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I am using Arduino Duemilanove 328,and running the AnalogReadSerial example sketch.

Even when nothing is connected to the Arduino board it still shows a value of about 350 out of 1023. Ideally it should have shown 0 out of 1023 when nothing is connected to Arduino.

So how can I solve this problem?

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    Connect it to something. E.g. a pull-down resistor if you want to read zero when nothing else is connected. – Edgar Bonet Jul 8 '15 at 9:00
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Why do you think it should show 0 when not connected to anything?

Does the antenna of your radio give a zero signal just because the end isn't connected to anything? An Arduino's input pin (either analog or digital) is just the same. If it isn't connected to anything then it just picks up noise - be that radio transmissions, induction from nearby mains wiring, neutrinos from Alpha Centauri... Basically if it's not connected to anything then there is no way you can possibly know what the value should be since there is no set value.

In order to read 0 on the ADC the ADC has to be connected 0V. That means connect it to ground.

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