Why is the product of a rational number and an irrational number irrational

Answers

Answer 1

Answer:

Great question. Let's let r be a rational number and s be irrational. Note r has to be nonzero for this to work. In other words, it's not true that when we multiply zero, a rational number, by an irrational number like π we get an irrational number. We of course get zero.

The question is: why is the product

$p = rs$

irrational?

In math "why" questions are usually answered with an illuminating proof. Here the indirect proof is enlightening.

Suppose p was rational. Then

$s = \frac p r$

would be rational as well, being the ratio of two rational numbers, so ultimately the ratio of two integers.

But we're given that s is irrational so we have our contradiction and must conclude our assumption that p is rational is false, that is, we conclude p is irrational.

Answer 2

Answer:

A proof by contradiction.

Let assume that the product of a rational number and an irrational number is rational.

Let $(a)/(b)$ and $(c)/(d)$ be rational numbers, where $a,b,c,d\in \mathbb{Z} \wedge b,d\not=0$ and $x$ an irrational number.

Then

$(a)/(b)\cdot x=(c)/(d)\nx=(bc)/(ad)$

Integers are closed under multiplication, therefore $bc$ and $ad$ are integers, making the number $x=(bc)/(ad)$ rational, which is contradictory with the earlier statement that $x$ is an irrational number.

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Answers

You multiple it by 20....
1:20
2:40
3:60
4:80
5:100
So it’s 80 inches by 100 inches (80x100) =800 square feet

I was going to try to answer but the guy above answered it perfect aha

Hi really need 1, 2 and 3!!!!! Thank you all in advance! Again just ONE, TWO, and THREE thanks!

Answers

1. is 19.24
2. is 8,16,24,32,40
3. is associative property of addition

1.) 19.24
2.) 8, 16, 24, 32, 40
3.) associative property

Naomi's pedometer recorded 43,498 steps in one week. Her goal is 88,942 steps. Naomi estimate she has about 50,000 more steps to meet her goal. Is Naomi's estimate reasonable?

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yes it is not because 40 thousand + 5 thousand = 90,000 which is close to her goal

If 3 pieces of candy cost $2.10,how much would four pieces cost?

Answers

We need to find the unit rate

2.10/3 = 0.70

0.70*4 = 2.80

Answer: 4 pieces would cost $2.80

210 divided by 3 equals 70

210+70= 280

$2.80

How do i round 3,176 to the nearest thousand

Answers

Here is the rule: 4 and below let it go (round down), 5 and above give it a shove (round up)
Look to the place after the thousandths in the number 3,176 (the 1)
Since it is below 5, you would round down to 3,000

Well you would look at the 176 in 3,176. Since it is less then half (500) you would round it down. So 3,176 rounded to the nearest thousand would be 3,000

A shelter has enough food to feed an average of 80 hungry people a day for 14 days. If an average of 32 people are at the shelter each day, how long will the food last?

Answers

Answer:

35 days.

Explanation

The capacity of the shelter is a constant. It is 80 people × 14 days.

That is an inverse relation: the product of the number of people and the number of days will be always the same.

As the number of people decreases the number of days that the food lasts increases, in a proportion such that the product of the days and the number of people is the same.

Thus, for 32 people the number of days the food lasts is x, such that: