Final answer:
Zero is a non-negative number because it is not less than zero and when added to other numbers, it does not decrease their value. Zero is smaller than any positive number because any positive value, no matter how small, is greater than zero.
Step-by-step explanation:
To prove that zero is a non-negative number but smaller than any positive number, we can rely on the fundamental properties of numbers and signs. In mathematics, numbers can have either a positive (+ve) or negative (-ve) sign, with positive numbers being greater than zero and negative numbers being less than zero.
Zero itself is considered non-negative because it is not less than zero; it is the point of reference from which positivity and negativity are defined.
When adding two positive numbers, the result remains positive, as seen with the equation 3 + 2 = 5. Similarly, when adding two negative numbers, the result remains negative, exemplified by -4 + (-2) = -6. In the case of zero, it does not change the value of the number it adds to because adding zero is equivalent to adding no value.
Therefore, 0 + a positive number equals the positive number itself, which means zero is neither increasing nor decreasing the value of a positive number, establishing it as non-negative.
Looking at the order of numbers, any number with a positive sign (> 0) is greater than zero. Therefore, even the smallest fraction of a positive number, like 0.00001, is larger than zero, meaning zero is less than any positive number.
This is reinforced by our understanding of scientific notation, where even a small number represented by a negative exponent (indicating its small magnitude) is still greater than zero. For example, 10-6 is greater than zero but represents a very small number.