Find one rational number and one irrational number between 0.3333.... and 0.4444....
To find a rational number between 0.3333... and 0.4444..., we can convert these decimal representations into fractions and find an appropriate fraction in between them.
0.3333... can be written as 1/3, while 0.4444... can be written as 4/9. Therefore, we have the fractions 1/3 and 4/9 representing 0.3333... and 0.4444... respectively.
To find a rational number between them, we can take the average of these two fractions:
(1/3 + 4/9) / 2 = (3/9 + 4/9) / 2 = 7/18
Hence, 7/18 is a rational number between 0.3333... and 0.4444...
Now, to find an irrational number between them, we can choose any irrational number within this range, such as the square root of 2 (approximately 1.414).
Hence, the square root of 2 is an irrational number between 0.3333... and 0.4444...
To find a rational number and an irrational number between 0.3333... and 0.4444..., we can start by converting these repeating decimals into fractions.
Let's define the decimal 0.3333... as x and the decimal 0.4444... as y.
To convert x into a fraction, we can represent it as follows:
x = 0.3333...
Multiply both sides of the equation by 10 to move the decimal point:
10x = 3.3333...
Now, subtract the original equation from the equation with multiplied sides to eliminate the repeating decimal:
10x - x = 3.3333... - 0.3333...
9x = 3
Divide both sides by 9:
x = 3/9
x = 1/3
Hence, the rational number between 0.3333... and 0.4444... is 1/3.
To find an irrational number between these two, we can use the fact that any number between two rational numbers is irrational. Since we know 1/3 is a rational number, we can find another rational number greater than 1/3, such as 2/3. Therefore, any number between 1/3 and 2/3 will be irrational.
For example, we can choose the average of 1/3 and 2/3:
(1/3 + 2/3) / 2 = 3/6
= 1/2
Hence, the irrational number between 0.3333... and 0.4444... is 1/2.