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1.BOSHLANG`ICH TUSHUNCHALAR

2016.08.25

BOSHLANG`ICH TUSHUNCHALAR I. NAZARIY QISM
Raqamlar

Matematika so`zi yunoncha so`zdan olingan bo`lib, “fan”, “bilim” degan ma`nolarni anglatadi. (Matematika fani real borliqning miqdoriy o`lchovlardagi obrazlarini va ular orasidagi munosabatlarni o`rganadi). Matematika fanining alifbosi raqamlardir. Sonlarni yozish uchun ishlatiladigan belgilarga raqamlar deyiladi. Ular: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Masalan, 105 sonini yozish uchun 1, 0, 5 raqamlaridan, 7 sonini yozish uchun esa 7 raqamidan foydalanamiz. Bu raqamlarni arab raqamlari, ayrim manbalarda hind
_raqa_m_l_ari_ham_d_e_y_i_l_ad_i_.

Natural sonlar

Tabiiy ehtiyoj tufayli, ya`ni narsa-buyumlarni sanash uchun kashf qilingan sonlarga
natural sonlar deyiladi. Ular: 1,2,3,4,5,6,7,… Barcha natural sonlar to`plami
N={1,2,…,n,…} ko`rinishda tasvirlanadi. 3 𝜖 N yozuvi 3 soni natural sonlar to`plamiga tegishli ekanligini, 0 ∉ N yozuvi esa 0 soni natural sonlar to`plamiga tegishli emasligini
^ang^l^a^t^a^dⁱ^.¹^,²^,^…^,ⁿ^,^{… k}^e^tm^a^-^k^e^t^li^k^k^a^na^t^ural^s^on^l^ar^qa^t^ori^de^yⁱ^l^adⁱ^.^N^a^t^u^r^a^l^s^oⁿ^l^ar
qatorida katta son kichigidan o`ngda joylashadi. Eng kichik natural son 1, eng kattasi esa
_m_a_v_j_ud_e_m_a_s_.

Toq va juft sonlar

2n–1, (n 𝜖 N) yoki 2n+1, (n=0, n 𝜖 N) ko`rinishda ifodalash mumkin bo`lgan natural sonlarga toq sonlar deyiladi. Masalan, 23=2∙11+1, demak, 23–toq son. 2n, (n 𝜖 N)
^ko^`rⁱⁿⁱ^s^hdaⁱ^f^o^d^a^l^a^s^h^m^u^m^kⁱⁿ^bo^`^l^g^an^na^t^u^r^a^l^s^oⁿ^l^a^r^ga^j^u^f^t^s^on^l^ar^de^y^il^adⁱ^.^M^a^s^a^l^an,
18=2∙9, demak, 18–juft son. Eslatib o`tamiz, o soni ham juft sondir.
^T^ub^s^on^l^ar

Natural bo`luvchilari faqat bir va o`zidan iborat bo`lgan birdan katta natural sonlarga tub sonlar deyiladi. Ular: 2,3,5,7,11,13,17,… Demak, tub sonlar 2 tagina natural bo`luvchilarga ega: 1 va shu sonning o`zi. Masalan, 19 ning natural bo`luvchilari: 1 va 19.

Murakkab sonlar

Tub bo`lmagan 1 dan katta natural sonlarga murakkab sonlar deyiladi. Ular:
4,6,8,9,10,12,14,15,… Demak, murakkab sonlar 2 tadan ortiq natural bo`luvchilarga
ega. Masalan, 9 ning natural bo`luvchilari: 1, 3, 9. 3 ta natural bo`luvchi.

_E_s_l_a_t_i_b_o_`_t_a_mi_z,₁_s_oni_t_ub_ha_m_,_m_ura_kk_ab_ham_e_m_a_s_.
Mukammal sonlar
O`zidan boshqa barcha natural bo`luvchilari yig`indisiga teng bo`lgan natural sonlarga mukammal sonlar deyiladi. Masalan, 6=1+2+3.
^M^uka^mm^al^s^oⁿ^f^o^r^m^u^l^a^sⁱ^:²p–1^∙⁽²p^–1⁾^,^bu^ye^r^da^p^–^t^ub^s^o^n.
^F^o^r^m^u^l^adan^f^o^y^da^l^an^is^h:¹^-^m^uk^a^mm^a^l^s^oⁿⁿⁱ^t^o^p^a^y^lⁱ^k,^bunⁱ^ng^u^chun^pⁿⁱ^ng^o^`rⁿⁱ^g^a¹^-^tub^s^oⁿⁿⁱ^,^y^a^`ⁿ^{i 2}ⁿⁱ^qo^`^y^a^mⁱ^z^p⁼²⁼^>²2–1^∙⁽²2^–¹⁾⁼²³⁼^6.²^-^m^uka^mm^a^l^m^uk^a^mm^a^l^s^oⁿ^:^p⁼³⁼^{> 2}3–1^∙⁽²3^–1⁾⁼⁴^∙⁷⁼^28.
Qo`shish va ko`paytirish qonunlari

Qo`shishning o`rin almashtirish qonuni: a+b=b+a. Masalan, 3+5=5+3=8;

2) Ko`payrishning o`rin almashtirish qonuni: a∙b=b∙a. Masalan, 3∙5=5∙3=15;

³⁾^Q^o^`^s^hⁱ^s^hnⁱⁿ^g^g^u^r^u^h^l^a^s^h^q^o^nuⁿⁱ^:⁽^a+b)+c=a+⁽^b+c).^M^a^s^a^l^an,⁽²⁺³⁾⁺⁴⁼²⁺⁽³⁺⁴⁾^=>
=> 5+4=2+7 => 9=9.
⁴⁾^K^o^`^p^a^y^tⁱ^rⁱ^s^hnⁱⁿ^g^g^u^r^u^h^l^a^s^h^q^oⁿ^uⁿⁱ^:⁽^a^∙^b⁾^∙^c=a^∙⁽^b^∙^c⁾^.^M^a^s^a^l^an,⁽²^∙³⁾^∙⁴⁼²^∙⁽³^∙⁴⁾^=>
=> 6∙4=2∙12 => 24=24.
⁵⁾^Q^o^`^s^hⁱ^s^hnⁱⁿ^g^k^o^`^p^a^y^tⁱ^rⁱ^s^h^g^aⁿⁱ^s^b^a^t^an^t^aq^s^im^o^t^q^o^nunⁱ^:^a^∙⁽^b+c)=a^∙^b+a^∙^c.^M^a^s^a^l^an,
2∙(3+4)=2∙3+2∙4 => 2∙7=6+8 =>14=14.
BOSHLANG`ICH TUSHUNCHALAR
mavzusidagi testlar
¹^-^mⁱ^s^o^l^.^I^f^o^daⁿⁱⁿ^g^qⁱ^y^m^a^tⁱⁿⁱ^t^o^pⁱⁿ^g:¹²^–⁶^:³⁺²^∙⁴^.

¹⁶^B^{) 10}^C⁾¹⁸^D^{) 48}^E⁾

Yechish: 12–6:3+2∙4=12–2+8=10+8=18. Javob: 18. (C)
^T^opshⁱ^rⁱ^q:¹⁵^–9^:³⁺⁴^∙^3.^A^{) 24}^B^{) 18}^C^{) 48}^D^{) 6}^E⁾
^Jav^o^b:^24.⁽^A⁾
²^-^mⁱ^s^o^l^.^I^f^o^daⁿⁱⁿ^g^qⁱ^y^m^a^tⁱⁿⁱ^t^o^pⁱⁿ^{g: 2}⁶^∙^25–25^∙²⁴⁺²⁴^∙^23–23^∙²²^–12^∙^8.
^A^{) 106}^B^{) 1}^C⁾⁵⁴^D^{) 8}^E^{) 0}
^Y^echⁱ^s^h:²⁶^∙^25–25^∙²⁴⁺²⁴^∙^23–23^∙^22–12^∙⁸⁼²⁵^∙⁽^26–24⁾⁺²³^∙⁽^24–22⁾^–⁹⁶⁼
=25∙2+23∙2–96=2∙(25+23)–96=2∙48–96=96–96=0. Javob: 0. (E)
^T^opshⁱ^rⁱ^q:^I^f^o^daⁿⁱⁿ^g^qⁱ^y^m^a^tⁱⁿⁱ^t^o^pⁱⁿ^g:
8∙36–16∙36+24∙27–25∙24–21∙5.
A) 45 B) 1 C) 0 D) 15 E) 115 Javob: 15. (D)
³^-^mⁱ^s^o^l^.^Hⁱ^s^o^b^l^aⁿ^g:²⁷⁰⁴⁸^∙^{27044–27047}^∙^27043.
^A^{) 60491}^B^{) 58051}^C⁾⁵⁷⁰⁹¹^D⁾⁵⁴⁰⁹¹^E^{) 56091}
Yechish: Sonlardan eng kichigini a=27043 deb belgilash kiritamiz =>
⁼^{> 27048}^∙²⁷⁰⁴⁴^–27047^∙²⁷⁰⁴³⁼⁽^a⁺⁵⁾⁽^a⁺^1)–⁽^a⁺⁴⁾^a⁼^a2⁺^a⁺⁵^a⁺^5–^a2^–4a=
=2a+5=2∙27043+5=54086+5=54091. Javob: 54091. (D)

^T^opshⁱ^rⁱ^q:^Hⁱ^s^o^b^l^aⁿ^g:⁴⁵⁸¹⁵^∙^{45818–45814}^∙^45816.
^A^{) 137446}^B^{) 137447}^C⁾²⁴¹⁵⁸⁴^D^{) 241586}^E^{) 24158}⁵^.Javob: 137446. (A)
4-misol. Ushbu 1234567891011…4950 sonning raqamlari yig`indisini toping.
A) 335 B) 330 C) 320 D) 315 E) 310

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