# Explaining Missing Decimal Numbers 3, 6 & 9 in Calculations

• Hippasos
In summary, the decimal expansion of 1/7 and its multiples do not contain the digit 3 because of the limited set of possible remainders in the long division process. This is due to the fact that 7 is unique as the only integer with the maximum possible number of unique digits in its decimal expansion. The long division process for 1/7 and its multiples involves a multiple of 10 divided by 7, which does not allow for a non-zero last digit that could result in a 3. This is an interesting observation in mathematics.
Hippasos
1/56=0,017857142857142857142857...142857

1/7=0,142857142857142857142857...142857

142857

142857/7=
20408,142857142857142857...142857

20408/7=
2915,42857142857142857...142857

2915/7=
416,42857142857142857142857...142857

416/7=
59,42857142857142857142857...142857

59/7=
8,42857142857142857142857...142857

8/7
1,142857142857142857142857...142857

Missing decimal numbers 3 - why are these"missing"? Not needed even during calculations - at all?

what's to explain? it's just the result of particular calculations.

Hey, try 1/3 = .333333 you don't need 1,2,4,5,6,7,8, or 9 Explain that !

phinds said:
what's to explain? it's just the result of particular calculations.

Hey, try 1/3 = .333333 you don't need 1,2,4,5,6,7,8, or 9 Explain that !

Yes of course my stupid mistake. I was *runk and it was late. Still interesting ...142857.. cycles there.

Hey! don't dismiss your observation so quickly. It is very interesting and if you look at the long division process you can see just why there are no 3 or multiples of, in the decimal expansion.

First of all observe that there are at most 6 digits possible. There is a limited set of possible remainders in the long division process{1,2,3,4,5,6}. A zero would terminate the process, 7 and greater means that you have not made a correct choice of a multiplier. This implies that there can be at most 6 digits in the expansion. 7 is unique in that it is the only integer with the max possible number of unique digits in the decimal expansion.

Now let's go through the long division process and see if we can see why 3 does not appear.

The first step:

10/7 = 1 r 3 : so 1 is the first digit the remainder sets up the second step, r*10
30/7 = 4 t 2 : so 4 is the 2nd digit.
20/7= 2 r 6 : so 2 is the 3rd digit.

now we can see why 3 does not appear, each of the steps involves a multiple of 10 divided by 7. But the only way to get a 3 would be if you could have a non zero as the last digit, this cannot happen. Since 7*3=21 and 7*4=28.

I also find this an interesting little tidbit. Hope I helped throw some light on it for you.

The missing decimal numbers 3, 6, and 9 in these calculations are not actually missing. They are simply not being shown in the decimal representation of the numbers. This is because these numbers have repeating decimal patterns that can be expressed in a shorter form using the bar notation (e.g. 0.142857142857... can be represented as 0.142857 with a bar over the repeating numbers).

In fact, all rational numbers (numbers that can be expressed as a ratio of two integers) have either a finite or repeating decimal representation. This is due to the fact that our base-10 number system is not able to accurately represent certain fractions. For example, 1/3 cannot be represented as a finite decimal in base-10 (it would be 0.333333... with an infinite number of 3s).

Therefore, the decimal numbers 3, 6, and 9 are not "missing" in these calculations, they are simply not necessary to show in the decimal representation. They are still present in the calculations and can be seen in the repeating patterns.

## 1. Why are there missing decimal numbers 3, 6, and 9 in calculations?

The missing decimal numbers 3, 6, and 9 in calculations are part of a mathematical concept called rounding. When we round numbers, we often round to the nearest whole number or to a certain number of decimal places. The numbers 3, 6, and 9 are commonly used as reference points for rounding because they are in the middle of each set of three consecutive whole numbers.

## 2. How do you determine which number to round to when using 3, 6, or 9?

The number to round to when using 3, 6, or 9 is determined by looking at the digit right after the last number you want to keep. If this digit is 5 or greater, you round up to the next number. If it is less than 5, you round down to the current number. For example, if you are rounding to the nearest whole number and the digit after the last number you want to keep is 7, you would round up to the next whole number.

## 3. Why do we use 3, 6, and 9 specifically for rounding?

We use 3, 6, and 9 specifically for rounding because they are evenly spaced between consecutive whole numbers. This makes it easier to determine which number to round to and helps to evenly distribute the rounding errors.

## 4. Are there other methods for rounding besides using 3, 6, and 9?

Yes, there are other methods for rounding, such as using 2, 5, and 8, or using a specific number of decimal places. The method used often depends on the context and the desired level of precision.

## 5. How does rounding affect the accuracy of calculations?

Rounding can affect the accuracy of calculations by introducing a small amount of error. However, this error is usually insignificant unless a large number of calculations are being performed or if very precise results are needed. In most cases, rounding is necessary to simplify calculations and make them more manageable.

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