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Classifying Real Numbers Solutions

• Every number that you have learned about so far is a    real number   . That includes square and cube roots, $\pi$, as well as positive or negative numbers.
• Real numbers are all the numbers that can be plotted on a    number line   . 

• To    classify    a real number is to identify the specific subsets to which it belongs.
• Each number set in the Real Number System is represented by a    capital script letter    that you can see next to the name of the set. 

• Real numbers (R)  have two main subsets:    rational (Q)    and    irrational (I)    numbers.

• Within the set of rational numbers, there are    integers (Z)   ,    whole numbers (W)   , and    natural numbers (N)   .
  • 
Natural numbers (N) are often thought of as “counting numbers” beginning at one. Written as a set:$\underline{\mathbf{ }\mathbf{ }\mathbf{ }\left\{1, 2, 3, ...\right\}\mathbf{ }\mathbf{ }\mathbf{ }}$.

  • Whole numbers (W) include all of the natural numbers and add the number 0. Written as a set: $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\left\{0, 1, 2, 3, ...\right\}\mathbf{ }\mathbf{ }\mathbf{ }}$.

  • Integers (Z) include all of the whole numbers and their opposites. Integers are positive and negative whole numbers. Written as a set: $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\left\{..., –3, –2, –1, 0, 1, 2, 3, ...\right\}\mathbf{ }\mathbf{ }\mathbf{ }}$.

• Rational numbers (Q) can be expressed as a $\underline{ \mathbf{ratio}\mathbf{ }\frac{\mathit{a}}{\mathit{b}}\mathbf{ }\mathbf{ }\mathbf{ }}$ where a and b are integers and b cannot equal zero.

Note

‘b’ cannot equal zero, because dividing by zero is undefined.

• Integers (Z) , whole numbers (W) and natural numbers (N) can be written as    ratios    when put over the number    one   .
• Rational numbers (Q) can be written as    fractions    or    decimals   .
• However, when rational numbers are written as decimals they will either    terminate    or    repeat   .

• Irrational numbers    cannot    be written as a ratio of two integers. This means it cannot be written    exactly    as a fraction or a decimal. 

• Irrational numbers when written as a decimal are    non-terminating    and    non-repeating   .

Each subset of the real number system can be represented in this diagram.

Example 1

Determine if the number is rational or irrational. Explain.

Example 2

Determine if the rational number is an integer, whole number, or natural number. It is possible to have more than one answer.  

Note

Q: Why are all natural and whole numbers integers?

A: The set of integers contains all whole numbers and their negatives.

Example 3

Place the numbers for A–E in the most specific set in the diagram.

Note

Q: Why is the $\sqrt{9}$ a natural number?

A: Because $\sqrt{9}=3.$