The Sanskrit word for "truth" is satyam and this is also the word for reality. The only reality is brahman. Ignorance is ignoring (literally "turning away from") this truth through identifying ourselves with a body, mind, belief, cause or whatever. We mistakenly take these things to be real in their own right instead of simply a form of one essential reality.
Another frequently used metaphor in Advaita is that of clay and a pot made from the clay. The clay exists before the pot is made. Whilst the pot is in use to hold something, it is still clay. And after the pot has been broken, the clay is still there. Advaita defines "real" as being that which exists in or transcends all three periods of time
(i.e. past, present and future) - trikAlAtIta, so that it is only actually the clay that is real by this definition. Yet whilst the clay is in the form of the pot, it would not be true to say that the pot does not exist. Clearly it has some reality but it cannot be described as real according to the definition. But neither is it false, since we can use it to carry water about, while the clay in the form of an amorphous lump is not much use for this purpose. The pot's reality is entirely dependent upon the clay and, moreover, it is always clay and nothing but clay whether it is in the form of the pot or not. Thus the pot has a "dependent reality." There is no English word to describe this – the Sanskrit word that is used is mithyA.
Similarly, the world did not exist a few billion years ago and will be swallowed up by the sun in few more. The reality upon which it depends is Brahman. Brahman exists before during and after the world. The world too, whilst it exists, is nothing but Brahman in essence. Brahman is the only reality; the world is neither completely real nor completely unreal – it is mithyA. And the same applies to every "thing" in the world, be it people, houses, minds, concepts, emotions etc.
With this explanation, then, another possible definition for ignorance is available – ignorance is pursuing mithyA instead of satyam. Having mixed up real and unreal, most people spend their lives trying to derive happiness from material objects or transient relationships instead of coming to the realization that our already existent essence is limitless consciousness. Because we already are That, there is not, strictly speaking, anything that we can do to become That. Once the ignorance of the fact is removed, the already existent truth is revealed. This means, somewhat surprisingly, that it is knowledge of mithyA that brings about enlightenment, not knowledge of satyam, because it is the mithyA that constitutes the apparent bondage.
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