 Erik Østergaard - Octal Number System    ## Octal Number System

### The Octal Number Base System

Although this was once a popular number base, especially in the Digital Equipment Corporation PDP/8 and other old computer systems, it is rarely used today. The Octal system is based on the binary system with a 3-bit boundary. The Octal Number System:

uses base 8
includes only the digits 0 through 7 (any other digit would make the number an invalid octal number)

The weighted values for each position is as follows:

 8^5 8^4 8^3 8^2 8^1 8^0 32768 4096 512 64 8 1

### Binary to Octal Conversion

It is easy to convert from an integer binary number to octal. This is accomplished by:

1. Break the binary number into 3-bit sections from the LSB to the MSB.
2. Convert the 3-bit binary number to its octal equivalent.

For example, the binary value 1010111110110010 will be written:

 001 010 111 110 110 010 1 2 7 6 6 2

### Octal to Binary Conversion

It is also easy to convert from an integer octal number to binary. This is accomplished by:

1. Convert the decimal number to its 3-bit binary equivalent.
2. Combine the 3-bit sections by removing the spaces.

For example, the octal value 127662 will be written:

 1 2 7 6 6 2 001 010 111 110 110 010

This yields the binary number 001010111110110010 or 00 1010 1111 1011 0010 in our more readable format.

### Octal to Decimal Conversion

To convert from Octal to Decimal, multiply the value in each position by its Octal weight and add each value. Using the value from the previous example, 127662Q, we would expect to obtain the decimal value 44978.

 1*8^5 2*8^4 7*8^3 6*8^2 6*8^1 2*8^0 1*32768 2*4096 7*512 6*64 6*8 2*1 32768 8192 3584 384 48 2

32768 + 8192 + 3584 + 384 + 48 + 2 = 44978

### Decimal to Octal Conversion

To convert decimal to octal is slightly more difficult. The typical method to convert from decimal to octal is repeated division by 8. While we may also use repeated subtraction by the weighted position value, it is more difficult for large decimal numbers.

### Repeated Division By 8

For this method, divide the decimal number by 8, and write the remainder on the side as the least significant digit. This process is continued by dividing he quotient by 8 and writing the remainder until the quotient is 0. When performing the division, the remainders which will represent the octal equivalent of the decimal number are written beginning at the least significant digit (right) and each new digit is written to the next more significant digit (the left) of the previous digit. Consider the number 44978.

 Division Quotient Remainder Octal Number 44978 / 8 5622 2 2 5622 / 8 702 6 62 702 / 8 87 6 662 87 / 8 10 7 7662 10 / 8 1 2 27662 1 / 8 0 1 127662

As you can see, we are back with the original number. That is what we should expect.

#### Working with Logarithms / Udregning ved hjælp af logaritmer

A logarithm is used when working with exponentiation. We all learned that the formula `X = YZ` means take the value `Y` and multiply it by itself the number of times specified by `Z`. For example, `23 = 8` (`2*2*2`). The value `Z` is the exponential value of the equation. As long as you know what the `Y` and `Z` values are in the equation, it is easy to calculate the value of `X`. / En logaritme bruges, når der udregnes med eksponent. Vi lærte alle, at formlen `X = YZ` betyder, tag værdien `Y` og multiplicer (gang) den med sig selv antallet af gange angivet af `Z`. For eksempel, `23 = 8` (`2*2*2`). Værdien `Z` er eksponentiel-værdien i ligningen. Så længe man kender, hvad `Y` og `Z` værdierne er i ligningen, er det nemt at beregne værdien af `X`.

Unfortunately, you may not always know the value of `Y` and `Z`. How do you determine `Z` if you know the value of `X` and `Y`? This is when you use a logarithm. A logarithm is the exponent value that indicates the number of times the value `Y` needs to be multiplied by itself to get the value `X`. The value that is multiplied (`Y`) is considered to be the base of the formula. / Uheldigvis kender man ikke altid værdien af `Y` og `Z`. Hvordan bestemmer man `Z`, hvis man kender værdien af `X` og `Y`? Det er da, man bruger en logaritme. En logaritme er eksponent-værdien, som angiver antallet af gange værdien `Y` behøver at blive multipliceret (ganget) med sig selv for at få værdien `X`. Værdien som er multipliceret (ganget) (`Y`) betragtes som grundtallet i formlen.

There are two basic types of logarithms: common and natural. A common logarithm uses a value 10 as the base value. Therefore, in the basic formula for exponentiation above, `X = YZ`, the value of `Y` is 10, and `Z` is the number of times that `Y` needs to be multiplied by itself to return the value indicated by `X`. / Der er to grundlæggende typer af logaritmer: sædvanlig og naturlig. En sædvanlig logaritme bruger en værdi 10 som grundtal. Derfor i den grundlæggende eksponent-formel ovenfor, `X = YZ`, er værdien af `Y` 10, og `Z` er antallet af gange som `Y` behøver at blive multipliceret (ganget) med sig selv for at returnere værdien angivet af `X`.

Natural logarithms use a base value of approximately 2.71828182845905, normally referred to as `e`. The mathematical notation `e` is Euler's constant, the base of natural algorithms, made common by the mathematician Leonhard Euler ( Basel, Switzerland April 15, 1707 - Russia September 18, 1783). VBScript provides two functions for working with logarithms: `Exp()` and `Log()`. Each of these functions assumes that the base value is `e`. The `Log()` function returns the natural logarithm of the supplied numeric expression, and the `Exp()` function raises the supplied numeric expression to `e`. The similar methods in JavaScript is called: `Math.exp()` and `Math.log()`. / Naturlige logaritmer bruger et grundtal på tilnærmelsesvis 2,71828182845905, i reglen henvist til som `e`. Den matematiske notation `e` er Eulers konstant, de naturlige algoritmers grundtal, gjort alminding af matematikeren Leonhard Euler ( Basel, Schweiz 15. april 1707 - Rusland 18. september 1783). VBScript har to funktioner til udregninger med logaritmer: `Exp()` og `Log()`. Hver af disse funktioner antager, at grundtallet er `e`. `Log()` funktionen returnerer den naturlige logaritme til det leverede numeriske udtryk, og `Exp()` funktionen opløfter det leverede numeriske udtryk til `e`. De lignende metoder i JavaScript kaldes: `Math.exp()` og `Math.log()`.

It is possible to use these VBScript functions or JavaScript methods if you have a different base value by using a simple formula. By dividing the natural log of the desired number (`X`) by the natural log of the desired base (`Y`), you can determine the desired logarithm value (`Z`) in VBScript: `Z = Log(X) / Log(Y)` or similar in JavaScript: `Z = ((Math.log(X)) / (Math.log(Y)));`. / Det er muligt at bruge disse VBScript funktioner eller JavaScript metoder, hvis man har et andet grundtal ved at bruge en simpel formel. Ved at dividere den naturlige log til det ønskede tal (`X`) med den naturlige log til det ønskede grundtal (`Y`), kan man bestemme den ønskede logaritme-værdi (`Z`) i VBScript: `Z = Log(X) / Log(Y)` eller lignende i JavaScript: `Z = ((Math.log(X)) / (Math.log(Y)));`.

##### JavaScript comments: / JavaScript bemærkninger:

The custom function `Pow2(NumDbl)`, which returns the base to an exponent power of 2, and the custom function `Log2(NumDbl)`, which calculates base-2 logarithms, can be seen in this page's source code. They uses respectively the JavaScript `Math.pow()` method, which returns base to the exponent power, that is, base exponent, and a formula based on the JavaScript `Math.log()` method, which returns the natural logarithm (base E) of a number. / Funktionen lavet på bestilling `Pow2(NumDbl)`, som returnerer grundtallet til en eksponent potens af 2, og funktionen lavet på bestilling `Log2(NumDbl)`, som beregner grundtal-2 logaritmer, kan ses i denne sides kildekode. De bruger henholdsvis JavaScript `Math.pow()` metoden, som returnerer grundtallet til en eksponent potens, det vil sige grundtal eksponent, og en formel baseret på JavaScript `Math.log()` metoden, som returnerer den naturlige logaritme (grundtal E) af et tal.

##### See the JavaScript by View Source / Se JavaScript'et via Vis Kilde

You can see the JavaScript by using View Source. / Man kan se JavaScript'et ved at bruge Vis Kilde.

##### My Sources / Mine kilder

Sources: Various books, the Internet, and various encyclopedias.

Kilder: Forskellige bøger, internettet og forskellige leksikoner.

### Computer Data Representation and Number Systems / Computer data repræsentation og talsystemer      ©1997 - 1999 Erik Østergaard, Copenhagen, Denmark.