#+TITLE: Mathematics
#+PROPERTY: ANKI_DECK math

** Matrix multiplication in summation notation                       :matrix:
   :PROPERTIES:
   :ANKI_NOTE_TYPE: Basic
   :ANKI_PREPEND_HEADING: t
   :ANKI_NOTE_ID: 1763204653615
   :ANKI_NOTE_HASH: daaf9ec7d2a4d14635f10c618f0bdde4
   :END:

\[X, Y \in \mathcal{M}^{n\times n}\]
\[XY = Z\]
\[Z_{ij} = ?\]

*** Back

\[Z_{ij} = \sum^n_{k=1} X_{ik} Y_{kj} \]

** What is the formula for the sum of all interior angles of a polygon? :geometry:
   :PROPERTIES:
   :ANKI_NOTE_TYPE: Basic
   :ANKI_NOTE_ID: 1763204636204
   :ANKI_NOTE_HASH: 8e81158ed3b7926514cbf7b6c2cfd616
   :END:

\[180^\circ (n -2)\]

** What is the formula for a single interior angle in a regular polygon? :geometry:
   :PROPERTIES:
   :ANKI_NOTE_TYPE: Basic
   :ANKI_NOTE_ID: 1763204636206
   :ANKI_NOTE_HASH: d0d25098eab756bc7974d3d3e20f1fd4
   :END:

\[\dfrac{180^\circ (n -2)}{n}\]

** What is the formula for the surface area of a cone? Do you remember the derivation? :geometry:
   :PROPERTIES:
   :ANKI_NOTE_TYPE: Basic
   :ANKI_NOTE_ID: 1763204833159
   :ANKI_NOTE_HASH: 94aa8470da63fe643266bc4692cba3ed
   :END:

\[SA = \pi r^2 + \pi r l\]

** What is the formula for compound interest?
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1763522858226
:ANKI_NOTE_HASH: 505c0419327311c4d52b675864597c1f
:END:

\[A_n = P(1+r)^n\]
where:
$A_n$ is the $n\textup{th}$ accumulation;
$P$ is the principal amount;
$r$ is the interest rate per unit time
$n$ is the /units of time/


** What is the formula for simple interest?
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1763522858228
:ANKI_NOTE_HASH: a1fe3c7a6c1e7d6590b48c64b0bf4a91
:END:

\[I = Prt\]
where:
$I$ is the /interest/ (not accumulated). 
$P$ is the principal amount;
$r$ is the interest rate per unit time
$t$ is the /units of time/

** Pi to 20 decimal places
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470918
:ANKI_NOTE_HASH: 9e98e2b8f2a813e9b3a00c6137f0f4bf
:END:

3.14159265358979323846

** $e$ to 10 decimal places
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470936
:ANKI_NOTE_HASH: 05b272bdb162d59b24993328d5e376b7
:END:

2.7182818285

** $\sqrt{2}$ to 5 decimal places
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470937
:ANKI_NOTE_HASH: 805ff7cdc5df5be01b5f0082130f902a
:END:

1.41421

** $14^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470939
:ANKI_NOTE_HASH: 63182e3d78731ae476a17ae9d3c7c86f
:END:

196

** $15^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470940
:ANKI_NOTE_HASH: 4c675c81648629aeec5f77a697c92595
:END:

225

** $16^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470941
:ANKI_NOTE_HASH: 51902dbb3880a03951a3c847c1c2072a
:END:

256

** $17^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470942
:ANKI_NOTE_HASH: 4bfb36b4f5789aa43e70da7d2fa94c34
:END:

289

** $18^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470943
:ANKI_NOTE_HASH: 55b37d0b92e0bc2968c0dfd143089815
:END:

324

** $19^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470944
:ANKI_NOTE_HASH: cb669865c17c3c66222dfcb7c710f173
:END:

361

** $20^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470946
:ANKI_NOTE_HASH: 549546724e16391413295cd48adcc77c
:END:

400

** $21^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470947
:ANKI_NOTE_HASH: c0e1f25ee4ef5429ca2116209cdd3ed8
:END:

441

** $22^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470948
:ANKI_NOTE_HASH: 438d2b2586ca7d7710e5b7293779309c
:END:

484

** $23^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470949
:ANKI_NOTE_HASH: 226b7397e383309755e4e936ba54de66
:END:

529

** $24^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470951
:ANKI_NOTE_HASH: c0d1d139e77265bc7015fb85d4bae3bb
:END:

576

** $25^2$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470952
:ANKI_NOTE_HASH: 8fc4adcc569b2ee5f10f433bb8ac92da
:END:

625

** $2^4$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470953
:ANKI_NOTE_HASH: 1fd762593fb160d3225d43c31dbf4084
:END:

16

** $2^5$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470954
:ANKI_NOTE_HASH: 233087c49101e696e66c0a1b9597a92e
:END:

32

** $2^6$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470955
:ANKI_NOTE_HASH: b194aea2e74606f111442b47a7cbfd52
:END:

64

** $2^7$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470956
:ANKI_NOTE_HASH: e5ca963310612156666317d6e1328fa8
:END:

128

** $2^8$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470957
:ANKI_NOTE_HASH: 70db2d407f492c277bd3aa50277e4648
:END:

256

** $2^9$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470958
:ANKI_NOTE_HASH: 009c10d032eed834fa92132a66e8dafa
:END:

512

** $2^{10}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470960
:ANKI_NOTE_HASH: c86a506a3dfe137304e4df3fc9501fe1
:END:

1024

** $2^{11}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470961
:ANKI_NOTE_HASH: 13c7d591da74304df095b9a947db9994
:END:

2048

** $2^{12}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470963
:ANKI_NOTE_HASH: 288fd36c80e7db317ec2dd8c069f7c21
:END:

4096

** $2^{13}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470964
:ANKI_NOTE_HASH: 9692756812cc29b9545d9df8f9ad9983
:END:

8192

** $2^{14}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470965
:ANKI_NOTE_HASH: 122accc3ee3da150789763188590d403
:END:

16384

** $2^{15}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470967
:ANKI_NOTE_HASH: c4b7e03fc3d4ff1d2eb8ef74ae42effd
:END:

32768

** $2^{16}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470968
:ANKI_NOTE_HASH: 2d48c4d6aa3d1f370ae649c525cb84c7
:END:

65536

** $2^{17}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470969
:ANKI_NOTE_HASH: ea99d8877adf125fcda1804dc1f67ced
:END:

131072

** $2^{18}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470970
:ANKI_NOTE_HASH: 13d69c3ed491372166d527511a1e42b1
:END:

262144

** $3^{3}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470971
:ANKI_NOTE_HASH: a56ca482f397bddbb9bc0ede5eca440a
:END:

27

** $3^{4}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470972
:ANKI_NOTE_HASH: d9d2da230e018e7c3ee3500d66df21e6
:END:

81

** $3^{5}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470973
:ANKI_NOTE_HASH: f87511bac53d89bc356b6dd2365091eb
:END:

243

** $3^{6}$
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1764225470974
:ANKI_NOTE_HASH: 6360766c34b6fa2556c535acb54d06b4
:END:

729

** What does an $m \times n$ matrix mean?
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1765171060301
:ANKI_NOTE_HASH: 8246ef6d990e7e9690d8b33982b4e48e
:END:

$m$ rows, $n$ columns.

** What is the *distance* $d$ from a point $\mathbf{x}_0$ to a hyperplane defined by $\mathbf{w}\cdot\mathbf{x} + b = 0$?
:PROPERTIES:
:ANKI_NOTE_TYPE: Basic
:ANKI_NOTE_ID: 1765171060303
:ANKI_NOTE_HASH: 0f5aaf2d679fc90768fcbbc16aa1f11e
:END:

\[d = \frac{|\mathbf{w}\cdot\mathbf{x}_0 + b|}{||\mathbf{w}||_2}\]

where
- $\mathbf{w}$ is the normal vector to the hyperplane
- $\cdot$ is the dot product (or inner product)
- $\mathbf{x}_0$ is the vector representing the given point in \(n\)-dimensional space.