Converting equations to English

1. The absolute value of a sum of n numbers, a-sub-1, a-sub-2 and so on, up to a-sub-n, divided by n, is less than or equal to the square root of the sum of the squares of those numbers, written a-sub-1 squared plus a-sub-2 squared, et cetera, through a-sub-n squared, divided by n. ("Et cetera", abbreviated "etc.", means "and so on.)

2. The nth root of the product of n numbers, a-sub-1, a-sub-2 and so on, up to a-sub-n, is less than or equal to the sum of those numbers divided by n.

3. The square root of one over n times the sum of the squares of n numbers, a-sub-i from i equals one through i equals n (or "from one to n"), is less than or equal to the nth root of the product of all those numbers a-sub-i.

4. Sine theta equals the tangent of theta times cosine of theta. Cosine theta equals cotangent theta times sine theta. (Sometimes we say "of theta" but sometimes we leave out the "of". We abbreviate tangent "tan", cosine "cos", cotangent "cot", arc tangent "arctan" or tan ^{-1}.)

5. The integral of x to the 4th power d x over x cubed plus one is one half x squared, plus one sixth times the natural logarithm of the expression x plus one squared divided by the expression x cubed minus x plus one, minus one over the square root of three times the arc tangent of two x minus one divided by the square root of three, plus a constant. (Note how the words don't make clear the grouping of terms very well - you need to point to the terms as you talk.)

6. The logarithm of x to base e is the natural logarithm of x, "l" "n" x. (Some people say "line x".) The logarithm of x to base 10 is the log of x. (Pronounced 'log' like a big piece of a tree trunk. We also say "natural log of x".)

7. x to the n (or "to the nth", or "to the nth power") times the nth derivative of y plus a-sub-1 times x to the power of n minus one times the n-minus-oneth derivative of y plus, and so on , continuing the series, a-sub-n-minus-one times x times the first derivative of y plus a-sub-n times y, equals f of x. (We generally write "f" on the left hand side of an equation defining it.)

8. The partial derivative of u with respect to x times the second (partial) derivative of u with respect to x plus the partial derivative of u with respect to y times the second (partial) derivative of u with respect to y plus u squared equals zero.

(For du/dx we often say "d" "u" "d" "x", because everyone understands the "with respect to". For partial derivatives, I don't remember what shortened version we might say. For du2/d2x, we are likely to say "the second derivative of u with respect to x" or "d" "u" squared "d" "x" squared.)

9. The integral of f of x dx from a to b [or from x equals a to x equals b) equals the integral of f of x dx from a to c plus the integral integral of f of x dx from c to b.