Lambarada koowaad waa kuwa kaliya waxay leeyihiin 2 qaybiye, maaddaama ay kaligood kala qaybsamayaan iyo unugga, taas oo ah, lambarka 1. Laakiin ka digtoonow! Maxay tani ka dhigan tahay? Aad u fudud. Tirada koowaad, tusaale ahaan 2, waxaa loo qaybin karaa oo kaliya 2, -2, 1, iyo -1.
Lambarada leh in ka badan 2 qaybiye ayaa la wacayaa tirooyin ka kooban. Haddii aan soo qaadanno tiro isku dhafan, tusaale ahaan, 10, waxaan arki doonnaa inaan u kala qaybin karno naftiisa iyo midnimada, taas oo ah, inta u dhexaysa 10 iyo 1, laakiin sidoo kale u dhexeeya 2 iyo 5. Sidaa darteed, 10 waa tiro isku dhafan.
Dhammaan tirooyinku ma yihiin kuwo asaasi ah ama isku jira?
Waxaa jira labo tirooyin "gaar ah" kuwaas oo aan ahayn kuwa ugu fiican ama isku dhafan: 0 iyo 1. Waayo? Aan aragno:
- Tirada 1 ayaa kaligeed loo qaybin karaa (1/1 = 1) iyo midnimo, taas oo ah, lambarka 1 (1/1 = 1). Si kastaba ha ahaatee, si lambar loo tixgeliyo ra'iisul -wasaare, waa inay lahaataa 2 qaybiye oo kala duwan. Lambarka 1 wuxuu leeyahay kaliya hal qaybiye, sidaa darteed ma aha ra'iisul iyo mid isku jira toona.
- 0 -ka kaligeed lama qaybin karo, maadaama natiijadu tahay mid aan la hubin.
Haddaba haddii aan liiska ka saarno 0 iyo 1, oo ka mid ah tirooyinka tirada badan ee hadhay, sideen ku garan karnaa kuwa muhiimka ah iyo kuwa aan ahayn?
Sida loo ogaado haddii lambar yahay mid muhiim ah
Waxa ugu badan ee caadiga ah waa inaad ka fekerto inaad ku samayso iska tuur, yacnii, inaad tagto imtixaanka ilaa aad ka hesho qaybiyeyaasha. Xisaabiyaha waa mid aad u dhakhso badan, laakiin haddii ay qasab tahay in aan xagga sare u rogno ama qalin iyo warqad, arrimuhu way yara adkaadaan. Waxaan ku baraynaa laba hab oo aad ku ogaato haddii lambar yahay mid muhiim ah iyo in kale.
Shaandhada Eratosthenes
Shaandhada Eratosthenes waa a farsamo lagu ogaanayo tirooyinka u dhexeeya 2, oo ah lambarka koowaad ee ugu horreeya, iyo tiro cayiman.
Habkani wuxuu ka kooban yahay samaynta miis iyo ka gudubka tirooyinka tirooyinka oo dhan. Marka hore waxaan tirtiri doonaa tirooyinka 2, ka dibna 3, iyo wixii la mid ah ilaa aan ka gaarnayno tirada labajibbaaran waxay ka weyn tahay tirada ugu dambeysa ee miiska.
Sida wax kasta oo xisaabta ku jira, shaandhada Eratosthenes ayaa si fiican loogu fahmaa tusaale ahaan:
- Waxaan samaynaa miis leh tirooyinka 2 ilaa 30.
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
- Waxaan ka gudubnaa tirooyinka 2 ee liiska, taas oo ah, waxaan ka gudubnaa 2 illaa 2: 4, 6, iwm. Iska ilaali! 2 -ka, oo kaliya lagu kala qaybin karo naftiisa iyo lambarka 1, kama tallaabsanayno, maadaama ay tahay lambarka koowaad.
| 2 | 3 | 5 | 7 | 9 | |||||
| 11 | 13 | 15 | 17 | 19 | |||||
| 21 | 23 | 25 | 27 | 29 |
- Waxaan soo qaadannaa lambarka xiga, 3, oo waxaan hubinnaa in labajibbaaranahu ka yar yahay tirada ugu badan ee miiska. Sida 32 <30, waxaan ku sii wadnaa shaandhada oo aan ka tallaabsannaa tirooyinkeeda: 6, 9, 12 ... Sidii tallaabadii hore, ma dhaafin lambarka 3, oo isna ah kan ugu horreeya.
| 2 | 3 | 5 | 7 | ||||||
| 11 | 13 | 17 | 19 | ||||||
| 23 | 25 | 29 |
- Waxaan ku celinnaa tallaabadii hore oo leh lambarka xiga jadwalka: 4 waa la dhaafay, marka waxaan qaadannaa 5. Sida 52 <30, waxaan ka gudubnaa tirooyinkooda.
| 2 | 3 | 5 | 7 | ||||||
| 11 | 13 | 17 | 19 | ||||||
| 23 | 29 |
- Waxaan sii wadaynaa lambarka soo socda annagoo aan ka tallaabin: 7. Sida 72 = 49, taas oo ah, labajibbaaranaha 7 wuxuu ka weyn yahay lambarka ugu dambeeya ee jadwalka, habku wuu dhammaanayaa, tirooyinka aan la isku haleynina waa tirooyinka koowaad.
- Gunaanad. Lambarada ugu muhiimsan ee u dhexeeya 2 iyo 30 waa: 2, 3, 5, 7, 11, 13, 17, 19, 23 iyo 29.
Shaandhada Eratosthenes waa hab deg deg ah oo sahlan si loo ogaado tirooyinka ugu waaweyn, laakiin ka warankawaran haddii tirada aan rabno inaan baranno ay aad u badan tahay, tusaale ahaan, 54657?
Sidaad fahantay, ma ahaan doonto wax la taaban karo in miiska laga sameeyo 2 ilaa 54657, sax? Maxaan samayn karnaa markaa? Aad u fudud: adeegso shuruudo lagu qaybsado.
Shuruudaha qaybsiga
Shuruudaha qaybsanaantu waa xeerarka si loo ogaado haddii hal lambar loo qaybin karo mid kale iyada oo aan la samayn qaybin.
Sidaas darteed, haddii aan isticmaalno xeerarkan oo aan u kuurgalno in lambar loo qaybin karo lambar kale oo aan ahayn isaga iyo cutubka, waxaan ogaan doonnaa inaysan ahayn mid muhiim ah.
- Halbeegga qaybinta lambarka 2. Lambar waxaa loo qaybin karaa 2 haddii ay tahay xitaa, taas oo ah, haddii ay ku dhammaato 0, 2, 4, 6 ama 8. Oo, halkan waa khiyaano: sida lambar kasta oo loo qaybin karo 4, 6 ama 8 ayaa sidoo kale loo qeybin karaa 2, uma baahnaan doonno inaan ogaano shuruudaha qeybsiga ee lambarada kale xitaa.
- Halbeegga kala qaybinta lambarka 3. Tirada waxaa loo qaybin karaa 3 haddii wadarta tirooyinkeeda ay tahay saddex tiro. Aan aragno tusaale:
267 -> 2 + 6 + 7 = 15
Maadaama 15 ay tahay 3, 267 waxaa loo qeybin karaa 3.
Intaa waxaa dheer, mar haddii tiro kasta oo loo qaybin karo 9 ay sidoo kale u qaybsami karto 3, waxay nagu filnaan doontaa inaan ogaano halbeeggan.
- Halbeegga kala -qaybsanaanta lambarka 5. Lambar waxaa loo qaybin karaa 5 haddii uu ku dhammaado 0 ama 5.
- Halbeegga qaybinta lambarka 7. Si aad u ogaato in lambar loo qaybin karo 7, waa in aan ka jarnaa lambarka oo aan lahayn lambarkii ugu dambeeyay iyo laba jeer lambarkii ugu dambeeyay. Haddii lambarka la helay uu yahay 0 ama ka badan yahay 7, lambarka hore waa loo qaybin karaa 7. Waxaad si fiican u fahmi doontaa tan tusaale ahaan, aan u gaarno!
378 -> 37 − (8 × 2) = 37 − 16 = 21
Maadaama 21 ay tahay 7, 378 waxaa loo qeybin karaa 7.
- Heerka qaybsiga lambarka 11. Haddii aan ka jarayno tirooyinka isle'eg iyo wadarta tirooyinka aan caadiga ahayn, oo tirada la helay ay tahay 0 ama tiro 11 ah, taasi waxay ka dhigan tahay in lambarka la bartay loo qaybin karo 11. Waa kan tusaale:
8591 -> (8 + 9) − (5 + 1) = 17 − 6 = 11
Maadaama 11 ay tahay 11, 8591 waxaa loo qeybin karaa 11.
Waana intaas! Hadda waa markiina: ma ogaan lahayd sida loo xisaabiyo haddii tiradaas sare, 54657, ay tahay ra'iisul -wasaare?
Liiska tirooyinka ugu muhiimsan 1 ilaa 10.000
Ugu dambayntii, haddii aad raadinayso liis lambarrada u dhexeeya 1 ilaa 10.000, sida 1 illaa 100 ama 1 illaa 1.000, halkan waa mid dhammaystiran oo la cusboonaysiiyay:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919
