I have a (HCSR04) ultrasonic distance module connected to arduino , i have connected all the four pins of the module properly and its working fine , then when i pull off the wire which is connected to ground in arduino , the module works totally fine . How the module works without connecting its ground pin to arduino ground ?

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It's not uncommon for part to "sort of" work when one of the supply rails is missing and an input signal of the some polarity somewhat substitutes for it, often through built-in ESD protection diodes. In this case, the trigger input when idle is low, and there's probably some capacitance on the board that could hold it through the brief trigger pulse.

It's a bad idea to rely on this though, both as it typically means stressing parts by operating them outside of specified limits, and can also produce not-quite reliable behavior.

For example, I once wired an SPI flash chip on a small carrier stuck underneath an MCU eval board, in order to test some software for a board I was designing. It worked great. Or I should say, it did right up until I tried to write to it, which would always fail. Eventually I discovered that I'd forgotten the ground connection when wiring it up. It was able to return enough current through the I/Os to perform read operations, but not to support a write.

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