The Ancient Greek aphorism "know thyself" (Greek: γνῶθι σεαυτὸν) is one of the Delphic maxims and was inscribed in the pronaos (forecourt) of the Temple of Apollo at Delphi according to the Greek periegetic (travelogue) writer Pausanias. The maxim, or aphorism, "know thyself" has had a variety of meanings attributed to it in literature. The Suda, a 10th-century encyclopedia of Greek knowledge, says: "the proverb is applied to those whose boasts exceed what they are", and that "know thyself" is a warning to pay no attention to the opinion of the multitude.
"Know thyself" - The House of the Temple, Washington D.C.
To begin with, to "know thyself" means you must become conscious of your thoughts, your likes and dislikes, your prejudices, habits, memories, etc., because the more you understand yourself, the greater is your understanding of humanity as well.
Knowing your true self is when you learn to step outside of your thinking - in other words, to be the "observer" and not get caught up in your emotions and all the above mentioned aspects that make up your "persona" or "personality" as we see it here on earth.
It is your intellect (thinking) which sees duality, or opposites. No we are not two halves. That is just the mind at work. It is the mind that is incapable of understanding wholeness, that the universe is Infinite and Eternal. Can your mind picture that? No, it cannot. Then you need to move forward, to still the mind, and get in touch with your inner world. Meditation is helpful, perhaps visualization is also helpful. You need to find what works for you.
Eventually you must realize your centre of awareness (consciousness) is really an emanation of the Central Consciousness, and there is no separation in Cosmic Mind.
Great souls come to realize the age-old saying, "I Am That", not "I am a person" but I AM the whole of Life.