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Answers
Answer:
Use “Bus stop” long division.
1/3 is the same as saying 1÷3, or “how many times does 3 go into 1”
3|1.000
How many times does 3 go into 1? Zero remainder one. We take the one over into the next column. So now we have:
3|0.1000
0
So, how many 3s go into 10? Three with one remaining. We take the one into the next column again.
3|0.0100
0.3
How many 3s go into 10? 3 again, remainder 1.
3|0.0010
0.33
This process continues until you end up with no remainder - but that doesn't ever happen in the decimal expansion of 1/3,
1/3=0.333… with recurring (infinitely repeating) 3s, and that's why we tend to prefer factions where we can use them.
13=0.3¯¯¯
It is a never ending decimal: 0.3333… , a zero followed by a decimal point and an infinite number of threes. As we do not have infinitely long life and patience, we use an approximation with as many 3's as we think is accurate enough for our needs.
The fact that it has infinite digits in a repeating pattern is a consequence of our base 10 numbering system. Because 10=2×5, any fraction whose denominator has prime factors other than 2 and 5 has infinite digits in its decimal form.
1/125=1/(5×5×5)=(1×2×2×2)/(5×5×5×2×2×2)=8/1000=0.008 has a finite number of digits in its decimal form, because we can multiply the numerator and denominator by the same combination of 2's and 5's and get an equivalent fraction whose denominator is a power of 10. No such luck with any denominator than cannot be written as a product of only 2's and/or 5's.
It makes sense to use a base 10 system because most of us have 10 fingers in our hands. If we had 12 fingers, we would use a system with base 12=2×2×3, and then the fractions yielding decimals fractions whose denominatorss could be written as products of 2's and 3's. However, with a base 12 system, 1/5 would have infinite digits in its decimal form.
1. 1
2. 1
3. 2
4. 9/8
5. 9/17
6. 20/81
Step-by-step explanation: