Waxaad telefoonka kula jirtaa saaxiib jooga USA maalin qabow oo qabow. "Halkan waa laga jaray \(40\) darajo!" labadiinuba isku mar waad qaylisaan. Caadi ahaan, tani waxay noqon doontaa arrin lagu caddeeyo cidda macnaheedu yahay Celsius iyo cidda macnaheedu yahay Fahrenheit - laakiin maaha heerkulkan gaarka ah. Waa maxay sababta? Halkaa marka ay marayso waa heerkulka kaliya ee ay ku heshiiyaan miisaanka Celsius iyo Fahrenheit!
\(−40\) darajo Fahrenheit waa sax \(−40\) darajo Celsius. Tani maaha wax iska dhacay, balse waa natiijo toos ah oo ka timid xidhiidhka tooska ah ee ka dhexeeya labada miisaan. Labada cabbir heerkulku waa isbeddellada affine (linear + shift) ee tiro jireed oo isku mid ah, "heerkulka." U beddelashada labadan miisaan inta badan waa caajis. Si kastaba ha ahaatee, waxaa jira meel xiiso leh oo labada miisaan ay leeyihiin qiime isku mid ah.
- Qiyaasta Celsius (°C):
\(0^\circ\mathrm{C}\) Barafka biyaha
\(100^\circ\mathrm{C}\) barta biyaha
Fogaanta u dhaxaysa qodobadan go'an: \(100\) darajo. - Cabirka Fahrenheit (°F):
\(32^\circ\mathrm{F}\) Barafka biyaha
\(212^\circ\mathrm{F}\) Meesha biyaha lagu kariyo
Fogaanta u dhaxaysa qodobadan go'an: \(212-32=180\) darajo.
Tani waxay go'aaminaysaa saamiga (jiirada) ee u dhexeeya miisaanka:
\[
\frac{180}{100}=\frac{9}{5}
\]
Dhibicda eber (offset) sidoo kale way ka duwan tahay: \(0^\circ\mathrm{C}\) waxay u dhigantaa \(32^\circ\mathrm{F}\) .
Si aan u soo saarno caanaha caadiga ah, waxaan raadineynaa khariidad affin ee foomka
\[
T_\mathrm{F}=a T_\mathrm{C}+b,
\]
halka \(a\) jiirada (cabirka miisaanka) iyo \(b\) waa kabitaanka.
Labadan shuruudood ee soo socdaa waa ku filan yihiin sababtoo ah khariidaynta affine ee laba dhibcood ayaa si gaar ah loo go'aamiyay:
- \(T_\mathrm{C}=0 \Rightarrow T_\mathrm{F}=32 \Rightarrow 32 = a\cdot 0 + b \Rightarrow b=32.\)
- \(T_\mathrm{C}=100 \Rightarrow T_\mathrm{F}=212 \Rightarrow 212 = a\cdot 100 + 32 \Rightarrow a=\frac{212-32}{100}=\frac{180}{100}=\frac{9}{5}.\)
Beddelka wuxuu keenayaa qaacidada caadiga ah:
\[
T_\mathrm{F}=\frac{9}{5} T_\mathrm{C}+32
\]
Ka soo horjeeda (laga bilaabo Fahrenheit ilaa Celsius) waxaa lagu helaa iyadoo la xalinayo \(T_\mathrm{C}\) :
\[
T_\mathrm{C}=\frac{5}{9}\left(T_\mathrm{F}-32\right)
\]
Hadda waxaan raadineynaa heerkulka \(T\) kaas oo qiimaha nambarada isku midka ah uu ka muuqdo labada miisaan:
\[
T_\mathrm{F}=T_\mathrm{C}\equiv T
\]
Hadda geli \(T_\mathrm{F}\) qaabka caadiga ah:
\[
T=\frac{9}{5}T+32 \Leftrightarrow T-\frac{9}{5}T=32
\]
iyo ugu dambeyntii
\[
\left(1-\frac{9}{5}\right)T=32 \quad\Rightarrow\quad \left(\frac{5}{5}-\frac{9}{5}\right)T=32 \quad\Rightarrow\quad -\frac{4}{5}T=32.
\]
Tani waxay keenaysaa in \(T\)
\[
T=-32\cdot\frac{5}{4}=-8\cdot5=-40
\]
iyo sidaas
\[
-40^\circ\mathrm{F} = -40^\circ\mathrm{C}.
\]
Si loo qiimeeyo qiimaha Celsius ee togan, \(T_\mathrm{F}=\tfrac{9}{5}T_\mathrm{C}+32\) had iyo jeer qiimo tiro ka weyn \(T_\mathrm{C}\) (tusaale ahaan. \(0^\circ\mathrm{C} \rightarrow 32^\circ\mathrm{F}\), \(20^\circ\mathrm{C}\rightarrow68^\circ\mathrm{F})\). Si loo helo qiimayaasha Celsius taban oo ku filan, \(32\) Darajooyinka bilowga cabirka Fahrenheit dhab ahaantii way ka hooseeyaan eber. Mar marka qaarkood, tani waxay magdhowaysaa qodobka miisaanka \(\frac{9}{5}\). Qodobkan dheelitirka waa dhab \(−40\): waxaa jira isbedel dheeraad ah \(+32\) kaliya weyn oo ku filan si labada qiyamka tirooyinka ay isku mid yihiin. Sawir ahaan, \(T_\mathrm{F}= \tfrac{9}{5}T_\mathrm{C}+32\) (xariiq toosan) iyo \(T_\mathrm{F}=T_\mathrm{C}\) (Diagonal) – barta is-goysyada xariiqahoodu waa xagga \((-40,-40)\).
Taas bedelkeeda, heerkulka saxda ah (tusaale, xisaabinta heerkulbeegyada) ayaa lagu bixiyaa Kelvin ama Rankine, halkaas oo aysan jirin wax ka-soo-baxa beddelka miisaanka (kaliya arrin miisaan saafi ah). Tusaale ahaan, inta u dhaxaysa Celsius iyo Kelvin \(T_\mathrm{K} = T_\mathrm{C} + 273{,}15\) ayaa quseysa. Jiritaanka kabitaankani waa sababta saxda ah ee khariidaynta Celsius-Fahrenheit ay tahay affine oo aan toos ahayn. Sinnaanta \(-40^\circ\mathrm{F}=-40^\circ\mathrm{C}\) waxay si toos ah uga socotaa xidhiidhka affine ee ka dhexeeya Fahrenheit iyo Celsius.
Haddii aad ku beddesho \(T_\mathrm{F}=T_\mathrm{C}\) \(T_\mathrm{F}=\tfrac{9}{5}T_\mathrm{C}+32\) oo aad xalliso, si cad ayaad u helaysaa \(T=-40\) . Tani waa meesha ay labada miisaan isdhaafsanayaan. Barta isgoysyada ee \(-40\) waa barta kaliya ee qiyamka tirooyinka ee labada miisaan ay isku mid yihiin. Tani waxay sabab u tahay dabeecadda toosan ee beddelka: laba xariiq oo aan isbarbar dhigin ayaa had iyo jeer isku xira hal dhibic. Marka marka xigta ee uu qof sheego \(-40\) darajo, uma baahnid inaad si cad u weydiiso miisaanka ay ula jeedaan.