# Can 9.99 Really Equal 10? An Analysis of the Math"

• _Mayday_
In summary, there is no real number between 9.9999... and 10, making them equivalent. This can be proven by multiplying the equation by 10 and using the properties of infinite series.
_Mayday_
I was told today that 9.99 = 10? I really don't understand that. Can anyone show me the proof for it.

Oh god, not this again.

Since you cannot find a real number between 9.999... (repeating nines) and 10, they are the same number. They're just a different way of representing the same number.

- Warren

Some people do it this way:
If x= 9.9999... then multiplying the equation by 10 gives you 10x= 99.99999... Notice that, since the "9"s in the first number go on indefinitely, the "9"s on the right of the decimal point in both go on indefinitely. Subtracting the first equation from the second, all of the "9"s on the right side canceL: you get 9x= 90 so x= 90/9= 10.

Of course, that assumes that you can multiply infinite numbers of decimal places just like finite ones and subtract the same way. That is true but should be proven. More correct is to argue that an infinite number of decimal place is by definition of "base 10 numeration", an infinite series. In particular, 9.9999... is, by definition, the infinite series
$$9+ 9/10+ 9/100+ 9/1000+ ...= \sum_{n=0}^\infty 9(1/10)^n$$
That is a geometric series and there is an easy formula for the sum of an infinite series:
$$\sum_{n=0}^\infty a r^n= \frac{a}{1- r}$$
as long as -1< r< 1.
For a= 9 and r= 1/10 (and, of course, -1< 1/10< 1)
$$\frac{9}{1- 1/10}= \frac{9}{\frac{9}{10}}= 10$$

## What is the purpose of the analysis "Can 9.99 Really Equal 10?"

The purpose of this analysis is to examine the mathematical concept of rounding and determine if the statement "9.99 equals 10" is true or false.

## What methodology was used in this analysis?

The analysis utilized basic mathematical principles and equations to compare the values of 9.99 and 10.

## What were the results of the analysis?

The results of the analysis showed that while 9.99 and 10 are very close in value, they are not equal. 9.99 is slightly less than 10, with a difference of 0.01.

## What factors contribute to the misconception that 9.99 equals 10?

This misconception may arise due to the use of rounding in everyday situations, where 9.99 is often rounded up to 10. Additionally, the decimal system we use may suggest that 9.99 is equivalent to 10, as the decimal point is in the same place for both numbers.

## What is the significance of this analysis?

This analysis highlights the importance of precision in math and serves as a reminder to be cautious when rounding numbers, as it can lead to incorrect conclusions.

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