How do you convert 5/9 to a decimal?

#color(blue)("Preamble")#

you carry out the action of #5-:9# and you end up with a repeating decimal that is

0.555555555..... going on for ever

You write this as #0.5bar5#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Calculated using long division - with a 'trick'")#

#color(brown)("Intro to concept")#

Consider the number 5 This is the same as #50xx1/10#
So if we use this approach whenever we are dividing a 'bigger' number into a 'smaller' number we can avoid using decimals until the very end. All we have to do at the and is multiply the answer by all of the #1/10#

#color(brown)("Answering the question")#

# color(white)("dddddd")50 color(green)(xx1/10) #
#color(magenta)(5)( 9)->ul(45larr"Subtract")#
#color(white)("ddddddd")5larr" Remainder"#
///////////////////////////////////////////////
#color(white)("dddddd")50color(green)(xx1/10)larr" Remainder "xx1/10#
#color(magenta)(5)( 9)->ul(45larr"Subtract")#
#color(white)("ddddddd")5larr" Remainder"#
//////////////////////////////////////////////////////////

Obviously the remainder of 5 is going to repeat in a continuous cycle.

#color(brown)("Putting together what we have so far")#

#color(magenta)(55)color(green)(xx1/10xx1/10) = 0.55#

As this cycles for ever we have:

#0.55555...#

Write as: #0.5bar5#

where the #bar5# means it goes on for ever.

#color(blue)("Very important teaching bit")#

Have you ever wandered why people say 'just divide the bottom into the top'?

THIS IS WHY!

Consider whole numbers. For example 2,3,7 and any others you so choose.

People do not do this (not often ):

You can and may write whole numbers in the format of type: #color(white)("dddddddddddddddd")2/1,3/1,7/1#

Consequently they are members of a set of numbers that have the name of 'Rational Numbers'. That is of form #a/b#

These may, if you so choose, be considered as ratio but written in fraction format.

So by ratio you have:

#color(white)(" d")("count of one thing")/("count of something else")#

And the connection between this and a fraction is:

# ("count of what you have got")/("count of how many of what you have got that will fit into 1")#

#color(white)("dddddddddddddddddddd")2/1,3/1,7/1, 1/2#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("So for this question the starting point is:")#

Using this principle of ratio; converting a fraction to a decimal is:

#5/9->(5-:9)/(9-:9)->(5-:9)/1#

Which is why you 'just divide the top by the bottom!!