A repeating decimal happens because division gets stuck in a loop.
When you divide one number by another, you keep producing remainders. If the division does not end cleanly, eventually the same remainder appears again. Once that happens, the same digit pattern repeats forever.
So the basic cause is:
same remainder → same next step → same digit pattern → repeating decimal
A decimal ends only when the division eventually has no remainder. If that never happens, the decimal repeats.
When you divide one number by another, you keep producing remainders. If the division does not end cleanly, eventually the same remainder appears again. Once that happens, the same digit pattern repeats forever.
So the basic cause is:
same remainder → same next step → same digit pattern → repeating decimal
A decimal ends only when the division eventually has no remainder. If that never happens, the decimal repeats.
therighttothink50 · 61-69, M
RachelLia2003 · 22-25, F
DrWatson · 70-79, M
@Tennessee
When the fraction in reduced form has a denominator containing a factor which is not a factor of the base, the decimal (or duodecimal) will repeat .
1/5 = 2/10 = .2 ( base ten)
1/3 = 4/12 = .4 ( base twelve)
But 1/5 can't be written with a denominator that's a power of 12, and 1/3 can't be written with a denominator that's a power of 10.
1/4 = 25/100 = .25 (base ten)
1/9 = 16/144 = .16 (base twelve)
When the fraction in reduced form has a denominator containing a factor which is not a factor of the base, the decimal (or duodecimal) will repeat .
1/5 = 2/10 = .2 ( base ten)
1/3 = 4/12 = .4 ( base twelve)
But 1/5 can't be written with a denominator that's a power of 12, and 1/3 can't be written with a denominator that's a power of 10.
1/4 = 25/100 = .25 (base ten)
1/9 = 16/144 = .16 (base twelve)
candycane · 36-40, F
ammonia scrubbers .oooooo9