What are the Smallest Numbers Formed by Using the Digits 4, 1, 0, 5, 6, and 2?

When dealing with the smallest numbers formed using the digits 4, 1, 0, 5, 6, and 2, there are several factors to consider:

Definitions and Assumptions

The term "numbers" can refer to both integers and decimals. "Formed" can imply concatenation of the digits in some order, with the condition that all digits must be used without repetition. Let's explore these scenarios in detail.

Small Integers: Using All Digits Without Leading Zero

When forming integers using all the digits 4, 1, 0, 5, 6, and 2, and ensuring there's no leading zero, the smallest positive integer is 102456. This is because any arrangement starting with zero would result in a smaller integer, but it wouldn't be considered a positive number under the given rules.

Smallest Integers Including Negative Numbers

The smallest integer, both positive and negative, formed using these digits is -654210. This takes into account that negative numbers are valued lower than positive ones. For instance, -654210 is smaller than any positive number formed using these digits.

Smallest Decimal Numbers

When considering decimal numbers, the smallest positive number is 0.012456. This is achieved by placing the smallest non-zero digit (1) in the tenths place and arranging the remaining digits in ascending order after the decimal point. The smallest negative number remains -654210, assuming the decimal point is not required.

Considering Arithmetic Operations

To further explore the smallest numbers, we can apply arithmetic operations. For example, the expression 0-2456-1 is calculated as follows:

First, evaluate the exponentiation: 456. This results in a very large number, specifically: 4100000-2*{large number} (which is effectively a subtraction of a even larger negative number from 2). Finally, subtract 1: -1.9E81-1, which is a negative number with 81 significant digits.

This expression demonstrates that the concept of forming the smallest number can be expanded to complex expressions involving mathematical operations. However, for practical purposes, the smallest integer formed by arranging the digits 4, 1, 0, 5, 6, and 2 is still -654210.