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Real Numbers, Absolute Value

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  1. Real Numbers
  2. Absolute Value

What are Real Numbers?

Real numbers are any numbers that can be represented on the number line.

real numbers

A number line showing the distance between -1 and 1.

The set of real numbers (also called the reals) includes both the set of rational numbers (numbers that can be written as ratios or fractions) and the set of irrational numbers (numbers that can’t be written as fractions, like pi).

There are an infinite number of reals, and there is an infinite number of real numbers between any two real numbers. For example, between 1 and 2 there are 0.1, 0.21, 0.239, and so on.

All whole numbers (1,2,3,4…) and integers (-2098, -2, 1, 2…) are also included in the set of reals.

The set of reals is denoted by the symbol ℝ .

What Number is Not a Real Number?

Imaginary numbers of the from a + ib, with i= √(-1) and b ≠ 0, are not reals.

Core Properties of Real Numbers.

Since every real number has its own place along the number line, reals can be ordered. Properties of real numbers include:

  • The Distributive Property: a(b + c)= ab +ac for every a, b, and c in ℝ.
  • The Commutative Property: ab = ba and a + b = b + a for a and b in ℝ.
  • The Associative Property: (ab)c = a(bc) and (a + b) + c= a + (b + c) for all a, b, and c in ℝ
  • The Closure Property: a + b, a – b, and ab are all reals if a and b are real numbers; that is to say, real numbers are closed under addition, subtraction, and multiplication.

The Identity Properties of Real Numbers

There is one multiplicative identity, 1, and one additive identity, 0, such that

1a = a and 0 + a = a
for every a in ℝ.

The Inverse Properties of Reals

Every real number a has an unique additive inverse –a such that a + (-a) = 0, with –a being a member of the set of real numbers. Every nonzero real number a also has a unique multiplicative inverse 1/a such that a(1/a) = 1, where 1/a is a real number.

What is Absolute Value?

Watch the video or read the article below:

Absolute Value on the Number Line

The absolute value is used often in probability and statistics and is a number’s positive distance from zero on the number line. As it is a positive distance, absolute value can’t ever be negative.

absolute value

The absolute value of -3 is 3, because it’s three spaces from zero on the number line.


The absolute value also refers to the magnitude of a number. You can easily figure out the magnitude of any number by removing the negative sign. For example, the absolute value of -10 is 10 and -1000 is 1000.

Absolute value of a number is sometimes called the “modulus” of a number and is denoted by vertical bars on each side of the number. For example:

  • |-10| = the abs. value of 10 = 10
  • |-3| = 3
  • |-22| = 22

It is similar to, but different from, the term absolute difference, |x-y,| which is the distance of two numbers on the number line. For example the absolute difference of -5 and 4 is 9:

  • |-5 – 4| = 9
  • |-22 – 1| = 23
  • |4 – -2| = 6

(If you ever wondered why “two negatives equal a positive” in algebra, this last example, 4 – -2 helps to explain why!).

According to Math Boys, the word absolute comes from a variant of the word absolve and has a meaning close to free from conditions or restriction. The term, as it relates to mathematics, was first found in 1950 in the elements of analytical geometry; comprehending the doctrine of the conic sections, and the general theory of curves and surfaces of the second order by John Radford Young (1799-1885). “we have AF the positive value of x equal to BA – BF, and for the negative value, BF must exceed BA, that is, F must be on the other side of A, as at F’, hence making AF’ equal to the absolute value of the negative root of the equation”.”


  1. The Improving Mathematics Education in Schools (TIMES) Project (2011). Retrieved June 15, 2018 from:
  2. Schechter, E. What are the “real numbers,” really? Retrieved June 15, 2018 from:
  3. Peters, C. (n.d.). The Real Numbers. Retrieved June 15, 2018 from:

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Real Numbers, Absolute Value was last modified: June 15th, 2018 by Stephanie