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# Proof: sum & product of two rationals is rational

Sal proves that the sum, or the product, of any two rational numbers will always be a rational number. Created by Sal Khan.

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he said that if an int is divided by an int then its rational right?
but 2/7 = 0.285714 recurring
I'm pretty certain that's not rational.
my teacher couldn't explain this to me

edit: would love if someone answered this quickly
• So what makes you think it is not rational? If a decimal is repeating, it should be rational because some people such as myself can relatively easily find the two whole numbers to create a fraction. All truncating and repeating decimals are rational because they meet the definition of being a ratio of two integers or whole numbers.
An irrational number has a decimal that NEVER repeats. So if it repeats, then it does not meet the qualification of NEVER.
• can anyone of you can tell if this is true or not?
1+2+3+4+5.....to infinity= -1/12?

want to know just upvote for this question
• No, because this is not a series whose terms are approaching zero. The terms are increasing by 1, and this is an arithmetic series. So, no, you can't calculate the sum of this series. It just approaches infinity.

In fancy terms, we would say that it 'diverges', that is, it evaluates to some really big number we can't bound.
• but when we multiply 1/7 and 22/1 we get 22/7 which is irrational and same with addition when we add 1/7 and 21/7, we get 22/7
• 22/7 is a rational number (a fraction created with 2 integers). It creates a repeating decimal. It just has a long repeat pattern of 6 digits.

If you think it is irrational because you think it is equal to Pi, that is a common point of confusion. 22/7 is just an approximation of Pi. Pi is irrational (a non-terminating and non-repeating decimal). Compare their decimal values. By the 3rd decimal digits, their values are different.

22/7 = 3.142857142857142857...
Pi = 3.14159265358979323846264338...

Hope this helps.
• How do you find out which is which? (Irrational & Rational)
• Rational Numbers: The real numbers which can be represented in the form of the ratio of two integers, say P/Q, where Q is not equal to zero are called rational numbers. Irrational Numbers: The real numbers which cannot be expressed in the form of the ratio of two integers are called irrational num
• Isn't it the case that this proof is false because sal's reasoning ends by assuming the proof is true (we can't know if "am" is a rational number because that's what we're trying to prove).
• Rational numbers are defined as the numbers that can be written as the ratio of two integers.

We take two rational numbers a/b and m/n
which means that a, b, m and n are integers
according to the definition of rational numbers.

We want to know if the product of two rational numbers is also a rational number, so we multiply a/b by m/n

which equals to (a*m)/(b*n)

a*m and b*n are both integers, because multiplying an integer by an integer gives us an integer.

So (a*m)/(b*n) is also a ratio of two integers,
which makes it a rational number, because that's how rational numbers are defined.
• How do we know the sum of two integers is an integer?
• For the result to not be an integer, it would have to create a fraction or decimal. If you add integers, you are adding whole units (no fractions or decimals). So, your result will never include a fraction or decimal.
• At , when Sal added bn + bn, how come he didn't write it as 2bn? Since he added both of them together and they are the same quantity, shouldn't he have written it as 2bn?
• Sal is adding 2 fractions. The "bn" are in the denominators. When we add fractions, we only add the numerators. For example: 5/8+2/8 = 7/8, not 7/16.

Hope this helps.
• What about irrational divide irrational? rational divide irrational? irrational divide rational? irrational to a rational number's power? irrational to irrational? rational to irrational?