MathML Accessibility in Mozilla

This table below provides a list of MathML constructions and how they are read in Mozilla applications for various platforms:

Windows: MathML support is provided by NVDA, which actually delegates math reading to the proprietary MathPlayer plugin. We started exposing generic accessible objects for MathML in bug 920547 and so these can be retrieved by NVDA starting with Gecko 27.0 (Firefox 27.0 / Thunderbird 27.0 / SeaMonkey 2.24). Mozilla and NVDA developers essentially do not have control on how mathematical formulas are read, please contact Design Science for any inquiries.
Mac: MathML support is provided by VoiceOver. In bug 1175269 and bug 1001641, we relied on the WebKit's NSAccessibility MathML tree to expose the main constructions. Hence basic support is available in Gecko 41.0 (Firefox 41.0 / Thunderbird 41.0 / SeaMonkey 2.38) and we are still trying to keep in sync with WebKit/VoiceOver. Note that VoiceOver is proprietary so we do not have control on Improvements to MathML accessibility in VoiceOver.
Linux: MathML support is provided by the development version of Orca. We try to keep in sync with WebKit/ATK/Orca developments. At the moment, it is recommended to use a Nightly build to ensure that the accessible tree expected by Orca is correctly exposed.
AccessFu: MathML support in Android and Firefox OS is currently being implemented in our AccessFu module and a patch is available on bug 1163374. The table gives the output in mode "utterance" and description spoken last. Note that this is still very experimental.

MathML	Windows	Mac	Linux	AccessFu
Diagonal of a regular pentagon	Video	Video	Video	/
$x + 2$	__________ x plus 2 __________	x+2	x plus 2	x + y
$\frac{a}{b}$	__________ a over b __________	fraction start, a over b, end of fraction	fraction start. a over b. fraction end.	a numerator b denominator fraction
$\frac a b$
$\binom{a}{b}$	__________ a over b __________	fraction start, a over b, end of fraction. See WebKit's bug 146454	fraction without bar, start. a over b. fraction end.	a numerator b denominator fraction without bar
$a_{b}$	__________ a sub b __________	a, subscript b	a. subscript b	a base b subscript scripted
$a^{b}$	__________ a raised to the b __________	a, superscript b	a. superscript b	a base b superscript scripted
$a_{b}^{c}$	__________ a sub b to the c __________	a, subscript b, superscript c	a. subscript b superscript c	a base b subscript c superscript scripted
$\underset{b}{a}$	__________ a with b below __________	a. See WebKit's bug 146407	a. underscript b	a base b underscript scripted
$\overset{b}{a}$	__________ a with b above __________	a, with b underneath. See WebKit's bug 146407	a. overscript b	a base b overscript scripted
$a_{b}^{c}$	__________ a with b below and with c above __________	a, with c underneath. See WebKit's bug 146407	a. underscript b overscript c	a base b underscript c overscript scripted
$\sqrt{a}$	__________ the square root of a __________	square root of a, end of root	square root of a. root end.	a square root
$\sqrt{a b c}$	__________ the square root of the quantity a b c end root __________	square root of a, end of root. See WebKit's bug 146452	square root of a b c. root end.	a b c square root
$\sqrt[2]{a}$	__________ the second root of a __________	square root of a, end of root	square root of a. root end.	a base root index root
$\sqrt[3]{a}$	__________ the third root of a __________	cube root of a, end of root	cube root of a. root end.	a base 3 root index root
$\sqrt[3 + 4]{a}$	__________ the 3 plus 4 root of a __________	3 + 4 root of a, end of root	3 plus 4 root of a. root end.	a base 3+4 root index root
$\sqrt[\frac{3}{4}]{a}$	__________ the 3 over 4 root of a __________	fraction start, 3 over 4, end of fraction root of a, end of root	fraction start. 3 over 4. fraction end. root of a. root end.	a base 3 numerator 4 denominator root-index fraction
$\sqrt[\sqrt{b}]{a}$	__________ the the square root of b root of a __________	square root of b, end of root root of a, end of root	square root of b. root end. root of a. root end.	a base b square root root-index root
$\sqrt{\frac{a}{b}}$	__________ the square root of the quantity a over b end root __________	square root of fraction start, a over b, end of fraction, end of root	square root of fraction start. a over b. fraction end. root end.	a numerator b denominator fraction square root
$\sqrt[3]{\frac{a}{b}}$	__________ the third root of a over b end root __________	cube root of fraction start a over b, end of fraction, end of root	cube root of fraction start. a over b. fraction end. root end.	a numerator b denominator fraction base 3 root-index root
$\sqrt[c + d]{\frac{a}{b}}$	__________ the c plus d root of a over b end root __________	c + d root of fraction start, a over b, end of fraction, end of root	c plus d root of fraction start. a over b. fraction end. root end.	a numerator b denominator fraction base c + d root-index root
$\sqrt[b]{a}$	__________ the b root of a __________	b root of a, end of root	b root of a. root end.	a base b root index root
$\begin{matrix} a & b & c \\ d & e & f \end{matrix}$	__________ table with 2 rows and 3 columns row 1 column 1 a column 2 b column 3 c row 2 column 1 d column 2 e column 3 f end table __________	Table start, Row 1, Column 1, a, Row 1, Column 2, b, Row 1, Column 3, c, Row 2, Column 1, d, Row 2, Column 2, e, Row 2, Column 3, f, table end	math table with 2 rows 3 columns. row 1. a. b. c. row 2. d. e. f. table end.	a column 1 row 1 b column 2 c column 3 d column 1 row 2 e column 2 f column 3 math table with 3 columns and 2 rows
${}_{d}^{e}a_{b}^{c}$	__________ multi-scripts with base a and 1 prescript and 1 post-script pre-scripts lower d upper e post-scripts lower b upper c end multi-scripts __________	Not supported yet. See bug 1176983 and WebKit's bug 146454	a. pre-subscript d pre-superscript e subscript b.superscript c	a base b subscript c superscript d presubscript e presuperscript scripted
$a_{b}^{c}_{d}^{e}$	__________ multi-scripts with base a and 2 post-scripts lower b upper c lower d upper e end multi-scripts __________	Not supported yet. See bug 1176983 and WebKit's bug 146454	a. subscript b superscript c subscript d superscript e	a base b subscript c superscript d subscript e superscript scripted
${}_{b}^{c}_{d}^{e}a$	__________ multi-scripts with base a and 2 prescripts and 0 post-scripts pre-scripts lower b upper c lower d upper e post-scripts __________	Not supported yet. See bug 1176983 and WebKit's bug 146454	a. pre-subscript b pre-superscript c pre-subscript d pre-superscript e	a base b presubscript c presuperscript d presubscript e presuperscript scripted
${}_{e}^{f}^{g}a_{b}_{c}^{d}$	__________ multi-scripts with base a and 2 prescripts and 2 post-scripts pre-scripts lower e upper f upper g post-scripts lower b lower c upper d end multi-scripts __________	Not supported yet. See bug 1176983 and WebKit's bug 146454	a. pre-subscript e pre-superscript f pre-superscript g subscript b subscript c superscript d	a base b subscript c subscript d superscript e presubscript f presuperscript g presuperscript scripted
$x + y$	__________ long division symbol enclosing x plus y end symbol __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a long division sign	x + y long division enclosed
$x + y$	__________ long division symbol enclosing x plus y end symbol __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a long division sign	x + y long division enclosed
$x + y$	__________ actuarial symbol enclosing x plus y end symbol __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: an actuarial symbol	x + y actuarial enclosed
$x + y$	__________ x plus y __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a phasor angle	x + y phasor angle enclosed
$x + y$	__________ the square root of the quantity x plus y end root __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a radical	x + y radical enclosed
$x + y$	__________ begin box x plus y end box __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a box	x + y box enclosed
$x + y$	__________ begin rounded box x plus y end rounded box __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a rounded box	x + y rounded box enclosed
$x + y$	__________ begin circle x plus y end circle __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a circle	x + y circle enclosed
$x + y$	__________ line left of x plus y end line __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a line on the left	x + y left enclosed
$x + y$	__________ line right of x plus y end line __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a line on the right	x + y right enclosed
$x + y$	__________ line above x plus y end line __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a line at the top	x + y top enclosed
$x + y$	__________ line below x plus y end line __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a line at the bottom	x + y bottom enclosed
$x + y$	__________ x plus y __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a line on the left and a line on the right	x + y right left enclosed
$x + y$	__________ x plus y __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a line at the bottom, a line on the left, a line on the right, and, a line at the top	x + y bottom top right enclosed
$x + y$	__________ cross out enclosing x plus y end crossout __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: an up diagonal strike	x + y up diagonal strike enclosed
$x + y$	__________ cross out enclosing x plus y end crossout __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a down diagonal strike	x + y down diagonal strike enclosed
$x + y$	__________ cross out enclosing x plus y end crossout __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a vertical strike	x + y up vertical strike enclosed
$x + y$	__________ cross out enclosing x plus y end crossout __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: a horizontal strike	x + y horizontal strike enclosed
$x + y$	__________ x plus y __________	Not supported. See WebKit's bug 146543.	x plus y Enclosed by: an arabic factorial symbol	x + y up madruwb enclosed
$(a, b, c, d, e)$	__________ open a comma b comma c comma d comma e close __________	Not supported. See bug 1176970 ; Safari+VoiceOver reads "(, abcde, )".	left paren. a comma. b comma. comma. d comma. e. right paren	Not supported.
$\{a; b; c; d; e\}$	__________ the set a ; b ; c ; d ; e end set __________	Not supported. See bug 1176970 ; Safari+VoiceOver reads "{, abcde, }".	left brace. a semicolon. b semicolon. c semicolon. d semicolon. e. right brace	Not supported.
$\{a; b; c; d; e\}$	__________ the set a ; b ; c comma d comma e end set __________	Not supported. See bug 1176970 ; Safari+VoiceOver reads "{, abcde, }".	left brace. a semicolon. b semicolon. c comma. d comma. e. right brace	Not supported.
$A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z$			Not exposed to Orca. See bug 1182469.
$A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z$			Not exposed to Orca. See bug 1182469.
$A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z$			Not exposed to Orca. See bug 1182469.
$(x ≪̸ y) \land (X \subset⃒ Y)$
Torture Test 1: $x^{2} y^{2}$	__________ x squared y squared __________	x squared y squared	x. superscript 2. y. superscript 2.	x base 2 superscript scripted y base 2 superscript scripted
Torture Test 2: ${}_{2}F_{3}$	__________ multi-scripts with base cap f and 1 prescript and 1 post-script pre-scripts lower 2 post-scripts lower 3 end multi-scripts __________	Not supported yet. See bug 1176983 and WebKit's bug 146454	F. pre-subscript 2 subscript 3	F base 3 subscript 2 presubscript scripted
Torture Test 3: $\frac{x + y^{2}}{k + 1}$	__________ fraction x plus y squared over k plus 1 end fraction __________	fraction start, x + y squared over k + 1, end of fraction	fraction start. x plus y. superscript 2. over k plus 1. fraction end.	x + y base 2 superscript scripted numerator k + 1 denominator fraction
Torture Test 4: $x + y^{\frac{2}{k + 1}}$	__________ x plus y raised to the fraction 2 over k plus 1 end fraction power __________	x + y, superscript fraction start, 2 over k + 1, end of fraction	x plus y. superscript fraction start. 2 over k plus 1. fraction end.	x + y base 2 numerator k + 1 denominator fraction superscript scripted
Torture Test 5: $\frac{a}{b / 2}$	__________ fraction a over b divided by 2 end fraction __________	fraction start, a over b / 2, end of fraction	fraction start. a over b slash 2. fraction end.	a numerator b / 2 denominator fraction
Torture Test 6: $a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + \frac{1}{a_{4}}}}}$	__________ a sub 0 plus fraction 1 all over a sub 1 plus fraction 1 all over a sub 2 plus fraction 1 all over a sub 3 plus fraction 1 over a sub 4 end fraction end fraction end fraction end fraction __________	a, subscript 0, + fraction start, 1 over a, subscript 1, + fraction start, 1 over a, subscript 2, + fraction start, 1 over a, subscript 3, + fraction start, 1 over a, subscript 4, fraction end, fraction end, fraction end, fraction end	a. subscript 0. plus fraction start. 1 over a. subscript 1. plus fraction start. 1 over a. subscript 2. plus fraction start. 1 over a. subscript 3. plus fraction start. 1 over a. subscript 4. fraction end. fraction end. fraction end. fraction end.	a base 0 subscript scripted + 1 numerator a base 1 subscript scripted + 1 numerator a base 2 subscript scripted + 1 numerator a base 3 subscript scripted + 1 numerator a base 3 subscript scripted + 1 numerator a base 4 subscript scripted denominator fraction denominator fraction denominator fraction denominator fraction
Torture Test 7: $a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + \frac{1}{a_{4}}}}}$	__________ a sub 0 plus fraction 1 all over a sub 1 plus fraction 1 all over a sub 2 plus fraction 1 all over a sub 3 plus fraction 1 over a sub 4 end fraction end fraction end fraction end fraction __________	a, subscript 0, + fraction start, 1 over a, subscript 1, + fraction start, 1 over a, subscript 2, + fraction start, 1 over a, subscript 3, + fraction start, 1 over a, subscript 4, fraction end, fraction end, fraction end, fraction end	a. subscript 0. plus fraction start. 1 over a. subscript 1. plus fraction start. 1 over a. subscript 2. plus fraction start. 1 over a. subscript 3. plus fraction start. 1 over a. subscript 4. fraction end. fraction end. fraction end. fraction end.	a base 0 subscript scripted + 1 numerator a base 1 subscript scripted + 1 numerator a base 2 subscript scripted + 1 numerator a base 3 subscript scripted + 1 numerator a base 3 subscript scripted + 1 numerator a base 4 subscript scripted denominator fraction denominator fraction denominator fraction denominator fraction
Torture Test 8: $(\binom{n}{k / 2})$	No output	( fraction start, n over k / 2, end of fraction, ) See WebKit's bug 146454	left paren fraction without bar, start. n over k slash 2. fraction end. right paren	( open-fence n numerator k / 2 denominator fraction without bar ) close-fence
Torture Test 9: $(\binom{p}{2}) x^{2} y^{p - 2} - \frac{1}{1 - x} \frac{1}{1 - x^{2}}$	No output	( fraction start, p over 2, end of fraction, ), x squared y, superscript p - 2, - fraction start, 1 over 1 - x, end of fraction, fraction start, 1 over 1 - x, squared, end of fraction See WebKit's bug 146454	left paren fraction without bar, start. p over 2. fraction end. right paren x. superscript 2. y. superscript p minus 2. minus fraction start. 1 over 1 minus x. fraction end. fraction start. 1 over 1 minus x. superscript 2. fraction end.
Torture Test 10: $\sum_{\binom{0 \leq i \leq m}{0 < j < n}} P (i, j)$	__________ sum over fraction 0 is less than or equal to i is less than or equal to m over 0 is less than j is less than n end fraction of cap pee of open i comma j close __________	∑ P (i, j) See WebKit's bug 146407	sum. underscript fraction without bar, start. 0 less than or equal to i less than or equal to m over 0 less than j less than n. fraction end. P left paren i comma j right paren
Torture Test 11: $x^{2 y}$	__________ x raised to the 2 y power __________	x, superscript 2 y	x. superscript 2 y.
Torture Test 12: $\sum_{i = 1}^{p} \sum_{j = 1}^{q} \sum_{k = 1}^{r} a_{i j} b_{j k} c_{k i}$	__________ the sum from i is equal to 1 to pee of the sum from j is equal to 1 to q of the sum from k is equal to 1 to r of a sub i j end sub b sub j k end sub c sub k i end sub __________	∑, with p underneath, ∑, with q underneath, ∑, with r underneath, a, subscript i j, b, subscript j k, c, subscript k i See WebKit's bug 146407	sum. underscript i equals 1. overscript p. sum. underscript j equals 1. overscript q. sum. underscript k equals 1. overscript r. a. subscript i j. b. subscript j k. c. subscript k i.
Torture Test 13: $\sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + x}}}}}}}$	__________ the square root of the quantity 1 plus the square root of the quantity 1 plus the square root of the quantity 1 plus the square root of the quantity 1 plus the square root of the quantity 1 plus the square root of the quantity 1 plus the square root of the quantity 1 plus x end root end root end root end root end root end root end root __________	square root of 1, end of root See WebKit's bug 146452	square root of 1 plus square root of 1 plus square root of 1 plus square root of 1 plus square root of 1 plus square root of 1 plus square root of 1 plus x. root end. root end. root end. root end. root end. root end. root end.
Torture Test 14: $(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) {\| φ (x + i y) \|}^{2} = 0$	__________ open the second partialderivative with respect to x of plus fraction the partial derivative of squared over the partial derivative of y squared end fraction close times the absolute value of phi open x plus i y close end absolute value squared is equal to 0 __________	( fraction start, ∂ squared over ∂ x squared, end of fraction, + fraction start, ∂ squared over ∂ y squared, end of fraction, ), \| ϕ ( x + i y ) \| squared = 0	left paren fraction start. partial differential. superscript 2. over partial differential x. superscript 2. fraction end. plus fraction start. partial differential. superscript 2. over partial differential y. superscript 2. fraction end. right paren vertical bar φ left paren x plus i y right paren vertical bar. superscript 2. equals 0
Torture Test 15: $2^{2^{2^{x}}}$	__________ 2 raised to the 2 raised to the 2 raised to the x power power __________	2 to the 2 to the 2, superscript x	2. superscript 2. superscript 2. superscript x.
Torture Test 16: $\int_{1}^{x} \frac{d t}{t}$	__________ the integral from 1 to x of times fraction d t over t end fraction __________	integral, subscript 1, superscript x, fraction stary, d t over t, end of fraction	integral. subscript 1. superscript x. fraction start. d t over t. fraction end.
Torture Test 17: $\iint_{D} d x d y$	__________ double integral over cap d of d x d y __________	∬, subscript D, dxdy WebKit's bug 146505	double integral. subscript D. d x d y
Torture Test 18: $f (x) = {\begin{matrix} 1 / 3 & if 0 \leq x \leq 1; \\ 2 / 3 & if 3 \leq x \leq 4; \\ 0 & elsewhere. \end{matrix}$	__________ f of x is equal to left brace table with 3 rows and 2 columns row 1 column 1 1 divided by 3 column 2 if 0 is less than or equal to x is less than or equal to 1 ; row 2 column 1 2 divided by 3 column 2 if 3 is less than or equal to x is less than or equal to 4 ; row 3 column 1 0 column 2 elsewhere. end table __________	f ( x ) = { Table start, Row 1, Column 1, 1 / 3, Row 1, Column 2, if 0 ≤ x ≤ 1;, Row 2, Column 1, 2 / 3, Row 2, Column 2, if 3 ≤ x ≤ 4;, Row 3, Column 1, 0, Row 3, Column 2, elsewhere., table end	f left paren x right paren equals left brace math table with 3 rows 2 columns. row 1. 1 slash 3. if 0 less than or equal to x less than or equal to 1 semicolon. row 2. 2 slash 3. if 3 less than or equal to x less than or equal to 4 semicolon. row 3. 0. elsewhere. table end.'
Torture Test 19: $\overset{\overset{k times}{⏞}}{x + ... + x}$	__________ modified x plus ... positive x with modified over brace with k times above above __________	x + ... + x, with ⏞, with k times underneath underneath See WebKit's bug 146407	x plus ... plus x. overscript top brace. overscript k times.
Torture Test 20: $y_{x^{2}}$	__________ y sub x squared end sub __________	y, subscrit x squared	y. subscript x. superscript 2.
Torture Test 21: $\sum_{p prime} f (p) = \int_{t > 1} f (t) d π (t)$	__________ sum over pee prime of f of pee is equal to integral over t is greater than 1 of f of t d pi of open t close __________	∑ f ( p ) = ∫, subscript t > 1, f ( t ) d π ( t ) See WebKit's bug 146407	sum. underscript p prime. f left paren p right paren equals integral. subscript t greater 1. f left paren t right paren d π left paren t right paren
Torture Test 22: ${\underset{\underset{k + ℓ elements}{⏟}}{\overset{\overset{k a's}{⏞}}{a, ..., a}, \overset{\overset{ℓ b's}{⏞}}{b, ..., b}}}$	__________ the set modified modified a comma ... comma a with modified over brace with k a apostraphes above above comma modified b comma ... comma b with modified over brace with script l b apostraphes above above with modified under brace with k plus script l elements below below end set __________	{ a , ... , a, with ⏞, with k a 's underneath, , b , ... , b with ⏞, with ℓ b's underneath underneath, } See WebKit's bug 146407 and WebKit's bug 146631.	left brace a comma ... comma a. overscript top brace. overscript k a 's. comma b comma ... comma b. overscript top brace. overscript l b 's. underscript bottom brace. underscript k plus l elements. right brace
Torture Test 23: $(\begin{matrix} (\begin{matrix} a & b \\ c & d \end{matrix}) & (\begin{matrix} e & f \\ g & h \end{matrix}) \\ 0 & (\begin{matrix} i & j \\ k & l \end{matrix}) \end{matrix})$	__________ matrix with 2 rows and 2 columns row 1 column 1 matrix with 2 rows and 2 columns row 1 a b row 2 c d end matrix column 2 matrix with 2 rows and 2 columns row 1 e f row 2 g h end matrix row 2 column 1 0 column 2 matrix with 2 rows and 2 columns row 1 i j row 2 k l end matrix end matrix __________	( Table start, Row 1, Column 1, ( Table start, Row 1, Column 1, a, Row 1, Column 2, b, Row 2, Column 2, c, Row 2 Column 2, d, table end, ), Row 1, Column 2, ( Table start, Row 1, Column 1, e, Row 1, Column 2, f, Row 2, Column 2, g, Row 2 Column 2, h, table end, ), Row 2, Column 1, 0, Row 2, Column 2, ( Table start, Row 1, Column 1, i, Row 1, Column 2, j, Row 2, Column 2, k, Row 2 Column 2, l, table end, ), table end, )	left paren math table with 2 rows 2 columns. row 1. left paren nested math table with 2 rows 2 columns. row 1. a. b. row 2. c. d. nested table end. right paren. left paren nested math table with 2 rows 2 columns. row 1. e. f. row 2. g. h. nested table end. right paren. row 2. 0. left paren nested math table with 2 rows 2 columns. row 1. i. j. row 2. k. l. nested table end. right paren. table end. right paren
Torture Test 24: $\det \| \begin{matrix} c_{0} & c_{1} & c_{2} & \dots & c_{n} \\ c_{1} & c_{2} & c_{3} & \dots & c_{n + 1} \\ c_{2} & c_{3} & c_{4} & \dots & c_{n + 2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ c_{n} & c_{n + 1} & c_{n + 2} & \dots & c_{2 n} \end{matrix} \| > 0$	__________ determinant with 5 rows and 5 columns row 1 column 1 c sub 0 column 2 c sub 1 column 3 c sub 2 column 4 dot dot dot column 5 c sub n row 2 column 1 c sub 1 column 2 c sub 2 column 3 c sub 3 column 4 dot dot dot column 5 c sub n plus 1 end sub row 3 column 1 c sub 2 column 2 c sub 3 column 3 c sub 4 column 4 dot dot dot column 5 c sub n plus 2 end sub row 4 column 1 vertical ellipsis column 2 vertical ellipsis column 3 vertical ellipsis column 4 column 5 vertical ellipsis row 5 column 1 c sub n column 2 c sub n plus 1 end sub column 3 c sub n plus 2 end sub column 4 dot dot dot column 5 c sub 2 n end sub end determinant is greater than 0 __________	det \| Table Start, Row 1, Column 1, c subscript 0, Row 1, Column 2, c subscript 1, Row 1, Column 3, c subscript 2, Row 1, Column 4, ..., Row 1, Column 5, c subscript n, Row 2, Column 1, c subscript 1, Row 2, Column 2, c subscript 2, Row 2, Column 3, c subscript 3, Row 2, Column 4, ..., Row 2, Column 5, c subscript n + 1, Row 3, Column 1, c subscript 2, Row 3, Column 2, c subscript 3, Row 3, Column 3, c subscript 4, Row 3, Column 4, ..., Row 3, Column 5, c subscript n + 2, Row 4, Column 1, c ⋮, Row 4, Column 2, ⋮, Row 4, Column 3, ⋮, Row 4, Column 4, blank, Row 4, Column 5, ⋮, Row 5, Column 1, c subscript n, Row 5, Column 2, c subscript n + 1, Row 5, Column 3, c subscript n + 2, Row 5, Column 4, ..., Row 5, Column 5, c subscript 2 n, table end, \| > 0	det vertical bar math table with 5 rows 5 columns. row 1. c. subscript 0. c. subscript 1. c. subscript 2. horizontal ellipsis. c. subscript n. row 2. c. subscript 1. c. subscript 2. c. subscript 3. horizontal ellipsis. c. subscript n plus 1. row 3. c. subscript 2. c. subscript 3. c. subscript 4. horizontal ellipsis. c. subscript n plus 2. row 4. vertical ellipsis. vertical ellipsis. vertical ellipsis. blank. vertical ellipsis. row 5. c. subscript n. c. subscript n plus 1. c. subscript n plus 2. horizontal ellipsis. c. subscript 2 n. table end. vertical bar greater than 0
Torture Test 25: $y_{x_{2}}$	__________ y sub x sub 2 end sub __________	y, subscript x, subscript 2,	y. subscript x. subscript 2.
Torture Test 26: $x_{92}^{31415} + π$	__________ x sub 92 to the 31400 fifteenth plus pi __________	x, subscript 92, to the 31415, + π	x. subscript 92. superscript 31415. plus π
Torture Test 27: $x_{y_{b}^{a}}^{z_{c}^{d}}$	__________ x with subscript y sub b to the a and with superscript z sub c to the d end sub-superscript __________	x, subscript y, subscript b, superscript a, superscript z, subscript c, superscript d	x. subscript y. subscript b. superscript a. superscript z. subscript c. superscript d.
Torture Test 28: $y_{3}^{‴}$	__________ y with subscript 3 and with superscript triple prime end sub-superscript __________	y, subscript 3, superscript ‴	y. subscript 3. superscript triple prime.	y base 3 subscript ‴ superscript scripted
${\frac {d}{t}}=\varphi ={\frac {1+{\sqrt {5}}}{2}}\ ,$
$d = t \times φ = R \sqrt{\frac{5 + \sqrt{5}}{2}} = 2 R \cos 18^{\circ} = 2 R \cos \frac{π}{10} \approx 1.90211303259 R .$

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