l>Number Systems

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Introduction

A number system defines a set of values used to representquantity. We talk about the number of people attending class, thenumber of modules taken per student, and also use numbers torepresent grades achieved by students in tests.

Quantifying values and items in relation to each other ishelpful for us to make sense of our environment. We do this at anearly age; figuring out if we have more toys to play with, morepresents, more lollies and so on.

The study of number systems is not just limited to computers.We apply numbers every day, and knowing how numbers work willgive us an insight into how a computer manipulates and storesnumbers.

Mankind through the ages has used signs or symbols torepresent numbers. The early forms were straight lines or groupsof lines, much like as depicted in the film Robinson Crusoe,where a group of six vertical lines with a diagonal line acrossrepresented one week.

Its difficult representing large or very small numbers usingsuch a graphical approach. As early as 3400BC in Egypt and 3000BCin Mesopotamia, they developed a symbol to represent the unit 10.This was a major advance, because it reduced the number ofsymbols required. For instance, 12 could be represented as a 10and two units (three symbols instead of 12 that was requiredpreviously).

The Romans devised a number system which could represent allthe numbers from 1 to 1,000,000 using only seven symbols

I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000

A small bar placed above a symbol indicates the number ismultiplied by 1000.

The number system in most common use today is the Arabicsystem. It was first developed by the Hindus and was used asearly as the 3rd century BC. The introduction of the symbol 0,used to indicate the positional value of digits was veryimportant. We thus became familiar with the concept of groups ofunits, tens of units, hundreds of units, thousands of units andso on.

In number systems, its often helpful to think of recurringsets, where a set of values is repeated over and over again.

Considering the decimal number system, it has a set of valueswhich range from 0 to 9. This basic set is repeated over andover, creating large numbers.

You are watching: What is the lowest base in which the number 10 could be a valid number?

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Note how the set of values 0 to 9 is repeated, and for eachrepeat, the column to the left is incremented (from 0 to 1, then2).

Each increase in value occurs, till the value of the largestnumber in the set is reached (9), at which stage the next valueis the smallest in the set (0) and a new value is generated inthe left column (ie, the next value after 9 is 10).

09, 10 - 19, 20 - 29, 30 -39 etcWe always write the digit with the largest value on theleft of the number

BaseValuesThe base value of a number system is the number of differentvalues the set has before repeating itself. For example, decimalhas a base of ten values, 0 to 9.

Binary = 2 (0, 1) Octal = 8 (0 - 7) Decimal = 10 (0 - 9) Duodecimal = 12 (used for some purposes by the Romans) Hexadecimal = 16 (0 - 9, A-F) Vigesimal = 20 (used by the Mayans) Sexagesimal = 60 (used by the Babylonians)

WeightingFactorThe weighting factor is the multiplier value applied to eachcolumn position of the number. For instance, decimal has aweighting factor of TEN, in that each column to the leftindicates a multiplication value increase of 10 over the previouscolumn on the right, ie; each column move to the left increasesin a multiply factor of 10.

200 = ----- 0 * 100 = 0 * 1 = 0 ------ 0 * 101 = 0 * 10 = 0 ------- 2 * 102 = 2 * 100 = 200 ----- 200 (adding these up) -----Lets consider another example of 312 decimal.

See more: Whats The Cube Root Of 8 ? How To Find The Cube Root Of 8

312 = ----- 2 * 100 = 2 * 1 = 2 ------ 1 * 101 = 1 * 10 = 10 ------- 3 * 102 = 3 * 100 = 300 ----- 312 (adding these up) -----DecimalNumber System This number system uses TENdifferent symbols to represent values. The set values used indecimal are

0 1 2 3 4 5 6 7 8 9with 0 having the least value and nine having the greatestvalue. The digit or column on the left has the greatest value,whilst the digit on the right has the least value.

When doing a calculation, if the highest digit (9) isexceeded, a carry occurs which is transferred to the next column(to the left).

Example of addition and exceeding the base set range8 + 4 8 9 +110 +2 Note 1:11 +312 +4Note1: When 9 is exceeded, we go back to the beginning of the set (0), and carry a value of 1 over to the next column on the left. Another example of addition and exceeding the base set range198 + 4198199 +1200 +2 Note 2:201 +3202 +4Note2: When 9 is exceeded, we go back to the beginning of the set (0), and carry a value of 1 over to the next column on the left. Thus themiddle column (9) has 1 added to it, the next value in the set is 0, andwe carry 1 (because the set was exceeded) to the next left column. Addingthe carry value of 1 to 1 in the left most column gives.Positional Values We probably got taught at school about positional values, in thatcolumns represent powers of 10. This is expressed to us ascolumns of ones (0 - 9), tens (groups of 10), hundreds (groups of100) and so on.

237 = (2 groups of 100) + (3 groups of 10) + (7 groups of 1) = (100 + 100) + (10 + 10 + 10) + (1 + 1 + 1 + 1 + 1 + 1 + 1) = (200) + (30) + (7) = 237Each column move to the left is 10 times the previous value.

BinaryNumber System The binary number system uses TWOvalues to represent numbers. The values are,

0 1

with 0 having the least value, and 1 having the greatestvalue. Columns are used in the same way as in thedecimal system, in that the left mostcolumn is used to represent the greatest value.

As we have seen in the decimal system,the values in the set (0 and 1) repeat, in both the vertical andhorizontal directions.

0 1 10 Note: goto value lowest in set, carry to left 11100 Note: goto value lowest in set, carry to left101110 Note: goto value lowest in set, carry to left111In a computer, a binary variable capable of storing a binaryvalue (0 or 1) is called a BIT.

In the decimal system, columns represented multiplicationvalues of 10. That was because there were 10 values (0 - 9) inthe set. In this binary system, there are only two values (0 - 1)in the set, so columns represent multiplication values of 2.

1011 = ---- 1 * 20 = 1 ----- 1 * 21 = 2 ------ 0 * 22 = 0 ------- 1 * 23 = 8 ---- 11 (in decimal)Number Ranges in Binary Using A Specified Number Of BitsHow many different values can be represented by a specific numberof bits?

number of different values = 2nwhere n is the number of bitseg.28= 256 different valuesRules For Binary Addition