2.5 Real Numbers, the Number Line & Properties of Operations
Key Takeaways
- Every point on the number line is a real number: integers, rationals (any ratio of integers, including every terminating or repeating decimal), and irrationals such as sqrt(2) and pi.
- "Number" never means "integer" on the GMAT. If a stem says x is a number, test 1/2 and -0.4, not only 1, 2, and 3.
- "Between 1 and 4" is 1 < x < 4; "between 1 and 4, inclusive" is 1 <= x <= 4, and the count of integers from a to b inclusive is b - a + 1.
- |x| is the distance from x to 0, so |x| is never negative and -x is positive whenever x is negative.
- The distributive property read right to left is factoring: 37 x 98 + 37 x 2 = 37(98 + 2) = 3,700 replaces two multiplications with one.
Where the Number Line Sits on the Official Outline
Official Guide Quantitative Review 2026–2027 Math Review 3.1 opens with Numbers and the Number Line and closes with Properties of Operations. Those two sub-topics bracket everything else in this chapter. Divisibility, GCF and LCM, remainders, and parity all assume you can place a number on the line, order it against another number, and rearrange an expression without changing its value. On a 21-question, 45-minute, no-calculator section, that rearranging is where the time is won.
The real-number map
Every point on the number line is a real number, and every real number is a point on the number line. GMAT stems name the sub-categories directly, so keep them straight.
| Set | Contains | Examples | Watch for |
|---|---|---|---|
| Integers | the whole numbers and their negatives | −7, 0, 12 | 0 is an integer, and "integer" never implies "positive" |
| Rational | any ratio of two integers with a nonzero denominator | 3/4, −5, 0.25, 0.333... | every terminating or repeating decimal is rational |
| Irrational | reals that are not a ratio of integers | sqrt(2), sqrt(3), pi | non-terminating and non-repeating decimals |
| Real | all of the above | everything on the line | GMAT Quant stays inside the real numbers |
Two traps GMAC uses constantly. First, "number" does not mean "integer." If a stem says x is a number and you only substitute 1, 2, and 3, you will miss the case that breaks the pattern, which is usually a proper fraction such as 1/2 or a negative such as −0.4. Second, "positive" does not mean "integer" either. In n is a positive integer, both words are load-bearing, and dropping either one is how a must-be-true item gets answered wrong.
Order, betweenness, and inclusivity
On the number line each number is less than every number to its right. That single sentence settles most sign comparisons: −9 < −2 because −9 sits farther left, even though 9 > 2.
The official wording matters for counting.
- "x is between 1 and 4" means 1 < x < 4. The endpoints are excluded.
- "x is between 1 and 4, inclusive" means 1 ≤ x ≤ 4. The endpoints count.
The same trap runs through integer counts. The number of integers from a to b inclusive is b − a + 1, not b − a. From 1 to 4 inclusive that is four integers, not three. Chapter 8 reuses this exact count for evenly spaced sets, so build the habit here.
Absolute value is distance, not "drop the sign"
|x| is the distance from x to 0 on the number line: |x| = x when x ≥ 0, and |x| = −x when x < 0. Because it is a distance, |x| is never negative, and |−3| = |3| = 3.
Reading absolute value as distance is what makes |x − 5| = 2 immediate: x is two units from 5, so x = 3 or x = 7. Chapter 5 builds the full inequality machinery on this definition; here it is enough to see it as a length.
One sign note that decides must-be-true items: −x is not automatically negative. If x = −6, then −x = 6. The minus sign reverses whatever sign x already had.
Properties of Operations: Your Licence to Rearrange
The official Math Review lists the algebra of ordinary arithmetic as tested content. You will never be asked to name a property on Problem Solving, but every fast solution uses one.
| Property | Statement | What it buys you on Quant |
|---|---|---|
| Commutative | x + y = y + x and xy = yx | reorder terms to pair friendly numbers |
| Associative | (x + y) + z = x + (y + z) and (xy)z = x(yz) | regroup to build a 10 or a 100 |
| Distributive | x(y + z) = xy + xz | pull a common factor out of an ugly sum |
| Additive identity | x + 0 = x | adding zero changes nothing |
| Multiplicative identity | x × 1 = x | multiply by a disguised 1, such as 5/5, to rewrite a fraction |
| Additive inverse | x + (−x) = 0 | cancel matched terms on sight |
| Zero product | if xy = 0 then x = 0 or y = 0, or both | the entire reason factoring solves equations |
Division by zero is undefined
x/0 has no value for any x, including x = 0. A denominator that could be zero is therefore a hidden restriction rather than a nuisance. Given (x² − 9)/(x − 3), the condition x ≠ 3 exists before you cancel anything, and GMAC writes answer choices for testers who cancel first and check later.
Sign rules, stated once
- positive + positive = positive; negative + negative = negative
- positive × positive = positive; negative × negative = positive; positive × negative = negative
- x − y = −(y − x), so flipping a subtraction flips the sign of the result
- if xy = 0 then at least one factor is 0; if xy > 0 the factors share a sign; if xy < 0 they do not
Order of operations
Parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right. The one that actually costs points is the minus sign in front of a power: −3² = −9, but (−3)² = 9. The exponent binds to the 3 alone unless parentheses capture the sign. Chapter 4 returns to this with negative bases.
Distribution as a No-Calculator Weapon
Read right to left, the distributive property is factoring, and factoring is how you avoid long multiplication on a section with no calculator.
Worked example. 37 × 98 + 37 × 2. Do not compute either product. Factor the shared 37: 37(98 + 2) = 37 × 100 = 3,700.
Worked example. 63 × 101. Split the awkward factor: 63(100 + 1) = 6,300 + 63 = 6,363.
Worked example. 48 × 25. Rewrite 25 as 100/4, which is associativity plus a disguised 1: 48 × 100 / 4 = 4,800 / 4 = 1,200.
Worked example. (873 × 46) − (873 × 45). The stem shows two four-digit multiplications; the distributive property turns it into one subtraction: 873(46 − 45) = 873 × 1 = 873.
Worked example. 19 × 21 + 19 × 79. Factor: 19(21 + 79) = 19 × 100 = 1,900.
The habit to carry into every later chapter: before you compute, look for a shared factor. A stem that displays large products almost never wants you to evaluate them.
If x is a real number with −1 < x < 0, which of the following must be true?
What is the value of 57 × 999 + 57?
For which values of x is the expression (x + 2) / ((x − 3)(x + 5)) undefined?