Intermediate Arithmetic

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1 Git Lerig Guides Iteredite Arithetic Nuer Syste, Surds d Idices Author: Rghu M.D.

2 NUMBER SYSTEM Nuer syste: Nuer systes re clssified s Nturl, Whole, Itegers, Rtiol d Irrtiol uers. The syste hs ee digrticlly show elow. I/R Q/R - I N, W - 0, 0 R E A L N U M B E R S Expltio: N-NATURAL NUMBERS- Nuers otied for coutig,,, 4 They hve ee represeted i the ier ost circle. A useful hit is tht ost turl thigs for exple people c oly e couted,, or ore d ot - or. WHOLE NUMBERS- The whole uers iclude ll turl uers together with 0. It is helpful to reeer the s ll turl uers plus whole (phooy hole eig zero). Whole uers hve ee show i the digr W or the secod circle. INTEGERS- Itegers iclude ll turl uers, zero d egtives of turl uers. They hve ee show i the iddle circle i the digr, ut they lso iclude the uers show i the ier circles. QUOTIENTS OR RATIONAL NUMBERS- Ay uer tht c e expressed s rtio of two turl uers is clled quotiet or rtiol uer. Exples of rtiol uers re Iteredite-Arithetic of

3 give i the fourth circle. Nturl uers d Itegers re lso rtiol uers s they c e expressed s rtio Exple Teritig Decils c lso e coverted to quotiets Exple 0. IRRATIONAL NUMBERS- The uers whe expressed s decil uers re either teritig or recurrig. For Exple π or, 4 is o teritig decil uer hece is irrtiol. However,,.. is recurrig decil whe expressed or quotiet ecoes 4 or 4 hece is rtiol. REAL NUMBERS- The coplete syste of uers, ely turl, whole, iteger, quotiet or rtiol d irrtiol uers re clled Rel uers. Rel uers c e represeted y uer lie. Every rel uer correspods to poit o the uer lie. A exple of uer lie is give elow Nuer lie ADDITION AND MULTIPLICATION OF REAL NUMBERS: Additio d ultiplictio of rel uers follow certi lws d rules Additio:. Closure Rule: The su of two rel uers is lwys rel uer. Associtive lw: ( + ) + c = + ( + c) if, d c re rel uers. Couttive lw: ( + ) = ( + c) for ll rel uers or d 4. Iverse Nuer Rule: Every rel uer there exists rel uer- such tht + (-) = (-) + = 0 d re clled the dditive iverse of ech other. Multiplictio:. Closer Rule: The product of two rel uers is lwys rel uer.. Associtive lw: ( ) c ( c) if, d c re rel uers. Couttive lw: if d re rel uers 4. Multiplictio Idetity Rule:, if is rel uer. Multiplictio Iverse Rule:, if is rel d ot equl to zero Uit I Worked exples: Exple. Prove is irrtiol uer? Aswer: Let e expressed s quotiet of itegers Iteredite-Arithetic of

4 or P or p If p q q q Becuse q is iteger q is lwyseve, hece p p But P q Hece q is eve. is eve the p hs to e eve uer ecuse squrer of eve uers c oly e eve. Let r e theiterger eve or odd whe douled is equl to P Hece P r p or q Hece 4r q 4r r is eve therefore q is eve If pd q re eve es pd q hve coo fctor of By defeitio of rtiol uer EXERCISES is ot rtiol uer d therefore is p d q should ot hve coo fctor errtiol uer. Clssify the followig uers s rtiol or irrtiol. () () ()0 (4) ()π () () 9 () 00 (9) (0) (). (recoccurig) ().. Drw uer lie d rk the followig uers () - () - (). (4) () () ()0 () Select the est swer. is : ) whole uer oly ) Nturl uer oly c) Nturl d whole uer oly d) Irrtiol uer Iteredite-Arithetic of

5 4. Quotiets or Rtiol uers iclude ) Irrtiol uers ) Rel uers c) d) Nturl d whole uers. Fid the correct swer ( ) c c ( ) c c c ( ) c ( c) ( ) c c. Which of the followig uers c e represeted s quotiet () () () π (4).. Fid the irrtiol uer: (). () () (4). Show tht is irrtiol? 9. Fid the followig 0. Su of two irrtiol uer which is rtiol. Product of two irrtiol uer which is irrtiol Iteredite-Arithetic 4 of

6 SURDS AND INDICES (Icludes rdicls d idices) Idices d Roots: Nuers c lso e clssified s Rdicls d Idices. writte s. Liewise uer c e ultipled y itself ties d for y uer is kow s the idex (or power). Oppositeprocess for squrig is kow squre rootig or i rootig for y other For exple squre root of 4 is writte s root is clled s the rdicl sie of idex. SURDS AND RADICALS If The For exple squre root of or lso equl to is irrtiol where is is rtiol uer d is For exple is clled squre of d 4 d is equtl to likewise positiveiteger. the swer writte s X or X X, uer. is secod order surd for d is the rdicl or th is clled surd or rdicl order of thesurd of th order d is kow s the rdicd LAWS OF INDICES AND SURDS A set of lws re pplicle for oth idices d surds. If is positive rtiol uer d is iteger, the followig lws re pplicle for idices. ) ) ( ) ) 4) ) d ( ) d d Lie wise the followig lws re pplicle for rdicls give is positive rtiol uer d is iteger. Iteredite-Arithetic of

7 () () () (4) Fors of Surds: Siple surds hve the sllest rdicl for e.g., etc. re cosidered s siple surds. is ot surd s it c e further siplified to 4. 4 Pure surds coti oly irrtiol ter for exple, d 0. Mixed surds coti product or su of rtiol uer with surd. Mixed surds re geerlly expressed with the rdicl i the uertor. For exple, isted of the the surd is writte s However the siplest for of surd cotis the lowest iteger s the rdicd for exple s writte s. EXAMPLES: Exple : Ne the followig s rtiol uers or surds fter siplifictio. ) surd ) c) d) 4 9 surd 9 Rtiol uer surd Exple : Write the followig surds i the siplest for ) ) c) d) ( ) Iteredite-Arithetic of

8 Exple : Stte which of the followig re surds d which re ot. ) 4 4 4, is surd ) c) d) is ot surd, is surd is surd. Exple 4: Rtiolize the followig surds. Note: Surds re ot geerlly expressed with the rdicl or root sie i the deoitor. The deoitor is will hve to e rtiol uer otherwise oth the uertor d deoitor will e ultiplied y the se surd i order to trsfer the surd to the uertor. This process is clled rtioliztio. Rtiolize the followig 4 ) 4 ( ) ( ) ) ( ) c) d) ( ) ( ) ( ( )( ) ( ) ) 4 Hit : Reeer the foruls ( ) ( )( ) EXERCISES. Fid the squre root of the followig uers. Fid the root 49 ( ) () 04 () (4) () () (4) ( ). Idetify with resos which of the followig re surds d which re ot 0 Iteredite-Arithetic of

9 ) ) ) 4) ) ) 0 ) 0 ) 9) 0) 4. Rtiolize the followig surds y retiig the root sig oly i the deoitor. 9 ) ) ) 4) ) ) ) ) 9) ( ) 0). Write the followig surds i the for of c where, d c re itegers ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) ) 4) 4 9 ) Iteredite-Arithetic of

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root Suer MA 00 Lesso Sectio P. I Squre Roots If b, the b is squre root of. If is oegtive rel uber, the oegtive uber b b b such tht, deoted by, is the pricipl squre root of. rdicl sig rdicl expressio rdicd