Positive and Negative Numbers
Numbers strictly greater than zero are positive, whereas numbers strictly less than zero are negative.
- The sum of two numbers with the same sign retains their common sign. The sum of two numbers with opposite signs takes the sign of the number with the greater absolute value.
- The product or quotient of two numbers with the same sign is always positive; the product or quotient of two numbers with opposite signs is always negative.
- Any non-zero real number raised to an even power yields a positive result. For an odd power, a positive base remains positive, while a negative base yields a negative result.
- The sign of the difference between two numbers is determined by their relative order (magnitude) rather than their individual signs, as shown below:
\begin{equation}a>b \Rightarrow a-b>0, \quad a<b \Rightarrow a-b<0\end{equation}
$$\Rightarrow \text{Let } a – b = x$$
$$\text{Then:}$$
$$ \text{If } a > b, \text{ then } x \text{ is positive.} $$
$$ \text{If } a < b, \text{ then } x \text{ is negative.} $$
Example:
Let $a$, $b$, and $c$ be integers such that $a \cdot b = 28$ and $b \cdot c = 35$. Find the maximum and minimum possible values for the sum $a+b+c$.
$$a \cdot b = 28 \implies 1 \cdot 28 = 2 \cdot 14 = 4 \cdot 7$$
$$a \cdot b = 28 \implies (-1) \cdot (-28) = (-2) \cdot (-14) = (-4) \cdot (-7)$$
$$b \cdot c = 35 \implies 1 \cdot 35 = 5 \cdot 7$$
$$b \cdot c = 35 \implies (-1) \cdot (-35) = (-5) \cdot (-7)$$
Since $b$ is a common factor in both equations, the possible integer values for $b$ are $1$, $-1$, $7$, or $-7$. Correspondingly, the values for $a$ are $28$, $-28$, $4$, or $-4$, respectively;
and the values for $c$ are $35$, $-35$, $5$, or $-5$, respectively. Thus, the possible values for the sum $a + b + c$ are analyzed to find the extrema:
The maximum value is: $$1+28+35=64$$ The minimum value is: $$ (-1)+(-28)+(-35)=-64 $$