Symbols were indeed introduced for convenience reasons. Because of them we can write math and perform computations on paper much faster than if we had to use words. Try writing the quadratic formula or the law of cosines entirely in words and you will see if your friends can figure out what the resulting sentences mean. They will likely be confused for a while.

As for the reasons for particular symbols, to make them easy to remember. For example = was chosen for equals because the two lines in it are of equal length, while the symbol used for square roots is a modified version of the letter r which is the first letter in the term radical which refers to nth roots.

Math didn't always use symbols--when it was first imported from Arabs, algebraic "sentences" were written out in words. For example, instead of "x," the phrase "the unknown thing" was used. Try writing your math notes in plain Elnglish for a day, and you may understand why symbols were invented.

One thing to think about is how confusing it might become if we used words instead of symbols. Think for a minute about the subtraction symbol. In words, we could use any of the following phrases:

subtract

less than

minus

take away

decreased by

etc.

With symbols, there is only one way to write a subtraction sign, which makes it much easier! (That's one of the reasons why word problems are usually more difficult for students. So many ways to word the same math statement!)

Also, math is independent of your own language. Whether you speak English, Korean, Spanish, or Swahili, 2+2=4. By using symbols, math demonstrates that it is universal.

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Math is timeless in a way; the symbols have been used in various eras and with various languages but the symbols remain.

Sometimes in order to explain what we have said a symbol does (like a + sign means add the first part to the second part) you would end up with a novel instead of a single sheet of paper.

Math has a lot of abstract and theoretical concepts. If we didn’t come up with the idea of variables (substitute what you don’t know right now or what could change with a ‘ x ’) how could we get to our theoretical answer?

There has to be more that can be said beyond these concepts, but I think this is a good start.