Notation¶
Commonly used math symbols in machine learning texts. Check more on the referenced sites 1 2.
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Basic math symbols¶
Symbol |
Symbol Name |
Meaning / definition |
Example |
= |
equals sign |
equality |
5 = 2+3 5 is equal to 2+3 |
≠ |
not equal sign |
inequality |
5 ≠ 4 5 is not equal to 4 |
≈ |
approximately equal |
approximation |
sin(0.01) ≈ 0.01, x ≈ y means x is approximately equal to y |
> |
strict inequality |
greater than |
5 > 4 5 is greater than 4 |
< |
strict inequality |
less than |
4 < 5 4 is less than 5 |
≥ |
inequality |
greater than or equal to |
5 ≥ 4, x ≥ y means x is greater than or equal to y |
≤ |
inequality |
less than or equal to |
4 ≤ 5, x ≤ y means x is less than or equal to y |
( ) |
parentheses |
calculate expression inside first |
2 × (3+5) = 16 |
[ ] |
brackets |
calculate expression inside first |
[(1+2)×(1+5)] = 18 |
plus sign |
addition |
1 + 1 = 2 |
|
− |
minus sign |
subtraction |
2 − 1 = 1 |
± |
plus - minus |
both plus and minus operations |
3 ± 5 = 8 or -2 |
± |
minus - plus |
both minus and plus operations |
3 ∓ 5 = -2 or 8 |
asterisk |
multiplication |
2 * 3 = 6 |
|
× |
times sign |
multiplication |
2 × 3 = 6 |
⋅ |
multiplication dot |
multiplication |
2 ⋅ 3 = 6 |
÷ |
division sign / obelus |
division |
6 ÷ 2 = 3 |
/ |
division slash |
division |
6 / 2 = 3 |
— |
horizontal line |
division / fraction |
# |
mod |
modulo |
remainder calculation |
7 mod 2 = 1 |
. |
period |
decimal point, decimal separator |
2.56 = 2+56/100 |
ab |
power |
exponent |
23 = 8 |
a^b |
caret |
exponent |
2 ^ 3 = 8 |
√a |
square root |
√a ⋅ √a = a |
√9 = ±3 |
3√a |
cube root |
3√a ⋅ 3√a ⋅ 3√a = a |
3√8 = 2 |
4√a |
fourth root |
4√a ⋅ 4√a ⋅ 4√a ⋅ 4√a = a |
4√16 = ±2 |
n√a |
n-th root (radical) |
for n=3, n√8 = 2 |
|
% |
percent |
1% = 1/100 |
10% × 30 = 3 |
‰ |
per-mille |
1‰ = 1/1000 = 0.1% |
10‰ × 30 = 0.3 |
ppm |
per-million |
1ppm = 1/1000000 |
10ppm × 30 = 0.0003 |
ppb |
per-billion |
1ppb = 1/1000000000 |
10ppb × 30 = 3×10-7 |
ppt |
per-trillion |
1ppt = 10-12 |
10ppt × 30 = 3×10-10 |
Algebra¶
Symbol |
Name |
Description |
Example |
\((f ∘ g)\) |
composite function |
a nested function |
(f ∘ g)(x) = f(g(x)) |
\(∆\) |
delta |
change / difference |
∆x = x_1 - x_0 |
\(e\) |
Euler’s number |
e = 2.718281828 |
s = frac{1}{1+e^{-z}} |
\(\sum\) |
summation |
sum of all values |
∑ x_i = x_1 + x_2 + x_3 |
\(\prod\) |
capital pi |
product of all values |
∏ x_i = x_1∙x_2∙x_3 |
\(\epsilon\) |
epsilon |
tiny number near 0 |
lr = 1e-4 |
Symbol |
Symbol Name |
Meaning / definition |
Example |
x |
x variable |
unknown value to find |
when 2x = 4, then x = 2 |
≡ |
equivalence |
identical to |
|
≜ |
equal by definition |
equal by definition |
|
:= |
equal by definition |
equal by definition |
|
~ |
approximately equal |
weak approximation |
11 ~ 10 |
≈ |
approximately equal |
approximation |
sin(0.01) ≈ 0.01 |
∝ |
proportional to |
proportional to |
y ∝ x when y = kx, k constant |
∞ |
lemniscate |
infinity symbol |
|
≪ |
much less than |
much less than |
1 ≪ 1000000 |
≫ |
much greater than |
much greater than |
1000000 ≫ 1 |
( ) |
parentheses |
calculate expression inside first |
2 * (3+5) = 16 |
[ ] |
brackets |
calculate expression inside first |
[(1+2)*(1+5)] = 18 |
{ } |
braces |
set |
|
⌊x⌋ |
floor brackets |
rounds number to lower integer |
⌊4.3⌋ = 4 |
⌈x⌉ |
ceiling brackets |
rounds number to upper integer |
⌈4.3⌉ = 5 |
x! |
exclamation mark |
factorial |
4! = 1*2*3*4 = 24 |
x |
|
vertical bars |
absolute value |
-5 | = 5
|
f (x) |
function of x |
maps values of x to f(x) |
f (x) = 3x+5 |
(f ∘ g) |
function composition |
(f ∘ g) (x) = f (g(x)) |
f (x)=3x,g(x)=x-1 ⇒(f ∘ g)(x)=3(x-1) |
(a,b) |
open interval |
(a,b) = {x | a < x < b} |
x∈ (2,6) |
[a,b] |
closed interval |
[a,b] = {x | a ≤ x ≤ b} |
x ∈ [2,6] |
∆ |
delta |
change / difference |
∆t = t1 - t0 |
∆ |
discriminant |
Δ = b2 - 4ac |
|
∑ |
sigma |
summation - sum of all values in range of series |
∑ xi= x1+x2+…+xn |
∑∑ |
sigma |
double summation |
# |
∏ |
capital pi |
product - product of all values in range of series |
∏ xi=x1∙x2∙…∙xn |
e |
e constant / Euler’s number |
e = 2.718281828… |
e = lim (1+1/x)x , x→∞ |
γ |
Euler-Mascheroni constant |
γ = 0.5772156649… |
Calculus¶
Symbol |
Name |
Description |
Example |
\(x'\) |
derivative |
first derivative |
(x^2)’ = 2x |
\(x''\) |
second derivative |
second derivative |
(x^2)’’ = 2 |
\(\lim\) |
limit |
function value as x approaches 0 |
|
\(∇\) |
nabla |
gradient |
∇f(a,b,c) |
Linear algebra¶
Symbol |
Name |
Description |
Example |
\([ ]\) |
brackets |
matrix or vector |
\(M = [1 3 5]\) |
\(\cdot\) |
dot |
dot product |
\((Z = X \cdot W\) |
\(\odot\) |
hadamard |
hadamard product |
\(A = B \odot C\) |
\(X^T\) |
transpose |
matrix transpose |
\(W^T \cdot X\) |
\(\vec x\) |
vector |
vector |
\(v = [1 2 3]\) |
\(X\) |
matrix |
capitalized variables are matrices |
\(X, W, B\) |
\(\hat x\) |
unit vector |
vector of magnitude 1 |
\(\hat x = [0.2 0.5 0.3]\) |
Probability¶
Symbol |
Name |
Description |
Example |
\(P(A)\) |
probability |
probability of event A |
P(x=1) = 0.5 |
Set theory¶
Symbol |
Name |
Description |
Example |
\({ }\) |
set |
list of distinct elements |
S = {1, 5, 7, 9} |
Statistics¶
Symbol |
Name |
Description |
Example |
\(μ\) |
population mean |
mean of population values |
|
\(\bar x\) |
sample mean |
mean of subset of population |
|
\(σ^2\) |
population variance |
variance of population value |
|
\(s^2\) |
sample variance |
variance of subset of population |
|
\(σ_X\) |
standard deviation |
population standard deviation |
|
\(s\) |
sample std dev |
standard deviation of sample |
|
\(ρX\) |
correlation |
correlation of variables X and Y |
|
\(\tilde x\) |
median |
median value of variable x |
References