Integers and Number Lines

Integers and the Number Line

The integers are the whole numbers together with their negatives: \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots. A number line is a picture of the integers (and, more generally, all real numbers) laid out in order on a line, increasing to the right. Number lines make it easy to compare numbers: whichever number sits further to the right is the larger one, no matter how negative or positive either number looks at first glance.

Absolute Value

The absolute value of a number is its distance from 0 on the number line, and distance is never negative. We write absolute value with vertical bars. For example, |5| = 5 and |-5| = 5, because both 5 and -5 sit exactly 5 units from 0. In general:

• |a| = a when a \geq 0
• |a| = -a when a < 0 (this makes the result positive)

Adding and Subtracting Signed Integers

Think of addition on the number line as movement: adding a positive number moves you right, and adding a negative number moves you left. Subtraction is the same as adding the opposite, so a - b always equals a + (-b). A few useful rules:

• Same signs: add the absolute values and keep the common sign. (-4) + (-7) = -11
• Different signs: subtract the smaller absolute value from the larger, and keep the sign of whichever number is farther from 0. (-9) + 3 = -6
• Subtracting a negative is the same as adding a positive: 8 - (-2) = 8 + 2 = 10

Multiplying and Dividing Signed Integers

For multiplication and division, only the signs matter, not the specific values:

• Positive \times positive = positive, and negative \times negative = positive (like signs give a positive result).
• Positive \times negative = negative, and vice versa (unlike signs give a negative result).
• The exact same rule applies to division: (-12) \div (-3) = 4, but (-12) \div 3 = -4.

A handy shortcut for a long product or quotient of signed numbers: count how many negative factors appear. An even number of negative factors gives a positive result; an odd number gives a negative result.

Order of Magnitude Intuition

Competition problems often reward a quick sanity check on the size of an answer before you trust an exact computation. Ask yourself: is this answer supposed to be around 10, around 100, or around 1{,}000? For instance, 29 \times 31 is close to 30 \times 30 = 900, so an answer like 899 looks reasonable while an answer like 89{,}900 should raise a red flag. Building this habit catches sign errors and misplaced decimal points quickly.

Example

Simplify -8 + 15 - (-6) - 20.

Step 1: Rewrite every subtraction as adding the opposite: -8 + 15 + 6 - 20

Step 2: Add the positive numbers and the negative numbers separately. Positives: 15 + 6 = 21. Negatives: -8 + (-20) = -28.

Step 3: Combine: 21 + (-28) = -7

So -8 + 15 - (-6) - 20 = -7.

Key Takeaways

• On a number line, larger numbers are farther right; absolute value measures distance from 0 and is always non-negative.
• To add/subtract signed numbers, rewrite subtraction as "add the opposite," then combine same-sign terms first.
• For multiplication and division, count the negative signs: an even count gives a positive answer, an odd count gives a negative answer.
• Estimate the size of an answer first to catch careless sign or magnitude errors.