Numerical methods (quads number system help please))

[buchi]

Consider a new computer system which stores data in “quads”, i.e. it has four states 0, 1,
2 and 3 (as opposed to binary which has only two, 0 and 1). Now imagine that numbers
are stored in a 12 quad format where there is one quad for the sign of the number, three
quads for the exponent, another one quad for the sign of the exponent and seven quads
for the normalised mantissa (ie. it is of the form 0.nqqqqqq, where n≥1). Find:

(a) The largest positive (decimal) number that can be represented in this system.
(b) The smallest positive (decimal) number that can be represented in this system.
(c) The machine epsilon, epsilonM .

i honestly have no idea how to start this question i was hoping if someone can lead me the right way on how to start it or send me some site about this number systems and how to figure this things out.

thank you in advance

 

[SammyS]

Do you know how to solve a similar problem in binary (base 2)?

Do you know how to solve a similar problem in decimal (base ten)?

 

[buchi]

no i don't really know how to do that either i am just reading about it now but it seems like i am stuck on the concept may be if you can tell me how to do it in another base i might be able to get some direction

 

[SammyS]

A similar problem in decimal would be for a ten state computer: states 0,1, ... ,8,9 .

The largest positive number in this case is: +.9999999×10⁺⁹⁹⁹.

Note: The base for the exponent is usually the base of the number system used. For binary computers, the base of the exponent is 2.​

 

[buchi]

+.9999999×10+999

doesn't this have 16 digits though? i am assuming for mine it would be +.33333*4^+333 does that look right the whole thing has 12 entry's and 3 being the largest number in that base system?
and would the smallest one be +.00001*4^+001 ?

 

[SammyS]

In both the largest and smallest numbers, you are missing the digit represented by the 'n' in the mantissa.

There are 12 'digits' including the signs, but excluding the decimal (or quad) point, the multiplication symbol the base and the exponentiation symbol, ^.

 

[buchi]

thank you so much i think i got it now.

 

[buchi]

could you comment on this for me for the machine epsilon

in the above scenario 1=.1000000*4^1 therefore the closest number we can write to one is .1000001*4^1 which equals 1.000001 therefore the machine epsilon is 4^-6?

is that right? i sense that the three spaces for the exponent value make this argument wrong but i can't quite see it.

Thanks in advance