 27 in decimal. The procedure is the same as that of decimal long division; here, the divisor 1012 goes into the first three digits 1102 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence:
1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
0 0 1
The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:
1 0 1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
1 1 1
− 1 0 1
-----
0 1 0
Thus, the quotient of 110112 divided by 1012 is 1012, as shown on the top line, while the remainder, shown on the bottom line, is 102. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2.
Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.
Square root
The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained here. An example is:
1 0 0 1
---------
√ 1010001
1
---------
101 01
0
--------
1001 100
0
--------
10001 10001
10001
-------
0
Representing real numbers
Non-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point (called a decimal point in the decimal system). For example, the binary number 11.012 means:
| 1 × 21 | (1 × 2 = 2) | plus |
| 1 × 20 | (1 × 1 = 1) | plus |
| 0 × 2−1 | (0 × = 0) | plus |
| 1 × 2−2 | (1 × = 0.25) |
For a total of 3.25 decimal.
All dyadic rational numbers have a terminating binary numeral—the binary representation has a finite number of terms after the radix point. Other rational numbers have binary representation, but instead of terminating, they recur, with a finite sequence of digits repeating indefinitely. For instance
The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in decimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the geometric series 2−1 + 2−2 + 2−3 + ... which is 1.
Binary numerals which neither terminate nor recur represent irrational numbers. For instance,
- 0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
- 1.0110101000001001111001100110011111110... is the binary representation of , the square root of 2, another irrational. It has no discernible pattern.
Base 5
Quinary /ˈkwaɪnəri/ (base-5 or pental) is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand.
In the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100 and sixty is written as 220.
As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers (4 and 6) guarantees that many recurring fractions have relatively short periods.
Today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a sub-base system, is sexagesimal, base 60, which used 10 as a sub-base.
Each quinary digit can hold log25 (approx. 2.32) bits of information.
Comparison to other radices
| × | 1 | 2 | 3 | 4 | 10 | 11 | 12 | 13 | 14 | 20 |
| 1 | 1 | 2 | 3 | 4 | 10 | 11 | 12 | 13 | 14 | 20 |
| 2 | 2 | 4 | 11 | 13 | 20 | 22 | 24 | 31 | 33 | 40 |
| 3 | 3 | 11 | 14 | 22 | 30 | 33 | 41 | 44 | 102 | 110 |
| 4 | 4 | 13 | 22 | 31 | 40 | 44 | 103 | 112 | 121 | 130 |
| 10 | 10 | 20 | 30 | 40 | 100 | 110 | 120 | 130 | 140 | 200 |
| 11 | 11 | 22 | 33 | 44 | 110 | 121 | 132 | 143 | 204 | 220 |
| 12 | 12 | 24 | 41 | 103 | 120 | 132 | 144 | 211 | 223 | 240 |
| 13 | 13 | 31 | 44 | 112 | 130 | 143 | 211 | 224 | 242 | 310 |
| 14 | 14 | 33 | 102 | 121 | 140 | 204 | 223 | 242 | 311 | 330 |
| 20 | 20 | 40 | 110 | 130 | 200 | 220 | 240 | 310 | 330 | 400 |
| Quinary | 0 | 1 | 2 | 3 | 4 | 10 | 11 | 12 | 13 | 14 | 20 | 21 | 22 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Binary | 0 | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1001 | 1010 | 1011 | 1100 |
| Decimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Quinary | 23 | 24 | 30 | 31 | 32 | 33 | 34 | 40 | 41 | 42 | 43 | 44 | 100 |
| Binary | 1101 | 1110 | 1111 | 10000 | 10001 | 10010 | 10011 | 10100 | 10101 | 10110 | 10111 | 11000 | 11001 |
| Decimal | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
| Decimal (periodic part) | Quinary (periodic part) | Binary (periodic part) |
| 1/2 = 0.5 | 1/2 = 0.2 | 1/10 = 0.1 |
| 1/3 = 0.3 | 1/3 = 0.13 | 1/11 = 0.01 |
| 1/4 = 0.25 | 1/4 = 0.1 | 1/100 = 0.01 |
| 1/5 = 0.2 | 1/10 = 0.1 | 1/101 = 0.0011 |
| 1/6 = 0.16 | 1/11 = 0.04 | 1/110 = 0.010 |
| 1/7 = 0.142857 | 1/12 = 0.032412 | 1/111 = 0.001 |
| 1/8 = 0.125 | 1/13 = 0.03 | 1/1000 = 0.001 |
| 1/9 = 0.1 | 1/14 = 0.023421 | 1/1001 = 0.000111 |
| 1/10 = 0.1 | 1/20 = 0.02 | 1/1010 = 0.00011 |
| 1/11 = 0.09 | 1/21 = 0.02114 | 1/1011 = 0.0001011101 |
| 1/12 = 0.083 | 1/22 = 0.02 | 1/1100 = 0.0001 |
| 1/13 = 0.076923 | 1/23 = 0.0143 | 1/1101 = 0.000100111011 |
| 1/14 = 0.0714285 | 1/24 = 0.013431 | 1/1110 = 0.0001 |
| 1/15 = 0.06 | 1/30 = 0.013 | 1/1111 = 0.0001 |
| 1/16 = 0.0625 | 1/31 = 0.0124 | 1/10000 = 0.0001 |
| 1/17 = 0.0588235294117647 | 1/32 = 0.0121340243231042 | 1/10001 = 0.00001111 |
| 1/18 = 0.05 | 1/33 = 0.011433 | 1/10010 = 0.0000111 |
| 1/19 = 0.052631578947368421 | 1/34 = 0.011242141 | 1/10011 = 0.000011010111100101 |
| 1/20 = 0.05 | 1/40 = 0.01 | 1/10100 = 0.000011 |
| 1/21 = 0.047619 | 1/41 = 0.010434 | 1/10101 = 0.000011 |
| 1/22 = 0.045 | 1/42 = 0.01032 | 1/10110 = 0.00001011101 |
| 1/23 = 0.0434782608695652173913 | 1/43 = 0.0102041332143424031123 | 1/10111 = 0.00001011001 |
| 1/24 = 0.0416 | 1/44 = 0.01 | 1/11000 = 0.00001 |
| 1/25 = 0.04 | 1/100 = 0.01 | 1/11001 = 0.00001010001111010111 |
Usage
Many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below:
| Number | Base 5 | Numeral |
|---|---|---|
| 1 | 1 | wanggany |
| 2 | 2 | marrma |
| 3 | 3 | lurrkun |
| 4 | 4 | dambumiriw |
| 5 | 10 | wanggany rulu |
| 10 | 20 | marrma rulu |
| 15 | 30 | lurrkun rulu |
| 20 | 40 | dambumiriw rulu |
| 25 | 100 | dambumirri rulu |
| 50 | 200 | marrma dambumirri rulu |
| 75 | 300 | lurrkun dambumirri rulu |
| 100 | 400 | dambumiriw dambumirri rulu |
| 125 | 1000 | dambumirri dambumirri rulu |
| 625 | 10000 | dambumirri dambumirri dambumirri rulu |
Biquinary
A decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are an early biquinary system. The numbers 1, 5, 10, and 50 are written as I, V, X, and L respectively. Seven is VII and seventy is LXX. The full list is:
| I | V | X | L | C | D | M |
| 1 | 5 | 10 | 50 | 100 | 500 | 1000 |
Note that these are not positional number systems. In theory a number such as 73 could be written as IIIXXL without ambiguity as well as LXXIII and it is still not possible to extend it beyond thousands. There is also no sign for zero. But with the introduction of inversions such as IV and IX, it was necessary to keep the order from most to least significant.
Many versions of the abacus, such as the suanpan and soroban, use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary. Units of currencies are commonly partially or wholly biquinary.
Bi-quinary coded decimal is a variant of biquinary that was used on a number of early computers including Colossus and the IBM 650 to represent decimal numbers.
Licensing
Content obtained and/or adapted from:
- Radix, Wikipedia under a CC BY-SA license
- Binary number, Wikipedia under a CC BY-SA license
- Quinary, Wikipedia under a CC BY-SA license
