If earth is the piece of sun?

The earth and the sun formed from the same huge cloud of interstellar gas and dust, but the sun grabbed most of everything, and had the strong gravity to hold in hydrogen and helium. The earth condensed separately in the huge disc that became our solar system, forming a rocky planet with an iron core. When the sun turned on, the solar radiation cleaned up the inner solar system and pushed out the gas and dust that remained.

So - the earth and the sun formed at the same time from the same dust cloud, but have very different compositions. And the earth was never physically a part of the sun.