How many fractions lie between
0 and 1? Think, discuss and
write your answer?
If you give me the correct answer I will mark you as the brainliest.
Answers
Answer:
infinitely many
Step-by-step explanation:
infinitely many fractions lie between two numbers
Answer:
Infinite
Step-by-step explanation:
Hey there.
Here's your explanation
Between any two whole numbers there is a fraction. Between 0 and 1 there is 12, between 1 and 2 there is 112=3/2, and so on. In fact, there are infinitely many fractions between any two whole numbers. Let’s see why.
Between 0 and 1 there are also 13, 14, 15, and any number that can be written as 1n where n is some whole number. In addition there are fractions like 23, 34, 45, and so on. More generally, if m and n are positive whole numbers and m is smaller than n, then mn is a fraction that lies between 0 and 1. In a similar way, there are also infinitely many fractions in between any other pair of whole numbers.
Numbers that can be written as fractions are called rational numbers. These include the whole numbers, since a whole number can be written as a fraction with 1 in the denominator: 1=11, 2=21, and so on. And as we have just seen, there are not only infinitely many rational numbers in the whole number line, but there are also infinitely many rational numbers in just the small interval between two consecutive whole numbers.
But we can go even further than this. If you give me any two rational numbers x and y on the number line, then no matter how close together they are, I can always find infinitely many other rational numbers that lie between them.
One way of doing this is as follows. Imagine all the numbers as lying along a ruler. Suppose you have given me x=1100 and y=2100. The two are very close together, the difference d is
d=y−x=2100−1100=1100.
The difference d is the length of the little interval that lies between x and y on the ruler.
Now suppose I chop that little interval between x and y in half. The number z that corresponds to the midpoint obviously lies between x and y. The distance from x to that number z is 1100×12=1200. So the number z is
z=x+1200=1100+1200=3200.
x, y and z represented on the axis of positive real numbers, equally spaced
Chopping the interval between x and y in half gives a new number z which is also a rational number.
It is also a rational number, since it can be written as a fraction. Any sum or product of two rational numbers is always itself a rational number, since when you add or multiply two fractions you always get another fraction.
So I have just found you a rational number z that lies in between x and y. But I could find infinitely many others in the same way. Rather than chopping the distance in half, I could have chopped it in three, four, five, a hundred, or any number n of pieces, giving me n little intervals. The distance from x to the right endpoint z of the first interval is then 1100×1n. The point z itself is
z=x+1n=1100+1n.
x, y and z represented on the axis of positive real numbers, z now a third respectively a quarter of the interval from y
Chopping the interval between x and y into three (top) and four (bottom) pieces.
It’s a rational number, since (as we noted above) products and sums of rational numbers are themselves rational numbers.
Since this works for any positive whole number n, and there are infinitely many of those, I have just managed to find infinitely many rational numbers that lie between x and y. The same kind of argument works for any two rational numbers x and y, not just our two specific examples. Therefore, whatever two rational numbers x and y you pick on the number line, I can always find infinitely many other rational numbers that lie between them. Fractions really are everywhere.
Hope it helps you :-)
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