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I know that air pressure and temperature are inversely proportional. Now I saw in a book that "Atmospheric pressure decreases as we go higher and higher."
But at greater heights the temperature becomes low, and so the air pressure would be high.
But it is given atmospheric pressure decreases with altitude. I understand that air pressure and atmospheric pressure are different.
But I can't understand how they are different.
Air pressure is the intrinsic pressure in a quantity of air. It can come from any number of sources. Perhaps there is a closed cylinder of air with a piston compressing it isothermally. Or perhaps the air in the tire in your car is under pressure due to the weight of the car and the surface tension of the tire.
Atmospheric pressure is air pressure due solely to the weight of the air above you in the atmosphere. It is typically around $10^5\ \mathrm{N/m^2}$ on Earth, but can vary with altitude and weather. It is just a "natural," ubiquitous special case of air pressure in physical situations taking place inside an atmosphere.
Note that this is a very subtle distinction, and interchanging the terms "air pressure" and "atmospheric pressure" should cause absolutely no confusion in practice.
I know that air pressure and temperature are inversely proportional.
You should not know that. This is the source of your misunderstanding.
The ideal gas law, $PV=nRT$, can be rewritten as $P=\frac R m \rho T$, where $m$ is the average mass of a molecule in the gas and $\rho$ is the density of the gas. The first term on the right is a constant for a well-mixed gas such as the lower part of the Earth's atmosphere.
Removing this constant, the gas law becomes $P\propto \rho T$: Pressure is directly proportional to the product of temperature and density. There is no inverse. Density in the atmosphere decreases monotonically with increased altitude, and in the lowest part of the atmosphere, temperature also decreases with altitude. The combined effect makes pressure drop even more sharply than either density or temperature in the troposphere.
If you simply go outside and hold in your hand something of standard area, like a coin, atmospheric pressure is nothing but the weight of all the air above that coin, in a very slender cylinder, going from the coin up to outer space. Of course, since some of the air can sneak in under it and push up, you don't feel that weight.
Temperature of air is like temperature of anything. If you heat it, it gets hotter, and if you cool it, it gets colder. Air gets heated because the sun shines on the ground, making it hot, which raises the temperature of the air.
Another way to make air hot is to squeeze it. Like if you take a bicycle tire pump and squeeze some air with it (increasing its pressure), you should notice the pump gets warm. The opposite happens too. If you let it expand, it gets cooler, but those only happen because the air in the pump is sort of insulated from the outside air. (There's a big word for that - "adiabatic".)
Now in real outside air, the air doesn't stay still. Some of it moves up, and some moves down. The air that moves up goes to a place of lower pressure because there's less air above it, so it expands and gets cooler. Likewise descending air gets warmer, for the same adiabatic reason. So, for example, the air is much warmer at the bottom of the Grand Canyon than at the top.
Put all these together and you'll start to understand how weather works.
Air pressure It can be measured by pressure gauge. It can be changed.
Atmospheric pressure. It is measured by barometer. It cannot be changed.
