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### Course: Algebra (all content) > Unit 1

Lesson 11: Sums and products of rational and irrational numbers- Proof: sum & product of two rationals is rational
- Proof: product of rational & irrational is irrational
- Proof: sum of rational & irrational is irrational
- Sums and products of irrational numbers
- Worked example: rational vs. irrational expressions
- Worked example: rational vs. irrational expressions (unknowns)
- Rational vs. irrational expressions

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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.

## Want to join the conversation?

- look, I'm genuinely confused about this rational thing.

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(10 votes)- 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.(16 votes)

- 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(5 votes)- 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.(5 votes)

- 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(2 votes)
- 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.(7 votes)

- How do you find out which is which? (Irrational & Rational)(4 votes)
- 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(2 votes)

- How do we know the sum of two integers is an integer?(3 votes)
- 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.(3 votes)

- 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).(2 votes)
- 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.(4 votes)

- At2:06, 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?(2 votes)
- 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.(4 votes)

- What about irrational divide irrational? rational divide irrational? irrational divide rational? irrational to a rational number's power? irrational to irrational? rational to irrational?(3 votes)
- is zero a rational number or a irrational number(1 vote)
- It is a rational number. 0 is an integer. All integers are rational numbers. Integers can be written as a fraction using 2 integers (that's the definition of a rational number):

0 = 0/1

Hope this helps.(5 votes)

- So If you do X + 34 + 9/33 x 44 = y =555 what would add 33+9/33x44(3 votes)

## Video transcript

What I want to do in
this video is think about whether the product or
sums of rational numbers are definitely going
to be rational. So let's just first
think about the product of rational numbers. So if I have one rational
number and-- actually, let me instead of writing
out the word rational, let me just represent it
as a ratio of two integers. So I have one rational
number right over there. I can represent it as a/b. And I'm going to multiply it
times another rational number, and I can represent that as a
ratio of two integers, m and n. And so what is this
product going to be? Well, the numerator,
I'm going to have am. I'm going to have a times m. And in the denominator, I'm
going to have b times n. Well a is an integer,
m is an integer. So you have an integer
in the numerator. And b is an integer
and n is an integer. So you have an integer
in the denominator. So now the product is a ratio
of two integers right over here, so the product is also rational. So this thing is also rational. So if you give me the product
of any two rational numbers, you're going to end up
with a rational number. Let's see if the same thing
is true for the sum of two rational numbers. So let's say my first
rational number is a/b, or can be represented as a/b, and
my second rational number can be represented as m/n. Well, how would I add these two? Well, I can find a
common denominator, and the easiest
one is b times n. So let me multiply
this fraction. We multiply this one times
n in the numerator and n in the denominator. And let me multiply
this one times b in the numerator and
b in the denominator. Now we've written
them so they have a common denominator of bn. And so this is going to
be equal to an plus bm, all of that over b times n. So b times n, we've
just talked about. This is definitely going to
be an integer right over here. And then what do
we have up here? Well, we have a times
n, which is an integer. b times m is another integer. The sum of two integers
is going to be an integer. So you have an integer
over in an integer. You have the ratio
of two integers. So the sum of two
rational numbers is going to give you another. So this one right over
here was rational, and this one is right
over here is rational. So you take the product
of two rational numbers, you get a rational number. You take the sum of
two rational numbers, you get a rational number.