Algebra was not an important invention to Islam as a religion. It was an important invention in mathematics that was discovered by a Muslim (Al-Khwarizmi). Algebra is important in mathematics because it establishes the ability to perform mathematical operations even if some or all of the quantities are unknown.
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Algebra exists to find solutions
Islamic algebra is the same as regular algebra. just like Arabic numerals is the same as 1 2 3. why is algebra important today? there is no difference between these questions.
The invention of algebra traces back to the ancient Babylonians. At the time, Egyptian, Greek, and Chinese mathematicians actually used geometry to solve math problems.
She did a lot but I think what you're looking for is her invention of the "variable" in algebra. That's right, all those x's, y's and z's that you hate are her invention.
Geometry is like a block from Algebra 1 to Algebra 2 and it contributes by teaching you the basics of the things you will learn in Algebra 2.