Nemeth Spatial Arrangements
This workshop will show rules and examples for spatial arrangements from elementary
through college level transcription.
Spatial Problems

See Nemeth Code Section 185.

There are blank lines above and below spatial problems, even when there is a
page change indicator or box above or below them. A blank line is required
between a running head and spatial problems that follow.

The separation line is made up of dots 25.

There must be one blank cell between separation lines when more than one
problem is written across the page when the problems are not numbered.

The separation line goes one cell to the right and left of the widest item in the
problem.

When spatial arrangements are identified by a number or letter, no symbol of one
spatial arrangement may be less than 3 cells distant from any symbol on any line
except separation lines.
Addition and Subtraction

Plus and minus signs are placed in the cell to the left of the widest number which
appears above the separation line. If they are deliberately misaligned, follow
print. Section 178 Nemeth Code.
===Q=
==HP=
=PPPP=

In the problem below, even though the separation line in the problem is longer in
print, it still extends only one cell to the right and left of the longest part of the
problem.
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 1 =====U=
====HP=
===PPPP=
====NN=
=

In the problems below, notice the one blank cell between the separation lines.
=
===NM====V====T====V=
==J=T===HT===JP===JP=
=PPPPP=PPPP=PPPP=PPPP=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 2 
When an arrangement contains fractions, fraction lines must be vertically aligned,
with each numerator right justified in the column for numerators and each
denominator must be left justified in the column for denominator. Fraction
indicators are also vertically aligned. Nemeth Code Section 178 e.
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 3 
The problem is in 1/3, with displayed material in 5/7.

The arrangement contains mixed fractions.

Notice the blank spaces in the 4 and 5th line of the spatial arrangement. This is
to align the fraction.

The numeric indicator is not used within a spatial problem, even after the equals.
@SR|Q=I^aa=>=tbfYqpQ=
==KIbLfj^qbP=
==@RVHOQHNTHOVHRU=Kh=@NUT=qlkp=
====KIbu^`qP=
=
======RU|\NLO|@=Kh==RU|\NOLOQ|@=
======OP|\RLU|@=Kh==OP|\NRLOQ|@=
======NS|\RLS|@=Kh==NS|\OMLOQ|@=
======OV|\NLQ|@=Kh==OV|\=SLOQ|@=
=====HRU|\NLP|@=Kh==RU|\=ULOQ|@=
====PPPPPPPPPPPPPPPPPPPPPPPPPPPP=
===================NUQ|\SNLOQ|@=
=
======Kh=@NUQHO|\NPLOQ|@=
======Kh=@NUS|\NPLOQ|@=
====I>=qlq^i=tbfYq=fp=@NUS|\NPLOQ|@===========
====qlkpQ=
=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 4 
The example below is aligned as shown in print.

The numeric indicator is not used within the spatial problem, not even after the
equals.

Author’s notes shown to the right of the problem are aligned at the right in
braille. If a problem won’t fit, it should be keyed with the key placed before the
problem.

Only the portions that won’t fit should be keyed. Literary numbers are used for
the key.

The example below is part of a displayed expression.
=
======IDIhbv=Smol@j=OiP=
=
@^=Ijriqfmiv=bnrIk=ENF=_v=J@R|QID=
=
===JRuJ=Rv=Kh=JOMR===@^=
====RuHNNv=Kh==PMT===EOF=
==PPPPPPPPPPPPPPPPP=
========Sv=Kh==NMO===I^aaQ=
=
==v=Kh=@NT=Iafsfab=_v=@S|Q=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 5 Multiplication

The multiplication cross is written in the cell immediately to the left of the first digit
(multiplier) above the line. Section 179 Nemeth Code.

The separation line goes 1 cell to the right and left of the overall width of the
problem.
===QVP=
===]GO=
==PPPPP=
=

The example below shows a series of problems across the page. There is one
blank cell between the separation lines of the problems.
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 6 ===O=====Q=====S=====U=====M=====V=
=]GM===]GM===]GM===]GM===]GT===]GN=
PPPPP=PPPPP=PPPPP=PPPPP=PPPPP=PPPPP=
=
===U=====O=====N=====P=
=]GN===]GN===]GR===]GN=
PPPPP=PPPPP=PPPPP=PPPPP================
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 7 
The Review and Name at the top of the page are left aligned in cell 1.

The heading for the worksheet is centered. A blank line is required before a
centered heading unless it follows a page change indicator. A blank line is
required after the centered heading.

Directions are in cell 5. A blank line is required below the directions because of
the spatial material that follows. Section 185, Nemeth Code.
IIobsfbt=Ij^\=I?tyq=@QJO=
IIobsfbt=IIC=IImo^`qf`b=
=
I?k=JJJJ=
=
====Ij=Ijriqfmif`Ik=C=IafsfKk=Ic^`qp=
=
====IcVa=>=molar`q=lo=nrlqfRqQ=
=
@N|Q====V===@O|Q====U===@P|Q====Q=
======]GP=========]GR=========]GV=
=====PPPPP=======PPPPP=======PPPPP=
=
@Q|Q====S===@R|Q====M===@S|Q====U=
======]GU=========]GV=========]GP=
=====PPPPP=======PPPPP=======PPPPP=
=
@T|Q====O===@U|Q====V===@V|Q====S=
======]GU=========]GN=========]GV=
=====PPPPP=======PPPPP=======PPPPP=
=
@NM|Q====U===@NN|Q====U===@NO|Q====T=
=======]GS==========]GU==========]GV=
======PPPPP========PPPPP========PPPPP=
=
@NP|Q====U===@NQ|Q====V===@NR|Q====U=
=======]GT==========]GV==========]GN=
======PPPPP========PPPPP========PPPPP=
=
@NS|Q====O===@NT|Q====Q===@NU|Q====T=
=======]GV==========]GU==========]GU=
======PPPPP========PPPPP========PPPPP=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 8 @NV|Q====V===@OM|Q====T===@ON|Q====S=
=======]GQ==========]GT==========]GT=
======PPPPP========PPPPP========PPPPP=
=
@OO|Q====V===@OP|Q====U===@OQ|Q====P=
=======]GS==========]GO==========]GV=
======PPPPP========PPPPP========PPPPP=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 9 
Comments are to the right of the problem. Since they fit on the page, they don’t
have to be keyed.

The problem is in 1/3, the displayed spatial problem is in 5/7.

Notice the alignment with the comma in the arrangement below.
@TP|Q=IcL=jriqfmiv=@VPRO=_v=@Q|Q=I>k=
==jriqfmiv=@VPRO=_v=@SI=j^hH=prob=SiVb=
==rm=>=qRpQ=I>k=jriqfmiv=@VPRO=_v=@OI=
==j^hH=prob=SiVb=rm=>=erkaoApQ=I>k=^aa=>=
==?mf^i=molar`qpQ=
=
==========VPRO=
=========]GOSQ=
====PPPPPPPPPPP=
========PT=QMU===@Q]GVPRO=
=======RSN=NO====@S]GVPRO=
=====N=UTM=Q=====@O]GVPRO=
====PPPPPPPPPPP=
=====OIQSUIVOU=
=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 10 Cancellation in Subtraction

When showing cancellation in spatial subtraction problems, the numeric indicator
isn't used for the numbers written above the cancellation. Refer to Section 60 of
the Nemeth Code.

Items that are individually cancelled in print should be individually cancelled in
braille.

The problem number goes on the main line of the problem, not where the
cancellation is.
Opening Cancellation x=
Closing Cancellation z=
========V=
===N==xNMz=NN==NN=
==xOzIx=Mzx=Ozx=Nz=
=J======R===T===S=
PPPPPPPPPPPPPPPPPPP=
========Q===Q===R=
=
=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 11 =
===============V===V=
==========R==xNMzxNMz=NM=
@ST|Q===SxSzIx=Mzx=Mzx=Mz=
=======JP=Q=I==Q===Q===Q=
======PPPPPPPPPPPPPPPPPPPP=
========P=N=I==R===R===S=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 12 Carried Numbers
=
=

When there are carried numbers in the arrangement, the carried number
indicator must be inserted. Nemeth Code Section 178 d.

The carried number iindicator must be the same length as the separation line.

Carried number indicator TTTTTTTT
====ILbm=@Q=
==I^aa=>=\ypCp=`lirjkN=V`iraH=>=obdoymA=
@N|Q=
=
====N=NO=
==TTTTTTTT=
=======NU=
======TMU=
====V=OUS=
===H==SPS=
==PPPPPPPP=
===NMISQU=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 13 Cancellation in Fractions
=

Cancelled fractions are done spatially. Refer to Section 60 of the Nemeth Code.
Remember to use the numeric indicator in the portion written above the
cancellation when a number stands alone. When a number begins the fraction
and is not within cancellation indicators, the numeric indicator is used.

Refer to page N5 of the Nemeth Revisions in the 2007 BANA Update. This
explains why the number indicator is used for cancelled fractions, but not in
numbers aligned for computation.

The numeric indicator is not used within cancellation indicators.

Individual units are cancelled separately.

The problem number is in cell 1. Divisions are in 3/5. The displayed expression
is in 7/9.

If there are runovers within cancelled fractions that don’t have numbers written
above the cancellation, the cancellation with runovers are centered above the
fraction line.
@NP|Q=
==X^|Q=Ibumi^V=>=afSzXb=V=>=qtl=
====^mmol^Gbp=rpA=Spfjmifcv=\OMLOU@|Q=
=
=======@N========@N==@N=
=======xQzGR=====xOzGxOzGR=
======\PPPPP@=C=\PPPPPPPPP@=
=======xQzGT=====xOzGxOzGT=
=======@N========@N==@N=
=
==X_|Q=I[b=>=obpriqp=>=p^jbU=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 14 @RP|Q=Itofqb=>=mofjb=c^`qlofwIk=E=_l\=
==krjz^qlo=C=abkljV^qloQ=I>k=afsfab=_l\=
==krjz^qlo=C=abkljV^qlo=M^kv=Jjlk=
==c^`qlopN=C=tofqb=^=@N=Mb^G=c^`qlo=q=
==e^p=_bR=afsfaAQ=IcVIvN=jriqfmiv=>=
==obj^VH=c^`qlop=V=_l\=krjz^qlo=C=
==abkljV^qloQ=
====\NSMLORS@=
=
============@N==@N==@N==@N==@N=
============xOzGxOzGxOzGxOzGxOzGR=
======Kh=\PPPPPPPPPPPPPPPPPPPPPPPPP@=
==========xOzGxOzGxOzGxOzGxOzGOGOGO=
==========@N==@N==@N==@N==@N=
=
=
======Kh=\RLU@=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 15 
This was part of a displayed problem. The cancelled part wouldn’t fit across the
page unless it was moved over to the margin. The preference is to keep the
spatial cancelled fraction on the same line and not have runovers within the
fraction.

Crossed out sections are cancelled together.
==I\=fp=?W=>=qbuq=fp=Oc=>=`KbiiIk=V=>=
co^`XkQ=
=
^Nu=
=
======J^N_N`OHx^O_N`NzH^N_O`NJx^O_N`Nz=
==Kh=\PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP@=
=================^N_OJ^O_N=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 16 
The problem is in 1/3, displayed equations are in 5/7.
@OR|Q=Itofqb=@PS=w=\PSLN@=C=jriqfmivQ=
====@PSG\RLU@G\VLNR@=
=
===========@V====@N====@P=
==========xPSz===xRz===xVz=
======Kh=\PPPP@G\PPP@G\PPPP@=Kh=\OTLO@=
===========@N====xUz===xNRz=
=================@O=====xPz=
========================@N=
=
======Kh=@NP|\NLO|@=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 17 Synthetic Division





In synthetic division, the numeric symbols in the synthetic dividend, synthetic
product, and synthetic quotient must be aligned by place value. Section 182,
Nemeth Code.
Symbols of operation must also be aligned when present.
Follow print for where the vertical bar appears—to the left or the right. It must be
unspaced from the dividend and divisior
The separation line must begin directly under the vertical line at one end and go
one cell beyond the overall product.
If the synthetic divisor appears in print as boxed on both sides, ignore this in
braille.
Synthetic Division Indicators
|=Dots (456) straight line
PPP=Dots (25) separation line
Left and right synthetic division are shown below.
=|=
=|=
=PPP=
=
=
===|=
===|=
=PPP=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 18 
The example below is a spatial arrangement. Shape indicators are not used in
spatial material, the full cell is used instead. Nemeth Code Section 58,
ILbm=@N=Itofqb=>=s^irb=E=Xh=C>=
==`lbSf`fRqp=E>=afsfaRa=V=loaz=E=abp`RaH=
==bumlkRqpQ=
=
====N|O=Q=JV=Z=
=====PPPPPPPPPP=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 19 =============IIorkkH=IIeNa=
==Ifc=tb=tofqb=>=@O=V=>=nrlqfRq=lk=>=
_lqqlj=iVbN=>=_lqqlj=iVb=dfsbp=_l\=>=
`lbSf`fRqp=E>=nrlqfRq=C>=obj^VazQ=I>=qlm=
iVb=kx=`=O=bifjV^qAN=C>=afsfKk=^mmb[p=w=
=
==JP|O=H=Q=J=P=HNM=
====|==J=S=JPM=JUN=
====PPPPPPPPPPPPPPP=
=====O==NM==OT==VN=
=
==I>=_lqqlj=iVb=M=l_q^VA=Mpr_qo^`qH=>=
jfQib=iVb=c=>=qlm=iVbQ=Ifc=tb=obmi^`b=>=
J@P=V=>=afsfplo=F=HPI=>=pfdkp=E=b^G=
krj_z=V=>=jfQib=iVb=t=O=obszpA=V=>=
afsfKk=mol`bppQ=I>k=>=_lqqlj=iVb=`=O=
l_q^VA=M^QfXkN=C=tb=e=>=cV^i=Zj=E>=
pvk>qf`=afsfKkQ=
=
=
=
=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 20 =============IIorkkH=IIeNa=
======IDIhbv=S>=mol@j=OiP=
=
@^=I>=`lbSf`fRqp=E>=afsfaRaQ=
@_=I>=`lbSf`fRqp=E>=nrlqfRq=C>=
==obj^VazQID=
=
==HP|O=H=Q=J=P=HNM===@^=
====|==H=S=HPM=HUN=
====PPPPPPPPPPPPPPP=
=====O==NM==OT==VN===@_=
=
I\rpN=
==\Ou{P?HQu{O?JPuHNMLuJP@=
====Kh=@Ou{O?HNMuHOTH\VNLuJP@=
=
=
=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 21 Itb=t=rpb=pvk>qf`=afsfKk=C=cVa=>=
obj^VazQ=
=
==f|N==JN===HN=JN=
===|====f=JNJf=HN=
===PPPPPPPPPPPPPPP=
====N=fJN===Jf==M=
=
IpV`b=>=obj^Vaz=fp=@MI=ImEfF=Kh=@M=C=Xf=
fp=^=wzl=E=ImEuF|Q=
==IcVa=ImEJfF|Q=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 22 Division

See Nemeth Code Section 180. When a problem only contains a divisor and
dividend, but no quotient or partial products, a spatial arrangement is not used.
Example:

The example below has a quotient, and is done spatially.

Symbols for the dividend and partial products and differences must be aligned as
in print.

The separation line begins in the column containing the division symbol and
extends one cell beyond the longest part of the problem.

The blue type in the remainder does not need to be shown in braille.
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 23 X_|Q=ISafsfab=ImEuF=_v=uJNI=tb=mol`bA=w=
==cliixpP=
=
========Pu{O?JOuHN=
=======PPPPPPPPPPPPPPPPPPP=
====uJNlPu{P?JRu{O?HPuJNM=
========Pu{P?JPu{O=
=======PPPPPPPPPPPPPPPPPPP=
=============JOu{O?HPu=
=============JOu{O?HOu=
=======PPPPPPPPPPPPPPPPPPP=
===================H=uJNM=
=====================uJ=N=
=======PPPPPPPPPPPPPPPPPPP=
======================J=V=
=
==Iklqb=q=>=obj^Vaz=fp=bnr^i=ql=ImENF|Q=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 24 
The example below is done this way for either a caret or an arrow to show
movement.
==Iafsfab=@PPTKR=_v=@NPKR=ScVa=>=rkfq=
o^qbQ=
=
===========O=RK=
========PPPPPPPPP=
==NPKR|YlPPTKR|Y=
=
==\PPTKR=jfibpKLNPKRLNPKR=d^iilkpKLNPKR@=
====Kh=\OR=jfibpLN=d^iilk@=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 25 
Since the key for expressions shown to the right won’t fit on the same braille
page as the division problem, using facing pages would be helpful for the reader.
=============IIorkkH=IIeNa=
@NR|Q=\PKNLMKMMS@=
=
======IDIhbv=S^r\loDp=klqbpP=
=
@^=IiVb=rm=ab`fj^i=mlVqpQ=
@_=Ijlsb=ab`fj^i=mlVq=V=afsfplo=C=
==afsfaRa=\obb=mi^`bpQ=Itofqb=@MM=V=
==afsfaRaQ=
@`=Itofqb=@M=V=afsfaRaQ=
@a=ILlm=C=oKa=^kptz=S>=kb[bL=erkaoA\QID=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 26 ==============RNSK=SSS===@^=
===========PPPPPPPPPPPP=
====MKMMS|YlPKNMM|YMMM===@_=
============P=M=
===========PPPPPPPPPPPP=
==============NM=
===============S=
===========PPPPPPPPPPPP=
===============QM=
===============PS=
===========PPPPPPPPPPPP=
================QM=======@`=
================PS=
===========PPPPPPPPPPPP=
=================QM======@`=
=================PS=
===========PPPPPPPPPPPP=
==================QM=====@`=
==================PS=
===========PPPPPPPPPPPP=
===================Q=====@a=
=
==I>=nrlqfRq=fp=@RNSKST=EoKaAF|Q======
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 27 
In the example below, there are arrowed notes to the right of the problem. The
arrows have been omitted.

Numbers are aligned as shown in the problem.
=====QuJR==============\JRuLu@=Kh=J@R=
===PPPPPPPPPPPPPPPPPP=
uHPl=Qu{O?=H=Tu=H=P=
===JEQu{O?=HNOuF=
===PPPPPPPPPPPPPPPPPP=
===========J=Ru=H=P=
=========JEJ=Ru=JNRF===J@REuHPF=
===PPPPPPPPPPPPPPPPPP====Kh=J@RuJNR=
=================NU=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 28 Have You Seen This?

The example below is shown in the Nemeth lesson manual, page 296. I forgot
that it was covered until this came along in a textbook.
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 29 =
====Ibu^jmib=@NM=IafsfaH=F^=Iobj^Vaz=
==Iafsfab=@NUMV=_v=@T|Q=
==Iafsfab=@T=Vql=@NU|Q=
=
=====O======\NULT@=Kh=@O=F=@Q=ibcq=lsz=
===PPPPPPP=
==TlNU=MV=
===TTTTTTT=
======Q=
==Iafsfab=@T=Vql=@QM|Q=
=
=====O=R=====\QMLT@=Kh=@R=F=@R=ibcq=lsz=
===PPPPPPPP=
==TlNU=M=V=
===TTTTTTTT=
======Q=R=
=
==Iafsfab=@T=Vql=@RV|Q=
=
=====O=R=U=Io?P===\RVLT@=Kh=@U=F=@P=ibcq=
===PPPPPPPPPPPPP====lsz=
==TlNU=M=V=
===TTTTTTTTTTTTT=
======Q=R=
=
I?t=Imol@j=@NM|Q=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 30 Enlarged Grouping for Notes

The note shown to the right of the enlarged brace can be divided between lines.
Runovers are indented 2 cells to the right for the note.

The enlarged grouping symbol can be continued on extra lines if the comment is
longer than the number of lines for the expression.

In this type of arrangement, if there isn’t an enlarged brace, but the notes apply
to a section, an enlarged brace can be inserted.
@NVj{R=C=@NQj{R=======KIF=Ibu^jmibp=E=
@Sv{VI=J@PTv{VI=C=v{V?KIF===i=qzjp=
@Pmn=C=J@Omn==========KIF=
@Ouv{O=C=Juv{O========KIF=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 31 For Discussion


What would you do with this?
Is this described well in the text, or would you need to draw the arrows or make
extra explanations?
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 32 For Discussion
What would you do with the problems below?

Will the notes fit across the page with the spatial division problem?

What are some of the issues with the second problem?
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 33 Systems of Equations

Since this type of equation is considered to be spatial, it requires a blank line
before and after it. The components should be aligned as shown in print.

If they are referred to as systems, use this format. See Section 184 for the rules.
Also see pages 316 and 317 in the lesson manual.

There is a blank line between the directions and the system of equations that
follows.

Notice that the comma goes on the first line of the equations.
==Iplisb=b^G=pvLbj=do^mef`IvQ=IO=prob=
SGb`h=vo=plirXkQ=Ifc=^=pvLbj=e^p=^k=
VcVfqb=krj_z=E=plirXkpN=rpb=pbqJ_rfiaz=
klqIk=Stofqb=>=plirXk=pbqQ=Ifc=^=pvLbj=
e^p=kl=plirXkN=L^qb=\Q=I?W=^mmolmof^qbN=
oKa=S>=kb[bL=erkaoA\Q=
=
@NT|Q=uJv=Kh=@PI=
======uHv=Kh=@R=
=
@NU|Q=uHv=Kh=@QI=
======uJv=Kh=@O=
=
@NV|Q=@PuH=v=Kh=@RI=
========uJOv=Kh=@Q=
=
@OM|Q=@OuJv=Kh=@QI=
======@RuJv=Kh=@NP=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 34 Unified Expressions

A unified system of equations is spatial. This requires a blank line between
directions and the system of equations that follow.

Opening enlarged grouping is used for the brace that is shown preceding each
system. Refer to Nemeth Code Section 126.

When enlarged grouping is used to unify an expression, each item begins right
after the grouping symbol. Align as shown in print.
====Irpb=>=do^meH=jb\la=Splisb=b^G=
==pvLbjP=
=
X^|Q=KIEPuHv=Kh=@N=
=====KIEJuHOv=Kh=@V=
=
X_|Q=KIEOuJPv=Kh=@Q=
=====KIEQu=Kh=J@QHSv=
=
X`|Q=KIEv=Kh=@QJu=
=====KIEOuHOv=Kh=@U=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 35 
In the example that follows, there is room on the first line for the rest of the
sentence. It goes on the top line of the system of equations.

The system is aligned as shown in print.
====Ibu^jmib=@N=
=
Iplisb=KIEu{O?Hv{O=Kh=@OR=Mdo^meHQ=
=======KIEOuHv=Kh=@NM=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 36 Matrices and Determinants

Refer to Section 183 of the Nemeth Code for more information.

In determinants or matrices each entry must be left-justified in the column that it
applies to.

One column of blank cells must be left between columns.

The numeric indicator is used with numeric entries in a determinant or matrix,
even when they are in direct contact with the opening grouping sign.

Runovers are indented 2 cells from the column margin.

Space can be saved by drawing the enclosure symbols.

Fractions may be represented spatially to save space. A blank line must be left
before and after when a spatial fraction is used.

The line for this arrangement is dots 456:
@OV|Q=]IE@N=J@U=|=@P==]IF\NLT@IoO=
======]IE@M=@T==|=J@NQ]IF=
=
==Kh=]IE@N=J@U=|=@P=]IF=
=====]IE@M=@N==|=J@O]IF=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 37 A determinant of a matrix represents a single number. We obtain this value by
multiplying and adding its elements in a special way. We can use the determinant
of a matrix to solve a system of simultaneous equations.
For example, if we have the (square) 2 × 2 matrix:
The determinant of this matrix is written within vertical lines as follows:
==I^=ID|=abqzjV^kq=E^=j^qofu=|ID=
obmobpRqp=^=pHib=krj_zQ=Itb=l_q^V=\=
s^irb=MjriqfmivH=C=^QH=up=bibXqp=V=^=
pmb`f^i=t^vQ=Itb=`=rpb=>=abqzjV^kq=E^=
j^qofu=Splisb=^=pvLbj=E=pfjriq^kbyp=
bnrIkpQ=
==IZ=bu^jmibN=fc=tb=e=>=Epnr[bF=@O]GO=
j^qofuP=
=
==IE@R=@T=IF=
==IE@O=J@PIF=
=
==I>=abqzjV^kq=E=\=j^qofu=fp=tofqqR=FV=
szqf`^i=iVbp=w=cliixpP=
=
==Iy@R=@T=Iy=
==Iy@O=J@PIy=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 38 
The example below shows a unified system of equations followed by a matrix.

The equations are displayed to narrative text, and are in 3/5.

In the last section, R1 and R2 require a multipurpose indicator to show that the
numbers are not subscripts to the R.

The equation (-2)R1+R2 goes on the top line of the matrix.

The equations are divided before the arrow, which is a sign of comparison.
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 39 =============IIorkkH=IIeNa=
=
====IIbu^jmib=@O=
==Iplisb=>=cliixH=pvLbj=Mqo^kpZjH=up=
pvLbj=j^qofu=VSox=bGbilk=Zj=C=_^`h=
pr_LfqrqHQ=
=
==KIEuHOvHPw=Kh=@Q=
==KIEOuJvJOw=Kh=@M=
==KIEuJPvJPw=Kh=J@O=
=
====IplirXk=
==I\=pvLbj=fp=obmobpRqA=M>=pvLbj=j^qofu=
=
==]IE@N=@O==@P==|=@Q=]IF=
==]IE@O=J@N=J@O=|=@M=]IF=
==]IE@N=J@P=J@P=|=J@O]IF=
=
Itb=t=obar`b=>=pvLbj=j^qofu=Sox=bGbilk=
ZjQ=Itb=`=rpb=>=@N=V=>=rmmz=ibcq=`lokz=
Swzl=y=>=obL=E>=cL=`lirjkQ=ISa=pN=tb=
jriqfmiv=>=cL=ox=_v=J@O=C=^aa=>=obpriq=
S>=pb`lka=oxQ=Itb=Vaf`^qb=\=lmzIk=F>=
klqIk=EJOFIo?NHIo?O=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 40 =
=============IIorkkH=IIeNa=
=
==]IE@N=@O==@P==|=@Q=]IFEJOFIo?NHIo?O=
==]IE@O=J@N=J@O=|=@M=]IF=
==]IE@N=J@P=J@P=|=J@O]IF=
=
====Al=]IE@N=@O==@P==|=@Q=]IF=
=======]IE@M=J@R=J@U=|=J@U]IF=
=======]IE@N=J@P=J@P=|=J@O]IF=
=
=
For discussion

What would you do with the portion with the lines drawn through them?
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 41 
The determinants below are spatial problems. A blank line is needed above and
below.

In print, red type is shown for parts of the problem. It is described well in the text,
and is not shown in braille.

The English letter indicator is not used with any letter or combination of letters in
a determinant or matrix.
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 42 EQF=v=Kh=\^cJb`L^aJ_`@=
==Itb=`=tofqb=>=s^irbp=E=Xu=C=Xv=V=
==IbnrIkp=@P=C=@Q=rpH=abqzjV^kqpQ=
=
==========Iyb=_Iy=
==========Iyc=aIy=
====u=Kh=\PPPPPPP@=Kh=\baJ_cL^aJ_`@=
==========Iy^=_Iy=
==========Iy`=aIy=
=
==========Iy^=bIy=
==========Iy`=cIy=
====v=Kh=\PPPPPPP@=Kh=\^cJb`L^aJ_`@=
==========Iy^=_Iy=
==========Iy`=aIy=
=
====Ifc=tb=Jm[b={>=Zjri^p=F>=lofdV^i=
==pvLbjN=
=
====KIE^uH_v=Kh=b=
====KIE`uHav=Kh=c=
=
==Itb=pbb=q=>=abkljV^qlop=[b=>=abqzjV^kq=
==E>=`lbSf`fRq=j^qofuP=
==============
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 43 IIorkkH=IIeNa=
====IabkljV^qlo=abqbojfk^kq=
=
======Kh=Iy^=_Iy=
=========Iy`=aIy=
=
====IScVa=>=krjz^qlo=abqzjV^kq=Z=XuI=tb=
==obmi^`b=>=X^=C=X`=V=>=cL=`lirjk=E>=
==abkljV^qlo=abqzjV^kq=F>=PL^kqp=Xb=C=
==Xc|Q=
====IScVa=>=krjz^qlo=abqzjV^kq=Z=XvI=tb=
==obmi^`b=>=X_=C=Xa=V=>=pb`lka=`lirjk=E>=
==abkljV^qlo=abqzjV^kq=F>=PL^kqp=Xb=C=
==Xc|Q=
=
==========Iyb=_Iy=
==========Iyc=aIy=
====u=Kh=\PPPPPPP@=
==========Iy^=_Iy=
==========Iy`=aIy=
=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 44 =============IIorkkH=IIeNa=
=
==========Iy^=bIy=
==========Iy`=cIy=
====v=Kh=\PPPPPPP@=
==========Iy^=_Iy=
==========Iy`=aIy=
=
====I\=jb\la=E=rpH=abqzjV^kqp=Splisb=
==pvLbjp=E=bnrIkp=fp=`^iiA=ID|=I`o^jzDp=
==oribQ=|ID=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 45 
The example below shows a determinant that won’t fit across the line.
Notice that the expression is centered above the fraction line. A blank line
is required before and after a spatial arrangement (determinant), Section
183, Nemeth Code.

I would put in a transcriber’s note before the section of the determinant
that has to be split between lines. Note: The numerator and determinant
of the matrix below each have been split into 2 parts.
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 46 =============IIorkkH=IIeNa=
=
====IIbu^jmib=@T=
==Irpb=I`o^jzDp=orib=Splisb=>=pvLbj=
=
==KIEOuJvHOw=Kh=@P=
==KIEuJvH=w=Kh=@O=
==KIEuHvHOw=Kh=@P=
=
====IplirXk=
==Ib^G=E>=s^irbp=XuI=XvI=C=Xw=fp=>=
nrlqfRq=E=qtl=@P]GP=abqzjV^kqpQ=I>=
abkljV^qlo=E=b^G=nrlqfRq=fp=>=abqzjV^kq=
PpfLH=E>=kVb=`lbSf`fRqp=E>=s[f^@pQ=I>=
krjz^qlop=Z=XuI=XvI=C=Xw=[b=jlafcfA=
`lmfbp=E=\=abkljV^qlo=abqzjV^kqQ=Itb=
pr_Lfqrqb=>=`lirjk=E=PL^kqp=Z>=
`lbSf`fRqp=E>=s[f^@=Z=W=tb=[b=plisHQ=
=
==KIEOuJvHOw=Kh=@P=
==KIE=uJvH=w=Kh=@O=
==KIE=uHvHOw=Kh=@P=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 47 =============IIorkkH=IIeNa=
=
========Iy@P=J@N=@OIy=
========Iy@O=J@N=@NIy=
========Iy@P=@N==@OIy=
==u=Kh=\PPPPPPPPPPPPP@=
========Iy@O=J@N=@OIy=
========Iy@N=J@N=@NIy=
========Iy@N=@N==@OIy=
=
========@PIyJ@N=@NIyJEJNFIy@O=@NIy=
==========Iy@N==@OIy=====Iy@P=@OIy=
=
===============HOIy@O=J@NIy=
=================Iy@P=@N=Iy=
====Kh=\PPPPPPPPPPPPPPPPPPPPPPPPPP@=
========@OIyJ@N=@NIyJEJNFIy@N=@NIy=
==========Iy@N==@OIy=====Iy@N=@OIy=
=
===============HOIy@N=J@NIy=
=================Iy@N=@N=Iy=
=
====Kh=\OLJN@=Kh=J@O=
=
=
Mary Denault 2012 CTEBVI Spatial Arrangements Workshop Page 48