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Real numbers and the number line
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AdnanBukhari13
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Real numbers and the number line
1.
R e a l n u mb e r s a n dt h e N u mb e r L i n e D e f i n i t i o n s A s e t i s a c o l l e c t i o n o f o b j e c t s , t y p i c a l l y g r o u p e dw i t h i nb r a c e s { } , w h e r e e a c h o b j e c t i s c a l l e da n e l e me n t . F o r e x a mp l e , { r e d , g r e e n , b l u e } i s a s e t o f c o l o r s . A s u b s e t i s a s e t c o n s i s t i n go f e l e me n t s t h a t b e l o n gt oa g i v e ns e t . F o r e x a mp l e , { g r e e n , b l u e } i s a s u b s e t o f t h e c o l o r s e t a b o v e . As e t w i t hn oe l e me n t s i s c a l l e dt h e e mp t y s e t a n dh a s i t s o w n s p e c i a l n o t a t i o n , { } o r ∅. Wh e n s t u d y i n gma t h e ma t i c s , w e f o c u s o ns p e c i a l s e t s o f n u mb e r s . T h e s e t o f n a t u r a l ( o r c o u n t i n g ) n u mb e r s , d e n o t e d N , i s { 1 , 2 , 3 , 4 , 5 , … } N a t u r a l N u mb e r s T h e t h r e e p e r i o d s ( … ) i s c a l l e da n e l l i p s i s a n di n d i c a t e s t h a t t h e n u mb e r s c o n t i n u e w i t h o u t b o u n d . T h e s e t o f w h o l e n u mb e r s , d e n o t e d W, i s t h e s e t o f n a t u r a l n u mb e r s c o mb i n e dw i t hz e r o . { 0 , 1 , 2 , 3 , 4 , 5 , … } Wh o l e N u mb e r s T h e s e t o f i n t e g e r s , d e n o t e d Z , c o n s i s t s o f b o t h p o s i t i v e a n dn e g a t i v e w h o l e n u mb e r s , a s w e l l a s z e r o . { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } N o t i c e t h a t t h e s e t s o f n a t u r a l a n d w h o l e n u mb e r s a r e b o t hs u b s e t s o f t h e s e t o f i n t e g e r s . R a t i o n a l n u mb e r s , d e n o t e d Q , a r e d e f i n e da s a n y n u mb e r o f t h e f o r m a b , w h e r e a a n d b a r e i n t e g e r s a n d b i s n o n z e r o . D e c i ma l s t h a t r e p e a t o r t e r mi n a t e a r e r a t i o n a l . F o r e x a mp l e , 0 . 7 = 7 / 1 0a n d0 . 3 ¯ = 0 . 3 3 3 3 ⋯= 1 / 3 T h e s e t o f i n t e g e r s i s a s u b s e t o f t h e s e t o f r a t i o n a l n u mb e r s b e c a u s e e v e r y i n t e g e r a/b
2.
c a nb e e x p r e s s e da s a r a t i o o f t h e i n t e g e r a n d 1 . I n o t h e r w o r d s , a n y i n t e g e r c a n b e w r i t t e no v e r 1 a n dc a n b e c o n s i d e r e da r a t i o n a l n u mb e r . F o r e x a mp l e , 5 = 5 / 1 I r r a t i o n a l n u mb e r s a r e d e f i n e da s a n y n u mb e r t h a t c a n n o t b e w r i t t e na s a r a t i o o f t w o i n t e g e r s . N o n t e r mi n a t i n gd e c i ma l s t h a t d on o t r e p e a t a r e i r r a t i o n a l . F o r e x a mp l e , π = 3 . 1 4 1 5 9 …a n d√ 2 = 1 . 4 1 4 2 1 … T h e s e t o f r e a l n u mb e r s , d e n o t e d R , i s d e f i n e da s t h e s e t o f a l l r a t i o n a l n u mb e r s c o mb i n e dw i t h t h e s e t o f a l l i r r a t i o n a l n u mb e r s . T h e r e f o r e , a l l t h e n u mb e r s d e f i n e ds of a r a r e s u b s e t s o f t h e s e t o f r e a l n u mb e r s . I n s u mma r y , F i g u r e 1 . 1 . 1 1 . 1 . 1 : R e a l N u mb e r s N u mb e r L i n e A r e a l n u mb e r l i n e , o r s i mp l y n u mb e r l i n e , a l l o w s u s t ov i s u a l l y d i s p l a y r e a l n u mb e r s b y a s s o c i a t i n gt h e mw i t h u n i q u e p o i n t s o na l i n e . T h e r e a l n u mb e r a s s o c i a t e dw i t ha p o i n t
3.
i s c a l l e da c o o r d i n a t e . Ap o i n t o nt h e r e a l n u mb e r l i n e t h a t i s a s s o c i a t e dw i t ha c o o r d i n a t e i s c a l l e di t s g r a p h . T oc o n s t r u c t a n u mb e r l i n e , d r a wa h o r i z o n t a l l i n e w i t h a r r o w s o nb o t h e n d s t oi n d i c a t e t h a t i t c o n t i n u e s w i t h o u t b o u n d . N e x t , c h o o s e a n y p o i n t t or e p r e s e n t t h e n u mb e r z e r o ; t h i s p o i n t i s c a l l e dt h e o r i g i n . Ma r k o f f c o n s i s t e n t l e n g t h s o n b o t hs i d e s o f t h e o r i g i n a n dl a b e l e a c ht i c k ma r k t o d e f i n e t h e s c a l e . P o s i t i v e r e a l n u mb e r s l i e t ot h e r i g h t o f t h e o r i g i n a n dn e g a t i v e r e a l n u mb e r s l i e t ot h e l e f t . T h e n u mb e r z e r o ( 0 ) i s n e i t h e r p o s i t i v e n o r n e g a t i v e . T y p i c a l l y , e a c h t i c k r e p r e s e n t s o n e u n i t . A s i l l u s t r a t e db e l o w , t h e s c a l e n e e dn o t a l w a y s b e o n e u n i t . I n t h e f i r s t n u mb e r l i n e , e a c h t i c k ma r k r e p r e s e n t s t w ou n i t s . I n t h e s e c o n d , e a c h t i c k ma r k r e p r e s e n t s 1 / 7 o f a u n i t .
4.
T h e g r a p h o f e a c h r e a l n u mb e r i s s h o w na s a d o t a t t h e a p p r o p r i a t e p o i n t o nt h e n u mb e r l i n e . Ap a r t i a l g r a p h o f t h e s e t o f i n t e g e r s Z f o l l o w s : E x a mp l e 1 . G r a p h t h e f o l l o w i n gs e t o f r e a l n u mb e r s : S o l u t i o n G r a p h t h e n u mb e r s o n a n u mb e r l i n e w i t ha s c a l e w h e r e e a c ht i c k ma r k r e p r e s e n t s 1 / 2 o f a u n i t . O r d e r i n g R e a l N u mb e r s Wh e n c o mp a r i n g r e a l n u mb e r s o n a n u mb e r l i n e , t h e l a r g e r n u mb e r w i l l a l w a y s l i e t o t h e r i g h t o f t h e s ma l l e r o n e . I t i s c l e a r t h a t 1 5 i s g r e a t e r t h a n 5 , b u t i t ma y n o t b e s o c l e a r t os e e t h a t − 1 i s g r e a t e r t h a n − 5 u n t i l w e g r a p h e a c hn u mb e r o na n u mb e r l i n e . We u s e s y mb o l s t oh e l pu s e f f i c i e n t l y c o mmu n i c a t e r e l a t i o n s h i p s b e t w e e nn u mb e r s o nt h e n u mb e r l i n e . T h e s y mb o l s u s e dt od e s c r i b e a n e q u a l i t y r e l a t i o n s h i p b e t w e e n n u mb e r s f o l l o w : = i s e q u a l t o≠ i s n o t e q u a l t o≈ i s a p p r o x i ma t e l y e q u a l t o T h e s e s y mb o l s a r e u s e da n di n t e r p r e t e di n t h e f o l l o w i n gma n n e r :
5.
5 = 5 0 ≠ 5 π ≈ 3 . 1 4 5 i s e q u a l t o 5 0 i s n o t e q u a l t o 5 p i i s a p p r o x i ma t e l y e q u a l t o 3 . 1 4 We n e x t d e f i n e s y mb o l s t h a t d e n o t e a n o r d e r r e l a t i o n s h i p b e t w e e n r e a l n u mb e r s . < L e s s t h a n > G r e a t e r t h a n ≤ L e s s t h a n o r e q u a l t o ≥ G r e a t e r t h a n o r e q u a l t o T h e s e s y mb o l s a l l o wu s t oc o mp a r e t w on u mb e r s . F o r e x a mp l e , − 1 2 0 < − 1 0 N e g a t i v e 1 2 0 i s l e s s t h a n n e g a t i v e 1 0 S i n c e t h e g r a p h o f − 1 2 0 i s t ot h e l e f t o f t h e g r a p h o f – 1 0 o nt h e n u mb e r l i n e , t h a t n u mb e r i s l e s s t h a n − 1 0 . We c o u l dw r i t e a n e q u i v a l e n t s t a t e me n t a s f o l l o w s : − 1 0 > − 1 2 0 N e g a t i v e 1 0 i s g r e a t e r t h a n n e g a t i v e 1 2 0 S i mi l a r l y , s i n c e t h e g r a p h o f z e r oi s t ot h e r i g h t o f t h e g r a p h o f a n y n e g a t i v e n u mb e r o n t h e n u mb e r l i n e , z e r oi s g r e a t e r t h a na n y n e g a t i v e n u mb e r . 0 > − 5 0 Z e r o i s g r e a t e r t h a n n e g a t i v e 5 0 T h e s y mb o l s < a n d > a r e u s e dt od e n o t e s t r i c t i n e q u a l i t i e s , a n dt h e s y mb o l s a n da r e u s e dt od e n o t e i n c l u s i v e i n e q u a l i t i e s . I ns o me s i t u a t i o n s , mo r e t h a n o n e s y mb o l c a n b e c o r r e c t l y a p p l i e d . F o r e x a mp l e , t h e f o l l o w i n gt w os t a t e me n t s a r e b o t ht r u e : − 1 0 < 0 a n d − 1 0 ≤ 0
6.
I na d d i t i o n , t h e “ o r e q u a l t o ” c o mp o n e n t o f a ni n c l u s i v e i n e q u a l i t y a l l o w s u s t o c o r r e c t l y w r i t e t h e f o l l o w i n g : − 1 0 ≤ − 1 0 T h e l o g i c a l u s e o f t h e w o r d“ o r ” r e q u i r e s t h a t o n l y o n e o f t h e c o n d i t i o n s n e e db e t r u e : t h e “ l e s s t h a n ” o r t h e “ e q u a l t o . ” E x a mp l e 2 F i l l i n t h e b l a n k w i t h < , = , o r > : − 2 _ _ _ _ − 1 2 . S o l u t i o n U s e > b e c a u s e t h e g r a p h o f − 2 i s t ot h e r i g h t o f t h e g r a p h o f − 1 2 o n a n u mb e r l i n e . T h e r e f o r e , − 2 > − 1 2 , w h i c hr e a d s “ n e g a t i v e t w oi s g r e a t e r t h a n n e g a t i v e t w e l v e . ” A n s w e r : − 2 > − 1 2 A b s o l u t e V a l u e T h e a b s o l u t e v a l u e o f a r e a l n u mb e r a , d e n o t e d | a | , i s d e f i n e da s t h e d i s t a n c e b e t w e e n z e r o( t h e o r i g i n ) a n dt h e g r a p h o f t h a t r e a l n u mb e r o nt h e n u mb e r l i n e . S i n c e i t i s a d i s t a n c e , i t i s a l w a y s p o s i t i v e . F o r e x a mp l e , | − 4 | = 4a n d| 4 | = 4 B o t h 4 a n d − 4 a r e f o u r u n i t s f r o mt h e o r i g i n , a s i l l u s t r a t e db e l o w :
7.
A l s o , i t i s w o r t hn o t i n gt h a t | 0 | = 0 T h e a b s o l u t e v a l u e c a nb e e x p r e s s e dt e x t u a l l y u s i n gt h e n o t a t i o n a b s( a ) . We o f t e n e n c o u n t e r n e g a t i v e a b s o l u t e v a l u e s , s u c ha s − | 3 | o r – a b s ( 3 ) . N o t i c e t h a t t h e n e g a t i v e s i g n i s i n f r o n t o f t h e a b s o l u t e v a l u e s y mb o l . I nt h i s c a s e , w o r k t h e a b s o l u t e v a l u e f i r s t a n dt h e n f i n dt h e o p p o s i t e o f t h e r e s u l t . − | 3 | − | − 3 | = − 3 = − 3 T r y n o t t oc o n f u s e t h i s w i t ht h e d o u b l e - n e g a t i v e p r o p e r t y , w h i c h s t a t e s t h a t − ( − 7 ) = + 7 . A t t h i s p o i n t , w e c a n d e t e r mi n e w h a t r e a l n u mb e r s h a v e a p a r t i c u l a r a b s o l u t e v a l u e . F o r e x a mp l e , | ? | = 5 T h i n k o f a r e a l n u mb e r w h o s e d i s t a n c e t ot h e o r i g i n i s 5 u n i t s . T h e r e a r e t w os o l u t i o n s : t h e d i s t a n c e t ot h e r i g h t o f t h e o r i g i n a n dt h e d i s t a n c e t ot h e l e f t o f t h e o r i g i n , n a me l y , { ± 5 } . T h e s y mb o l ( ± ) i s r e a d“ p l u s o r mi n u s ” a n di n d i c a t e s t h a t t h e r e a r e t w oa n s w e r s , o n e p o s i t i v e a n do n e n e g a t i v e . | − 5 | = 5 a n d| 5 | = 5 N o wc o n s i d e r t h e f o l l o w i n g : | ? | = − 5 H e r e w e w i s ht of i n da v a l u e f o r w h i c ht h e d i s t a n c e t ot h e o r i g i n i s n e g a t i v e . S i n c e n e g a t i v e d i s t a n c e i s n o t d e f i n e d , t h i s e q u a t i o n h a s n o s o l u t i o n . I f a ne q u a t i o nh a s Before
absolute modes there's not negative sign which means only a value within mode should be in such a way that the result will be -5 which is not possible.
8.
n o s o l u t i o n , w e s a y t h e s o l u t i o n i s t h e e mp t y s e t : ∅. R e a l , I ma g i n a r y a n dC o mp l e x N u mb e r s R e a l n u mb e r s a r e t h e u s u a l p o s i t i v e a n dn e g a t i v e n u mb e r s . I f w e mu l t i p l y a r e a l n u mb e r b y i , w e c a l l t h e r e s u l t a n i ma g i n a r y n u mb e r . E x a mp l e s o f i ma g i n a r y n u mb e r s a r e : i , 3 i a n d− i / 2 . I f w e a d do r s u b t r a c t a r e a l n u mb e r a n da n i ma g i n a r y n u mb e r , t h e r e s u l t i s a c o mp l e x n u mb e r . We w r i t e a c o mp l e x n u mb e r a s z = a +i bw h e r e a a n db a r e r e a l n u mb e r s C o mp l e x n u mb e r s a r e n u mb e r s t h a t c o n s i s t o f t w op a r t s —a r e a l n u mb e r a n da n i ma g i n a r y n u mb e r . C o mp l e x n u mb e r s a r e t h e b u i l d i n gb l o c k s o f mo r e i n t r i c a t e ma t h , s u c h a s a l g e b r a . T h e y c a n b e a p p l i e dt oma n y a s p e c t s o f r e a l l i f e , e s p e c i a l l y i n e l e c t r o n i c s a n de l e c t r o ma g n e t i s m. T h e s t a n d a r df o r ma t f o r c o mp l e x n u mb e r s i s a +b i , w i t ht h e r e a l n u mb e r f i r s t a n dt h e i ma g i n a r y n u mb e r l a s t . B e c a u s e e i t h e r p a r t c o u l db e 0 , t e c h n i c a l l y a n y r e a l n u mb e r o r i ma g i n a r y n u mb e r c a n b e c o n s i d e r e da c o mp l e x n u mb e r . C o mp l e x d o e s n o t me a n c o mp l i c a t e d ; i t me a n s t h a t t h e t w ot y p e s o f n u mb e r s c o mb i n e t of o r ma c o mp l e x , l i k e a h o u s i n gc o mp l e x —a g r o u po f b u i l d i n g s j o i n e dt o g e t h e r . R e a l n u mb e r s a r e t a n g i b l e v a l u e s t h a t c a nb e p l o t t e do n a h o r i z o n t a l n u mb e r l i n e , s u c h a s f r a c t i o n s , i n t e g e r s o r a n y c o u n t a b l e n u mb e r t h a t y o u c a nt h i n k o f . I ma g i n a r y n u mb e r s a r e a b s t r a c t c o n c e p t s t h a t a r e u s e dw h e ny o u n e e dt h e s q u a r e r o o t o f a n e g a t i v e n u mb e r . A d d i n ga n dS u b t r a c t i n gC o mp l e x N u mb e r s I f w e w a n t t oa d do r s u b t r a c t t w oc o mp l e x n u mb e r s , z 1= a +i ba n dz 2= c +i d , t h e r u l e i s t oa d dt h e r e a l a n di ma g i n a r y p a r t s s e p a r a t e l y : z 1+z 2= a +i b+c +i d= a +c +i ( b+d ) z 1- z 2= a +i b- c - i d= a - c +i ( b- d )
9.
E x a mp l e ( a ) ( 1+i ) +( 3+i ) =1+3+i ( 1+1 ) =4+2 i ( b ) ( 2+5 i ) - ( 1- 4 i ) =2+5 i - 1+4 i = 1+9 i
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