l>Number Systems

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Introduction

A number system specifies a collection of values provided to representquantity. We talk around the variety of people attending class, thenumber that modules taken per student, and likewise use numbers torepresent grades accomplished by college student in tests.

Quantifying values and also items in relation to each various other ishelpful for united state to make sense of ours environment. We do this at anearly age; figuring out if us have an ext toys come play with, morepresents, much more lollies and so on.

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

Mankind v the periods has used indications or icons torepresent numbers. The beforehand forms were directly lines or groupsof lines, lot like as illustrated in the film Robinson Crusoe,where a group of six vertical lines with a diagonal heat acrossrepresented one week.

Its an overwhelming representing big or very small numbers usingsuch a graphical approach. As at an early stage as 3400BC in Egypt and 3000BCin Mesopotamia, they developed a symbol to stand for the unit 10.This was a major advance, since it diminished the number ofsymbols required. For instance, 12 might be stood for as a 10and two units (three symbols rather of 12 that was requiredpreviously).

The Romans devised a number mechanism which can represent allthe numbers from 1 come 1,000,000 using only seven symbols

ns = 1 V = 5 X = 10 together = 50 C = 100 D = 500 M = 1000

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

The number device in most common use this particular day is the Arabicsystem. The was an initial developed by the Hindus and also was provided asearly together the third century BC. The development of the price 0,used to indicate the positional worth of digits was veryimportant. Us thus came to be familiar with the concept of groups ofunits, 10s of units, hundreds of units, countless units andso on.

In number systems, its often beneficial to think that recurringsets, whereby a collection of values is recurring over and also over again.

Considering the decimal number system, it has actually a collection of valueswhich selection from 0 to 9. This basic collection is repeated over andover, creating big numbers.

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

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Note exactly how the set of worths 0 to 9 is repeated, and also for eachrepeat, the shaft to the left is incremented (from 0 to 1, then2).

Each boost in worth occurs, till the value of the largestnumber in the set is got to (9), in ~ which stage the next valueis the the smallest in the set (0) and also a brand-new value is generated inthe left pillar (ie, the following value after ~ 9 is 10).

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

BaseValuesThe base value of a number device is the number of differentvalues the collection has before repeating itself. For example, decimalhas a basic 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 through the Mayans) Sexagesimal = 60 (used through the Babylonians)

WeightingFactorThe weighting element is the multiplier value used to eachcolumn position of the number. For instance, decimal has aweighting aspect of TEN, in the each shaft to the leftindicates a multiplication value increase of 10 end the previouscolumn on the right, ie; every column move to the left increasesin a multiply variable of 10.

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

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312 = ----- 2 * 100 = 2 * 1 = 2 ------ 1 * 101 = 1 * 10 = 10 ------- 3 * 102 = 3 * 100 = 300 ----- 312 (adding this up) -----DecimalNumber system This number system uses TENdifferent signs to represent values. The collection values offered indecimal room

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

When doing a calculation, if the greatest digit (9) isexceeded, a carry occurs i beg your pardon is moved to the next column(to the left).

instance of addition and exceeding the base collection range8 + 4 8 9 +110 +2 note 1:11 +312 +4Note1: once 9 is exceeded, us go back to the start of the set (0), and also carry a worth of 1 end to the following column top top the left. another example of enhancement and exceeding the base set range198 + 4198199 +1200 +2 note 2:201 +3202 +4Note2: when 9 is exceeded, we go back to the start of the set (0), and carry a value of 1 over to the next column top top the left. Hence themiddle pillar (9) has 1 included to it, the next value in the collection is 0, andwe carry 1 (because the set was exceeded) come the following left column. Addingthe carry value that 1 to 1 in the left most column gives.Positional values We probably got taught in ~ school about positional values, in thatcolumns stand for powers that 10. This is express to us ascolumns of people (0 - 9), tens (groups the 10), hundreds (groups of100) and so on.

237 = (2 teams of 100) + (3 teams of 10) + (7 teams of 1) = (100 + 100) + (10 + 10 + 10) + (1 + 1 + 1 + 1 + 1 + 1 + 1) = (200) + (30) + (7) = 237Each column relocate to the left is 10 times the vault value.

BinaryNumber mechanism The binary number mechanism uses TWOvalues to stand for numbers. The worths are,

0 1

with 0 having the least value, and 1 having actually the greatestvalue. Columns are used in the same means as in thedecimal system, in the 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 also 1) repeat, in both the upright andhorizontal directions.

0 1 10 Note: goto value lowest in set, lug to left 11100 Note: goto value lowest in set, lug to left101110 Note: goto worth lowest in set, bring to left111In a computer, a binary variable capable of save a binaryvalue (0 or 1) is dubbed a BIT.

In the decimal system, columns represented multiplicationvalues that 10. That was because there to be 10 values (0 - 9) inthe set. In this binary system, there are only two values (0 - 1)in the set, for this reason columns stand for multiplication worths of 2.

1011 = ---- 1 * 20 = 1 ----- 1 * 21 = 2 ------ 0 * 22 = 0 ------- 1 * 23 = 8 ---- 11 (in decimal)Number varieties in Binary using A Specified number of BitsHow countless different values can be stood for by a details numberof bits?

number of different values = 2nwhere n is the variety of bitseg.28= 256 various valuesRules because that Binary Addition