GMAT Preparation Online: Core Number Properties

In the vast landscape of mathematics, certain numbers serve as the bedrock upon which more complex concepts are built. Foremost among these are integers. Far more than just simple counting numbers, integers form a foundational set that underpins everything from basic arithmetic to advanced algebra, and they are absolutely crucial for success in quantitative reasoning. This comprehensive lesson will take you on an in-depth journey through the world of integers, their unique characteristics, their precise placement on the number line, and the critical concept of absolute value, ensuring a robust understanding vital for GMAT preparation.

I. Defining Integers: What's In and What's Out?

At its core, the set of integers encompasses all "whole" numbers, whether they are positive, negative, or zero.

• Positive Integers: These are the numbers we use for basic counting, also known as natural numbers or counting numbers. They start from 1 and extend indefinitely: 1, 2, 3, 4, 5, ...

• Negative Integers: These are the "opposites" of the positive integers, representing values less than zero. They extend indefinitely in the negative direction: ..., -5, -4, -3, -2, -1.

• Zero (0): This unique integer serves as the central point. It is neither positive nor negative. It signifies the absence of quantity or a neutral state.

The complete set of integers, denoted by the symbol Z (from the German word "Zahlen" for numbers), can be listed as:

..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...

What Integers ARE NOT:

It's equally important to understand what does not belong to the set of integers. Integers are strictly whole numbers. Therefore:

• Fractions (e.g., 1/2, -3/4, 5/2) are NOT integers.

• Decimals (e.g., 0.5, -2.7, 3.14) are NOT integers, unless they simplify to a whole number (e.g., 2.0 is an integer, but 2.1 is not).

• Numbers with radicals that don't simplify to a whole number (e.g., √2, √5) are NOT integers.

II. The Number Line: Visualizing Order and Magnitude

The number line is an indispensable tool for understanding integers. It's a straight line where every point corresponds to a real number, and conversely, every real number has a unique point on the line. Integers are specifically marked at equal, discrete intervals along this continuous line.

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                                    -5     -4     -3     -2     -1      0      1      2      3      4      5

• The Origin (Zero): The number 0 is positioned at the center of the number line. It acts as the reference point.

• Positive Direction: Numbers to the right of zero are positive. As you move further right, the value of the number increases. The arrow to the right indicates that the positive integers extend infinitely.

• Negative Direction: Numbers to the left of zero are negative. As you move further left, the value of the number decreases (it becomes "more negative"). The arrow to the left indicates that the negative integers extend infinitely.

• Discrete vs. Continuous: While the number line itself is continuous (meaning there are infinitely many points, including fractions and decimals, between any two given points), integers are discrete points – there are no other integers between, say, 2 and 3.

III. Ordering Integers: Understanding "Greater Than" and "Less Than"

The number line provides an intuitive and unambiguous way to compare integers.

The Golden Rule:

• Any number positioned to the right of another number on the number line is greater than (>) that number.

• Any number positioned to the left of another number on the number line is less than (<) that number.

Examples:

• 7 > 3 (7 is to the right of 3)

• -2 > -8 (-2 is to the right of -8. This can be counter-intuitive; think of temperature: -2 degrees Celsius is warmer than -8 degrees Celsius.)

• 0 > -5 (0 is to the right of -5)

• 4 < 9 (4 is to the left of 9)

• -1 < 0 (-1 is to the left of 0)

• -10 < -1 (-10 is to the left of -1)

Comparing Multiple Integers (Ascending/Descending Order):

You can arrange a set of integers in ascending order (from smallest to largest) by placing them as they appear from left to right on the number line. For descending order, arrange them from right to left.

• Example: Arrange {-6, 3, 0, -2, 5} in ascending order. On the number line: -6 is furthest left, then -2, then 0, then 3, then 5. Ascending order: -6, -2, 0, 3, 5.

IV. Absolute Value: Measuring Distance from Zero

The concept of absolute value allows us to measure the "size" or "magnitude" of an integer without considering its direction (positive or negative). It represents the distance of an integer from zero on the number line. Since distance is always a non-negative quantity, the absolute value of any integer will always be non-negative (positive or zero).

Notation: The absolute value of an integer x is denoted by |x|.

Formal Definition:

• If x is positive or zero (x ≥ 0), then |x| = x.

• If x is negative (x < 0), then |x| = -x (which makes it positive).

Examples:

• |7| = 7 (The distance from 0 to 7 is 7 units).

• |-7| = 7 (The distance from 0 to -7 is also 7 units).

• |0| = 0 (The distance from 0 to 0 is 0 units).

• | -2.5 | is not an integer's absolute value (but would be 2.5).

Absolute Value and Equations/Inequalities:

Problems involving absolute value often have two possible solutions or ranges, one positive and one negative.

• Equation Example: Solve |x| = 4.

  This means x is 4 units away from 0. So, x can be 4 or x can be -4.

  Solutions: x = 4 or x = -4.

• Inequality Example: Solve |x| < 3.

  This means x is less than 3 units away from 0. On the number line, this includes all integers between -3 and 3 (but not including -3 or 3 themselves).

  Solutions (for integers): -2, -1, 0, 1, 2.

V. Why Integers and the Number Line are Crucial for the GMAT

A deep understanding of integers forms the basis for many GMAT quantitative concepts:

• Number Properties: Parity (even/odd), prime and composite numbers, factors, and multiples all rely on a solid grasp of integers.

• Algebra: Solving equations and inequalities, especially those involving absolute values, requires proficiency with integer operations and comparisons.

• Problem Solving: Many word problems involve quantities that must be integers (e.g., number of people, items), and understanding positive/negative values is key (e.g., profit/loss, temperature changes).

• Data Sufficiency: Often, understanding the properties of integers (like whether a variable must be an integer, or its range) is enough to determine sufficiency.

VI. Common Pitfalls to Avoid

• Negative Number Comparison: The most frequent error. Remember that for negative numbers, the one closer to zero has a greater value. -1 is much larger than -100.

• Misinterpreting Absolute Value: Do not confuse |-x| with -x. |-x| will always be positive (or zero), whereas -x will be positive only if x is negative (e.g., if x = -5, then -x = 5).

• Fractions/Decimals are NOT Integers: Be vigilant about question wording. If a question states "x is an integer," it implicitly rules out non-whole numbers.

• Zero's Position: Remember zero is an integer, and it's neither positive nor negative.

Interactive Check Your Understanding:

1. Is -3.5 an integer? Why or why not?

2. Which integer is greater: -12 or -8?

3. Evaluate |4 - 9|.

4. List all integers x such that |x| ≤ 2.

GMAT Practice Questions:

1. Place the following integers in ascending order: {-15, 7, -2, 0, 11, -9}.

2. If an integer K is less than -3 but greater than -7, list all possible integer values for K.

3. A diver is at -25 feet relative to sea level. A bird is flying at an altitude of 18 feet. What is the absolute difference in their vertical positions?

4. Which of the following statements is true?

   a) |-6| < -5

   b) |2 - 7| = 5

   c) | -3 | = -3

   d) 0 < |-1|

5. On a number line, point A is at -10 and point B is at 6. What is the integer exactly halfway between points A and B?

Solutions to Practice Questions:

1. Ascending Order:

   {-15, -9, -2, 0, 7, 11}

2. Integer K values:

   K is less than -3 (K < -3), and K is greater than -7 (K > -7).

   So, -7 < K < -3.

   The integers satisfying this are: -6, -5, -4.

3. Absolute difference in vertical positions:

   Diver's position: -25 feet

   Bird's position: 18 feet

   Difference = 18 - (-25) = 18 + 25 = 43.

   Absolute difference = |43| = 43 feet.

4. True Statement:

   a) |-6| < -5 evaluates to 6 < -5, which is False.

   b) |2 - 7| = 5 evaluates to |-5| = 5, which is True.

   c) | -3 | = -3 evaluates to 3 = -3, which is False.

   d) 0 < |-1| evaluates to 0 < 1, which is True. (Wait, there could be multiple true statements. Let me re-evaluate this as a typical GMAT question usually has only one correct option for "Which of the following statements is true?" Unless it's "Which of the following statements are true?").

   Assuming single correct answer, b is definitively true. Let's re-check d. 0 < |-1| -> 0 < 1. This is also true.

   Self-correction: In a GMAT context, there would be only one correct answer unless specified. Both B and D are mathematically true. For the purpose of this exercise, I will highlight both as true.

5. Integer halfway between A and B:

   Point A = -10

   Point B = 6

   Distance between A and B = |6 - (-10)| = |6 + 10| = |16| = 16 units.

   Halfway point = (-10 + 6) / 2 = -4 / 2 = -2.

   Alternatively, from -10, move half the distance (16/2 = 8 units) to the right: -10 + 8 = -2.

   From 6, move half the distance (8 units) to the left: 6 - 8 = -2.

   The integer exactly halfway between points A and B is -2.