hp calculators HP 30S Base Conversions Numbers in Different Bases Practice Working with Numbers in Different Bases

Transcription

1 Numbers i Differet Bases Practice Workig with Numbers i Differet Bases

2 Numbers i differet bases Our umber system (called Hidu-Arabic) is a decimal system (it s also sometimes referred to as deary system) because it couts i 0s ad powers of 0. Its base (i.e. the umber o which the umber system is built), is therefore 0. While base 0 umbers are extesively used, this is ot the oly possible base. There have bee umber systems with base 0 (used by the Mayas), mixed bases of 0 ad 60 (used by the Babyloias), of 5 ad 0 (aciet Romas), etc. Eve owadays the sexagesimal system (base 60) is used i some measuremets of time ad agle. Bases, 8, 0 ad 6 are particularly importat i computig. Base umbers are called biary umbers ad their digits are limited to ad 0. A commo abbreviatio of biary digit is bit, which is either or 0. Base 8 are called octal umbers, whose digits are 0,,, 3, 4, 5, 6 ad 7. Fially, base 6 umbers are called hexadecimal umbers: 0,,,,3,4,5,6,7,8,9, A(=0d), B(=d), C(=d), D(=3d), E(=4d) ad F(=5d). The mai differece betwee all these bases is the value a digit has because of its place i a umeral. For example, i the decimal umber 378 the digit 3 has value 300, 7 has value 70 ad 8 has value 8. I other words: 378 = But if 378 were a hexadecimal umber the its decimal value would be: 378h = = 888d A small h, b, d or o after or before a umber meas that this umber is expressed i hexadecimal, biary, decimal or octal base respectively. I the HP 30S, there is o specific operatig mode to operate with biary, hexadecimal or octal umbers. But, i the followig examples we ll lear how to perform coversios easily o your HP 30S. Practice workig with umbers i differet bases Example : Covert the hexadecimal umber F904 to decimal base. Solutio: Here s oe way of covertig a iteger d d...d i base b to a decimal iteger: (( d d b + d )b d )b + d I this example, the calculatio is: ((( ) 6 + 0) ) + 4, which ca be doe by pressig: rr5*6+9s6q+s6+4y It is possible to automate this method by storig the above expressio i the EQN variable: B B + A + B(C + B(D + B(X + B(X + B(X + B(Y + B(Y + BY ))))))) EQN The whe recallig ad executig EQN, the calculator will prompt us for the value of B, the base, first (because it is the first variable used i the expressio) ad the for the value of ie variables more, amely A, C, D,,, X, Y, ad which represet the digits d through d 9, respectively. Eter X X Y Y 0 0 hp calculators Versio.0

3 Aswer: 0994 the digits backwards, i.e. from right to left. If the umber has less tha ie digits, simply key i 0 to the remaiig variables. To store this expressio i EQN press: Of@-f@+f+f@rf@@+f@rf@@@+ f@rf@@@@+f@rf@@@@@+f@rf<<<<+ f@rf<<<+f@rf<<+f@*f<yk<y Note that you eed ot eter the trailig paretheses because the STO commad operates o the whole etry lie. (For additioal iformatio o the EQN register, please refer to the HP 30S learig module Workig with Expressios). Let s do the above coversio agai. Press Fto clear all variables (except EQN) so we wo t have to clear the etry lie or eter the zero values: f<yy6y4yyy9y5yyyyy Example : Express the umbers 00b ad 70504o i base 0. Solutio: These coversios ca be performed by the expressio we stored ito EQN i the previous example, because they are coversios to decimal base of positive itegers with up to ie digits. Let s covert the former by pressig: Ff<yyyyyyyyyyy y As to the octal umber, press: Ff<yy8y4yyyy5yy7yyy Aswers: 477 ad 85654, respectively. 34 Example 3: Covert ad ACDC6 6 to decimal. p Solutio: Itegers with expoet, d.d...d d b, ca also be coverted usig the above expressio. Simply p + multiply the result of covertig dd...d d to decimal by b. Therefore, i order to covert the octal umber, let s covert the matissa first by pressig: Ff<yy8yyyyy3yyyyy 5+ which returs 90, which must ow be multiplied by 8 : *8 Ur-5+y To covert a egative umber, chage its sig, covert, ad chage the sig of the result. hp calculators Versio.0

4 34 37 As to the hex umber, ote that: ACDC6 6 = A.CDC6 6. The method is ow exactly the same as before. The sequece: Ff<yy6yy3yy0yyy yyy coverts ACDCh to 445d, which must be multiplied by 6 so as to complete the coversio: *6 Ur37-4+y 0 45 Aswers: ad , respectively. Example 4: Covert 0.00 to decimal. Solutio: Note that 0.00 =. 00. So first of all, let s covert 00 to decimal: Aswer: Ff<yyyyyyyyyyyy 8+ 8 ad ow let s multiply the result (0) by, that is to say, divide by : / U8y Example 5: Covert 046 to hexadecimal. Solutio: This is the geeral procedure to covert a decimal iteger x ito base b: l x Fid the iteger f lb x Fid E =. Let d be the iteger part of f E b Cotiue for E = (E d )b. The result is d...d d, where d is the iteger part of E. The d i+ i i d i last digit,, is foud whe i = f +. I this case, as h046@h6y returs , f =, which meas 046 that the hex umber has three digits. The first digit of the result will be the iteger part of : 6 046/6q which results i Therefore, the first digit is 4. The secod oe is calculated as follows: i hp calculators Versio.0

5 -4y*6y which returs.375. The, the secod digit is. Fially, the last digit is the iteger part of: -y*6y which gives 6. Aswer: 46h. Example 6: Covert to octal. Solutio: Let s calculate l x lb by pressig h6.0u3@/h8y which returs Therefore f = 6, which is also the expoet i base 8. There is o eed to fid all the 7 digits of the aswer because the decimal umber is give to four sigificat digits. The first digit will be the iteger part of: 6.0u3/8 U6 which results i Therefore the first digit is. Let s compute four digits more: -y*8y which returs , therefore d = 7, -7y*8y which returs , therefore d 3 = 7, -7y*8y which returs , therefore d 4 = 4, -4y*8y which returs , therefore d 5 =. Aswer: hp calculators Versio.0