Real Numbers

While preparing for the Quantitative reasoning section in GRE, Real numbers contribute to a significant part of the GRE syllabus. This article would help in understanding some of the concepts of Real numbers.

Want to understand the concepts of real numbers?

Let’s play a game. Imagine any number, any number that you can think of; now ask whether it is a real number or not. The answer you will be getting is YES. Amazed? This is the truth; real numbers have an infinite range that can include any number within it. Imagine real numbers as a superset, whose subsets are natural numbers, whole numbers, rational and irrational numbers, integers, fractions and decimals. The only number that is not a part of the real number is the imaginary number; which is√−1.

Real numbers can be represented on the real number line.

Number Line

\(\frac{1}{2} > \frac{-1}{2}\)


\(\frac{\sqrt{3}}{2} > \frac{1}{2}\)


\(-1 < 0 < \sqrt{982}\)

Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. On the number line, all numbers to the left of 0 are negative and all numbers to the right of 0 are positive. Only the number 0 is neither negative nor positive.

Properties of Real Numbers

  • \(p + q = q + p\)
  • \(pq = qp\)
  • \((p + q) + r = p + (q + r)\)
  • \((pq)r = p(qr)\)
  • \(p (q + r) = pq + qr\)
  • \(p + 0 = p\)
  • \((p)(0) = 0\)
  • \((p)(1) = p\)
  • If pq = 0, then either p=0 or q=0, or both, p, q = 0.
  • Dividing any number by 0 is not defined
  • Both, p + q and pq will be positive if p and q is positive.
  • Both, p + q and pq will be negative if p and q is negative.
  • pq will be negative if either p or q is negative
  • Triangle Inequality: \(|p + q| \leq |p| + |q|\)
  • \(|p||q| = |pq|\)
  • \(if \; p > 1, then \; p^{2} > p\)
  • \(if \; 0 < p < 1, then \; p^{2} < p\)

These properties of Real numbers would assist you in your preparation in the GRE Quantitative Reasoning section.

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