Let's analyze each option step-by-step to determine which number is irrational.
### Option A: 0.45
0.45 is a terminating decimal. Any terminating decimal can be expressed as a fraction of two integers. For example:
[tex]\[ 0.45 = \frac{45}{100} = \frac{9}{20} \][/tex]
Since 0.45 can be expressed as a ratio of two integers (9 and 20), it is a rational number.
### Option B: [tex]\(\sqrt{6}\)[/tex]
To determine whether [tex]\(\sqrt{6}\)[/tex] is rational, we need to check if 6 is a perfect square. The square root of a non-perfect square is always irrational. Since 6 is not a perfect square (as the integers between which it lies are 4 (2²) and 9 (3²)), [tex]\(\sqrt{6}\)[/tex] cannot be expressed as a ratio of two integers. Hence, [tex]\(\sqrt{6}\)[/tex] is an irrational number.
### Option C: 0.636363...
0.636363... is a repeating decimal. Any repeating decimal can be converted into a fraction. In this case:
[tex]\[ 0.636363\ldots = \frac{63}{99} = \frac{7}{11} \][/tex]
Since it can be expressed as a fraction (7/11), this number is rational.
### Option D: [tex]\(\sqrt{25}\)[/tex]
To determine whether [tex]\(\sqrt{25}\)[/tex] is rational, we recognize that 25 is a perfect square:
[tex]\[ \sqrt{25} = 5 \][/tex]
Since 5 is an integer, [tex]\(\sqrt{25}\)[/tex] is a rational number.
### Conclusion
Among the given options, [tex]\(\sqrt{6}\)[/tex] is the number that is irrational.
Therefore, the irrational number is:
[tex]\[ \boxed{2} \][/tex]
