Question

rational vs. irrational numbers
directions: select two colors for your key. color all of the rational numbers one color and all of the irrational numbers a different color.
key: square - rational square - irrational
\frac{12}{5} quad sqrt{56} quad -3.178925 quad sqrt{121} quad -\frac{3}{4}
sqrt{16} quad 21.6 quad -23.4567overline{89} quad \frac{299}{2} quad 63.767689...
.overline{45} quad pi quad -48 quad 2.141414... quad sqrt{49}
sqrt{2} quad -8.542176... quad sqrt{34} quad 19.457312... quad 3.14
-98.742018... quad 17\frac{2}{9} quad 4.232 quad sqrt{87} quad sqrt{84}
what is the difference between rational and irrational numbers? explain.

Explanation:

Step1: Identify rational numbers

Rational numbers can be written as $\frac{p}{q}$ where $p,q$ are integers, $q
eq0$; they include terminating decimals, repeating decimals, integers, fractions, and perfect square roots:
$\frac{12}{5}$, $-3.178925$, $\sqrt{121}=11$, $-\frac{3}{4}$, $\sqrt{16}=4$, $21.6$, $-23.4567\overline{89}$, $\frac{299}{2}$, $. \overline{45}$, $-48$, $2.141414...$, $\sqrt{49}=7$, $3.14$, $17\frac{2}{9}=\frac{155}{9}$, $4.232$

Step2: Identify irrational numbers

Irrational numbers cannot be written as $\frac{p}{q}$; they are non-terminating, non-repeating decimals, and non-perfect square roots:
$\sqrt{56}$, $63.767689...$, $\pi$, $\sqrt{2}$, $-8.542176...$, $\sqrt{34}$, $19.457812...$, $-98.742018...$, $\sqrt{87}$, $\sqrt{84}$

Step3: Explain the difference

Rational numbers can be expressed as a ratio of two integers (denominator non-zero), and their decimal forms are either terminating or repeating. Irrational numbers cannot be written as such a ratio, and their decimal forms are non-terminating and non-repeating.

Answer:

Categorization:

Rational Numbers:

$\frac{12}{5}$, $-3.178925$, $\sqrt{121}$, $-\frac{3}{4}$, $\sqrt{16}$, $21.6$, $-23.4567\overline{89}$, $\frac{299}{2}$, $. \overline{45}$, $-48$, $2.141414...$, $\sqrt{49}$, $3.14$, $17\frac{2}{9}$, $4.232$

Irrational Numbers:

$\sqrt{56}$, $63.767689...$, $\pi$, $\sqrt{2}$, $-8.542176...$, $\sqrt{34}$, $19.457812...$, $-98.742018...$, $\sqrt{87}$, $\sqrt{84}$

Difference Explanation:

Rational numbers can be represented as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$; their decimal expansions either terminate or repeat. Irrational numbers cannot be written as such a fraction, and their decimal expansions are non-terminating and non-repeating.