To show that a recurring decimal is rational, we need to express it as a fraction.
(1) 0.124 = 0.1 + 0.02 + 0.004
= 1/10 + 2/100 + 4/1000
= (1/10) + (2/10^2) + (4/10^3)
= (1/10) + (2/10^2) + (4/10^2 * 1/10)
= (1/10) + (2/10^2) + (4/10^2) * (1/10)
= (1/10) + (2/10) * (1/10^2 + 4/10^2)
= (1/10) + (2/10) * (14/10^2)
= (1/10) + (2/10) * (7/50)
= (1/10) + (2/10) * (7/50) * (10/10)
= 1/10 + 14/10^2 + 7/10^2 * (2/50)
= 1/10 + 14/100 + 7/100 * 2/5
= 1/10 + 14/100 + 7/100 * 2/5 * 20/20
= 1/10 + 14/100 + 14/100
= 1/10 + 28/100
= 1/10 + 28/100 * 10/10
= 1/10 + 280/1000
= 1/10 + 280/1000 = 1/10 + 280/1000 * 10/10
= 1/10 + 280/1000 + 280/1000
= 1/10 + 560/1000
= 56/1000
= 7/125
Therefore, 0.124 is a rational number.
(2) -1.124 = -1 + 0.124 = -1 + 7/125 = -132/125 + 7/125 = (-132 + 7)/125 = -125/125 = -1
Therefore, -1.124 is also a rational number.
show that the following recurring decimals are rational.
(1) 0,124 (the dot is on 1 and 4)
(2) -1,124 (the dot is on 2 and 4)
1 answer