**Contents:**

## What are Real Numbers?

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

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.

## Magnitude

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!).

**History**

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”.”

## References

- The Improving Mathematics Education in Schools (TIMES) Project (2011). Retrieved June 15, 2018 from: http://amsi.org.au/teacher_modules/Real_numbers.html
- Schechter, E. What are the “real numbers,” really? Retrieved June 15, 2018 from: https://math.vanderbilt.edu/schectex/courses/thereals/
- Peters, C. (n.d.). The Real Numbers. Retrieved June 15, 2018 from: https://www.math.uh.edu/~charles/Real-nos.pdf

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