Forming the Smallest and Largest Numbers

When it comes to forming numbers using a specific set of digits exactly once, arranging the digits in ascending or descending order provides a straightforward strategy for finding the smallest or largest number. Let's delve into the process, including various scenarios and considerations that might arise.

Understanding the Basics of Smallest and Largest Numbers

The smallest number can be formed by arranging all the digits in ascending order. For example, if we have the digits 4, 7, 2, and 5, we arrange them as follows:

Smallest Number

The digits in ascending order are: 2, 4, 5, 7. Therefore, the smallest number that can be formed is 2457.

Considerations and Beyond Regular Digits

Things get more interesting when non-numerical symbols like fractions, decimals, factorials, and superscripts are introduced. These symbols can significantly change the value of the number formed. For instance, we can consider the use of a fraction line or decimal point.

Using Fractions and Decimal Points

A fraction can drastically reduce the value of a number. Here are some examples:

2/754 is a smaller number than 2457. .2457 is smaller than 2457, but it's not as small as some fractions like 2/754.

Adding a decimal point or a superscript can also modify the value. For instance, 2/7^5^4 might be even smaller, provided the syntax is correctly formatted.

Another intriguing possibility is using a decimal point in conjunction with a superscript to represent a root of a number. For example:

0.2457 is a very small number, but it can be even smaller using a factorial or a square root.

Using Factorials to Get Even Smaller Numbers

Factorials (denoted by !) can reduce the number further. For example, if factorials are allowed in the format of a decimal number, the smallest number could be:

-7542! would be much smaller than simple decimal forms.

Combining Digits and Other Symbols for Smaller Numbers

If the task allows the use of additional digits or more complex operations, the smallest number can be further reduced. Here are a couple of examples:

The smallest whole number from the digits 4, 7, 2, and 5 is 2457. Allowing the use of a decimal point and digit 1, the smallest number would be .012457.

In conclusion, the smallest number formed using the digits 4, 7, 2, and 5 is typically 2457 unless additional digits or non-numerical symbols are allowed, in which case the smallest number can vary significantly.