Number lines and signed quantities

A signed number carries two pieces of information: its distance from zero and its direction from zero. Positive values lie to the right of zero and negative values lie to the left. The sign is not an instruction to subtract; it is part of the number’s name. Thus $-5$ is a single number five units left of zero.

Order follows position. A number farther right is greater, so $-2 > -7$ even though $7$ has the larger absolute value. Absolute value measures distance from zero and therefore cannot be negative: $|-7| = 7$. Opposites have equal absolute value and lie on opposite sides of zero.

Subtraction asks for a difference or change. Rewriting $a - b$ as $a + (-b)$ makes the direction explicit and gives one reliable rule for every sign combination. For example, $4 - (-3) = 4 + 3 = 7$. Check signed arithmetic by estimating direction and by reversing the operation.

Signed arithmetic becomes much easier when a number and an operation are kept separate. In $-7,$ the minus sign belongs to the number and locates it seven units left of zero. In $4 - 7,$ subtraction is the operation being performed. Parentheses make the distinction visible: $4 + (-7)$ means start at $4$ and apply a change of negative $7$. This language works for temperature, elevation, debt, direction, and any other quantity with a meaningful zero. Before calculating, name what zero means and what the positive direction means; otherwise a numerically correct result can still be interpreted backward.

To add signed numbers, first ask whether the changes point in the same direction. If both are positive, or both are negative, combine their distances and keep the common direction. If the signs differ, the movements compete: subtract the smaller absolute value from the larger absolute value and keep the direction of the movement with greater magnitude. This is not a separate collection of sign tricks. It is exactly what happens on a number line. For $-9 + 4,$ the four-unit rightward movement cancels four of the nine leftward units, leaving a position of $-5$.

Subtraction can describe removal, comparison, or change, but all three meanings agree with adding an opposite. The expression $a - b$ asks for the signed difference that carries $b$ to a; rewriting it as $a + (-b)$ turns that difference into a direction. This explains the easily memorized but often misunderstood statement that subtracting a negative becomes addition. The opposite of $-6$ is $+6,$ so $2 - (-6) = 2 + 6$. The two adjacent minus signs are not magically multiplying; one names subtraction and the other belongs to the number being subtracted.

Absolute value is useful for distance because distance ignores direction. The distance between a and $b$ is |a $-$ b|, not |a| $-$ |b|. For points $-4$ and $7,$ the distance is $|7 - (-4)| = |11| = 11$. Order, however, does depend on direction: a point farther right is greater. This is why $-3 > -10$ even though $|-10| > |-3|$. When a comparison feels uncertain, draw a short number line, mark zero, and place the two values. The picture prevents absolute magnitude from being mistaken for numerical order.

A reliable signed-number check uses the story as well as the arithmetic. Predict the direction before calculating: a withdrawal should lower $a$ balance, an upward movement should raise elevation, and a fall larger than the starting positive temperature should cross zero. After calculating, reverse the change. If a balance of $-\$18$ receives $a \$25$ deposit and becomes $\$7,$ subtracting the deposit should recover $-\$18$. This reverse-operation check catches copied signs and direction errors. It also builds the habit used later when equations are checked by substituting a proposed solution into the original statement.

Signed numbers are especially useful when a quantity changes repeatedly. Instead of restarting a story after every event, keep one running total: initial value $+$ first change $+$ second change $+$ ⋯. A bank balance of $\$12$ followed by $a \$20$ purchase, $a \$5$ refund, and $a \$3$ fee is $12 + (-20) + 5 + (-3) = -6$. Grouping the positive and negative changes gives $(12 + 5) + [(-20) + (-3)] = 17 - 23 = -6$. The regrouping is legal because addition is associative and commutative; it is not legal to rearrange an unrevised chain of subtractions as though every symbol were addition.

Pay special attention to words that reverse the viewpoint. “How much greater is a than $b$?” asks for $a -$ b, while “the change from a to $b$” asks for $b -$ a. Those expressions have opposite signs unless the values are equal. Write a short arrow from the initial value to the final value before choosing an expression. The arrow identifies direction, and its length identifies absolute change. This habit separates three answers that students often blur together: final position, signed change, and distance traveled. They may have the same magnitude in a one-step problem, but they do not represent the same quantity.