Show that the following recurring decimal are rational numbers. a) 19,45 i 19,45 b) \( 0,12 \dot{4} \)

To show that recurring decimals are rational numbers, let’s take a closer look at your examples! For 19.454545..., we can express it as a fraction. Set \( x = 19.454545...\). Multiplying by 100 gives \( 100x = 1945.454545...\). Subtracting the first equation from the second results in \( 99x = 1945 - 19\), which simplifies to \( x = \frac{1926}{99}\), confirming it’s rational. Now, let’s tackle \( 0.12\dot{4} \) (which is 0.124444...). Set \( y = 0.124444...\). Multiplying by 1000 gives \( 1000y = 124.444...\) and multiplying \( y \) by 10 gives \( 10y = 1.24444...\). Subtracting these equations leads us to \( 990y = 123.2\), ultimately yielding \( y = \frac{1232}{9900}\), a rational number!