Verify associativity of addition of rational numbers i.e., (x+y)+z=x+(y+z)
when:
(i) x=12,y=23,z=−15
(ii) x=−25,y=43,z=−710
(iii) x=−711,y=2−5,z=−322
(iv) x=−2,y=35,z=−43
Verify associativity of addition of rational numbers i.e., (x+y)+z=x+(y+z)
when:
(i) x=12,y=23,z=−15
(ii) x=−25,y=43,z=−710
(iii) x=−711,y=2−5,z=−322
(iv) x=−2,y=35,z=−43
(i) x=12,y=23,z=−15(x+y)+z=(12+23)+−15=3+46+−15=76+−15=76−15=35−630=2930and x+(y+z)=12+(23+−15)=12+10−315=12+715=15+1430=2930∴(x+y)+z=x+(y+z)
(ii) x=−25,y=43,z=−710∴(x+y)+z=(−24+43)+−710=−6+2015+−710=1415+−710LCM of 15,10=30=28−2130=730and x+(y+z)=−25+(43+−710)=−25+40−2130=−25+1930=−12+1930=730∴(x+y)+z=x+(y+z)
(iii) x=−711,y=2−5,z=−322⇒x=−711,y=2×(−1)−5×(−1),z=−322⇒x=−711,y=−25,z=−322Now(x+y)+z(−711+−25)+−322=−35−2255+−322=−5755+−322 (LCM of 22,55=110)=−114−15110=−129110and x+(y+z)=−711+(−25+−322)=−711+−44−15110=−711+−59110=−70−59110=−129110∴(x+y)+z=x+(y+z)
(iv) x=−2,y=35,z=−45∴(x+y)+z=(−21+35)+−45=−10+35+−45=−75+−45=−7−45=−115and x+(y+z)=−2+(35+−45)=−2+3−45=−21+−15=−10−15=−115∴(x+y)+z=x+(y+z)
(i) x=12,y=23,z=−15(x+y)+z=(12+23)+−15=3+46+−15=76+−15=76−15=35−630=2930and x+(y+z)=12+(23+−15)=12+10−315=12+715=15+1430=2930∴(x+y)+z=x+(y+z)
(ii) x=−25,y=43,z=−710∴(x+y)+z=(−24+43)+−710=−6+2015+−710=1415+−710LCM of 15,10=30=28−2130=730and x+(y+z)=−25+(43+−710)=−25+40−2130=−25+1930=−12+1930=730∴(x+y)+z=x+(y+z)
(iii) x=−711,y=2−5,z=−322⇒x=−711,y=2×(−1)−5×(−1),z=−322⇒x=−711,y=−25,z=−322Now(x+y)+z(−711+−25)+−322=−35−2255+−322=−5755+−322 (LCM of 22,55=110)=−114−15110=−129110and x+(y+z)=−711+(−25+−322)=−711+−44−15110=−711+−59110=−70−59110=−129110∴(x+y)+z=x+(y+z)
(iv) x=−2,y=35,z=−45∴(x+y)+z=(−21+35)+−45=−10+35+−45=−75+−45=−7−45=−115and x+(y+z)=−2+(35+−45)=−2+3−45=−21+−15=−10−15=−115∴(x+y)+z=x+(y+z)
