I understand that voltage is a force. I am confused about voltage whether voltage is positive, negative, or both. For example in some circuits I have seen people have shown "negative voltage". Please explain.
Don't think of voltage as a "force". It is a difference in potential.
Many people try to explain voltage and current like water pressure. Perhaps using air pressure would be a better way.
Hold both of your palms out. There is air in each palm, but nothing can happen between them because there is no difference in potential (pressure). Hold a blowgun in one hand and point it at the other, and you have a difference in potential. You can make a fan spin, blow dust, make soap bubbles, etc; because of the difference in pressure (potential). The cfm of the air flow can be used to show current sort-of- current is from - to +, but too much airflow (current) will collapse the air pressure (voltage).
To explain negative voltage, again, it's potential, just substitute a vacuum for pressure.
Positive and negative voltages are entirely arbitrary and rely totally on your frame of reference. Voltage is referred to as potential because that's exactly what it is - a potential energy difference between two points in a circuit.
When you make a voltage measurement/calculation, you have to first decide your reference point. That means that any point in the circuit with the same potential as your reference would be called 0V. Anything with higher potential would be called a positive voltage and anything with a lower potential would be a negative voltage.
With respect to Arduino, the reference point virtually always the ground but it doesn’t have to be. Compared to ground, the 5V pin on your Uno is positive 5 volts but if you were to call the 5V circuit your reference point then the ground would actually be measured as negative 5 volts. You can also see this with a multimeter by taking a measurement and them swapping your measurement probes around. The value will be the same but the sign will be different.
That's a really quick and dirty explanation but I hope it helps.
What Saximus posted is true. You could use an input to your arduino of 1KVdc on the positive, as long as the ground was 988-995Vdc. The difference in potential is still only 5-12Vdc. I wouldn't recommend touching it though, as you are probably at ground potential to the power source and would feel the entire 1KV.
You can make a split supply using a voltage divider of 2 equal value resistors. Use the node where the 2 resistors are connected together as ground, and the other 2 nodes would be + and - with respect to "ground". Of course that would only apply to the circuits that follow the divider, and that ground reference cannot be tied back to the power supply ground reference.
Another way is to take two batteries - connect the+ of one to the - of the other. Connect your multimeter COM (black) lead to that and call it "GND". Use your red lead to make measurements.
The - of the first battery will then measure as -9V, and the + of the other will measure +9V.
If you powered an op-amp with this +9, -9, and Gnd, you could feed AC into it and have an output that could swing - and + with respect to Gnd.
(If you measure from the - of the first battery to the + of the other battery you will see 18V, 9V in the middle, as this effectively moves GND to the lowest voltage reference point vs a middle reference point.)
Here’s an awesome example of negative voltage in logic circuits from Wikipedia:
When I was college intern my first summer job was testing with this high speed logic, experimenting with termination resistors and looking at reflections on signal traces. Fast stuff!
“Power supplies and logic levels
The ECL circuits usually operate with negative power supplies (positive end of the supply is connected to ground) in contrast to other logic families in which negative end of the supply is grounded. This is done mainly to minimize the influence of the power supply variations on the logic levels as ECL is more sensitive to noise on the VCC and relatively immune to noise on VEE. Because ground should be the most stable voltage in a system, ECL is specified with a positive ground. In this connection, when the supply voltage varies, the voltage drops across the collector resistors change slightly (in the case of emitter constant current source, they do not change at all). As the collector resistors are firmly “tied up” to ground, the output voltages “move” slightly (or not at all). If the negative end of the power supply was grounded, the collector resistors would be attached to the positive rail. As the constant voltage drops across the collector resistors change slightly (or not at all), the output voltages follow the supply voltage variations and the two circuit parts act as constant current level shifters. In this case, the voltage divider R1-R2 compensates the voltage variations to some extent. The positive power supply has another disadvantage - the output voltages will vary slightly (±0.4 V) against the background of high constant voltage (+3.9 V). Another reason for using a negative power supply is protection of the output transistors from an accidental short circuit developing between output and ground (but the outputs are not protected from a short circuit with the negative rail).”
(explanation goes better if one looks at the Wikipedia diagrams at the same time)