How to read the math

The symbols in this course, said in plain words. Come back whenever one stops you.

Math notation is shorthand, nothing more. Each symbol below stands for a short sentence. You’ll meet each one again later, most with a worked example, so a quick look now is plenty.

Symbol	Say it	What it means, with an example
$a_1,\ a_2,\ a_i$	“a one”, “a sub i”	The first, the second, or the i-th number in the list $a$ . The little letter i stands for any position. If $a=(2,1,3)$ , then $a_1=2$ and $a_3=3$ .
$w_1x,\ 2b$	“w one x”, “two b”	Two things written side by side are multiplied. So $w_1x$ means $w_1\times x$ . And $2b$ means $2\times b$ .
$\mathbf a\cdot\mathbf b$	“a dot b”	The dot product of the lists $\mathbf a$ and $\mathbf b$ : multiply them pair by pair, then add (lesson 3).
$W_{ij}$	“W i j”	The number in row i, column j of the grid $W$ .
$\sum$	“sum” (capital sigma)	Add them all up. Small print under and over it says where to start and stop: $\sum_{i=1}^{3}a_i=a_1+a_2+a_3=2+1+3=6$ .
$e$	“e”	A fixed number, about 2.718, the way $\pi$ is about 3.142. Your calculator’s $e^x$ key raises $e$ to a power: $e^2\approx7.389$ .
$2^3,\ e^{x}$	“2 to the 3rd”, “e to the x”	Multiply that many copies of the number together: $2^3=2\times2\times2=8$ . Anything to the power 0 is 1. A minus sign in the power means “1 divided by”: $2^{-3}=1/8$ . For powers that aren’t whole numbers, like $e^{1.5}$ , use your calculator’s key.
$\log_2,\ \ln$	“log base 2”, “L N”	Undo a power. $\log_2 8$ asks “2 to what power gives 8?” The answer is 3. ln asks the same about $e$ . Most calculators have ln but no log₂ key, so use $\log_2 x=\ln x\div\ln 2$ .
$\sqrt{\ },\ \lt,\ \gt$	“square root of”, “less than”, “more than”	$\sqrt{16}=4$ , because $4\times4=16$ . $2\lt3$ and $3\gt2$ ; a negative number is $\lt0$ .
$\|\mathbf a\|$	“the length of a”	How long the arrow $\mathbf a$ is: square each number, add them up, take the square root. $\|(3,4)\|=\sqrt{9+16}=5$ .
$\theta,\ \varphi$	“theta”, “phi”	Greek letters used as names for angles.
$\cos,\ \cos^{-1}$	“cosine”, “inverse cosine”	cos turns an angle into a number from −1 to 1. cos⁻¹ turns it back; it’s also called arccos or acos, and here the −1 is part of the name, not “1 divided by”. For example, $\cos 60^\circ=0.5$ , $\cos^{-1}0.5=60^\circ$ . Set the calculator to DEG for degrees or RAD for radians. One radian is about 57.3°.
$0.5\times(2,4)$	a number times a list	Multiply each number in the list: $0.5\times(2,4)=(1,2)$ . Two lists add position by position: $(1,2)+(3,0)=(4,2)$ .
$K^{\top}$	“K transpose”	Write each row of the grid $K$ as a column: row 1 becomes column 1, row 2 becomes column 2, and so on.
$\sigma(z)$	“sigma of z” (small sigma)	The sigmoid, a smooth yes or no. It turns any number into a chance between 0 and 1, and $\sigma(0)=0.5$ . The formula is $\sigma(z)=1/(1+e^{-z})$ .
$\hat y,\ t$	“y-hat”, “t”	$\hat y$ is the network’s answer; $t$ is the truth it should have said.
$\leftarrow$	“becomes”	Replace the left side with the right side. $w\leftarrow w-1$ means w goes down by 1.
$\eta$	“eta”	The step size used in training.
$\dfrac{\partial A}{\partial B}$	“the slope of A with respect to B” (experts say “the partial”)	How fast $A$ changes when $B$ moves a tiny bit, with everything else held still. You can check one yourself: nudge $B$ by 0.001, see how much $A$ moves, and divide by 0.001. You’ll get very nearly the slope.
$\delta$	“delta”	One neuron’s shared slope, used in backpropagation (idea 5).
$\Rightarrow,\ \approx$	“so”, “about”	$\Rightarrow$ : so it follows that. $\approx$ : approximately equal.

The two sigmas are different letters: big $\sum$ adds things up, small $\sigma$ is the sigmoid.