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HomeTechnologyWhat is a rational number? What is an irrational number?

What is a rational number? What is an irrational number?

Definitions and formulas for rational and irrational numbers are important knowledge in mathematics that students must understand to have a solid mathematical foundation. The article below would like to introduce to you the definitions, properties, and mathematical forms of rational numbers and irrational numbers, please refer to them.

Rational numbers, irrational numbers

What is a rational number?

– Rational numbers are a set of numbers that can be written as fractions (quotients). That is, a rational number can be represented by an infinite repeating decimal.

– Rational numbers are written as \frac{a}{b}where a and b are integers but b must be non-zero.

\mathbb{Q} is the set of rational numbers.

=> Set of rational numbers: \mathbb{Q} = \left\{ {\frac{a}{b}|a,b \in \mathbb{Z},b \ne 0} \right\}.

For example: -3; -1.75, 4\frac{1}{5}… are rational numbers.

  • Any integer a is a rational number because the integer a can be written in the form \frac{a}{1}.

For example: I have 1; -3; 0 are rational numbers.

We have:

\begin{matrix}1 = \dfrac{1}{1} \Rightarrow 1\mathbb{\in Q} \\- 3 = \dfrac{- 3}{1} \Rightarrow - 3\mathbb{\in Q} \\0 = \dfrac{0}{1} \Rightarrow 0\mathbb{\in Q} \\end{matrix}

Comment: \mathbb{Z =}\left\{ 0; \pm 1; \pm 2;... \right\}are all rational numbers.

Rational numbers

Classification of rational numbers

Rational numbers are divided into two types: negative rational numbers and positive rational numbers. Specifically:

  • Negative rational numbers: Includes rational numbers less than 0.
  • Positive rational numbers: Includes rational numbers greater than 0.

Note: The number 0 is neither a negative rational number nor a positive rational number.

Rational numbers

Nature

  • The set of rational numbers is a countable set.
  • Commutative property: x + y = y + x
  • Properties plus 0: x + 0 = y + 0
  • Combined properties: (x + y) + z = x + (y + z)

Represent rational numbers on the number line

– To represent rational numbers on the number line, we follow the following factors:

Step 1: Write rational numbers as fractions \frac{a}{b}

Step 2: Divide the unit line segment into b equal parts, we get the new unit segment \frac{1}{b} old unit.

Step 3: Rational numbers \frac{a}{b} is represented by point A, which is a new unit distance from point 0.

  • A is to the left of point 0 if it is negative.
  • A is to the right of point 0 if it is a positive number.

For example: In the figure, point P represents the rational number:

Rational numbers

Instruct

The unit line segment is divided into 6 equal parts (the new unit is 1/6 of the old unit)

Point P is located at a distance of 7 new units from point O

And point P is to the right of point O, so P is a positive rational number

So P represents a rational number \frac{7}{6}.

Add and subtract rational numbers

i) Rules for adding and subtracting two rational numbers

We can add and subtract two rational numbers x and y by writing them as two fractions and then applying the rules for adding and subtracting fractions.

With x = \frac{p}{m};y = \frac{q}{m};\left( p,q,m\mathbb{\in Z},m > 0 \right)” width=”251″ height=”29″ data-type=”2″ data-latex=”x = \frac{p}{m};y =
\frac{q}{m};\left( p,q,m\mathbb{\in Z},m > 0 \right)” class=”lazy” data-src=”https://tex.vdoc.vn?tex=x%20%3D%20%5Cfrac%7Bp%7D%7Bm%7D%3By%20%3D%0A%5Cfrac%7Bq%7D%7Bm%7D%3B%5Cleft(%20p%2Cq%2Cm%5Cmathbb%7B%5Cin%20Z%7D%2Cm%20%3E%200%20%5Cright)”/></span>  we have:</p>
<p><span class=x + y = \frac{p}{m} + \frac{q}{m} = \frac{p + q}{m}

x - y = \frac{p}{m} - \frac{q}{m} = \frac{p - q}{m}

ii) Properties

– Addition of rational numbers has the properties of addition of fractions: commutation, combination, addition with 0, addition with opposite numbers.

– With a,b,c\mathbb{\in Q}we have:

a) Commutative property: a + b = b + a

b) Combined properties: a + (b + c) = (a + b) + c

c) Add the number 0: a + 0 = 0 + a = a

d) Add the opposite number: a + (-a) = 0

iii, Rules for shifting sides

When moving a term from one side to the other side of an equality, we must change the sign of that term.

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In Q we have an algebraic sum, in which we can swap terms and place parentheses to arbitrarily group terms like algebraic sums in the set of integers.

Multiply and divide rational numbers

i) Rules for multiplying and dividing two rational numbers

– We can multiply and divide two rational numbers by writing them as fractions and then applying the rules for multiplying and dividing fractions.

  • With x = \frac{a}{b};y = \frac{c}{d};(b,d \neq 0)we have: xy = \frac{a}{b}.\frac{c}{d} = \frac{ac}{bd}
  • With y \neq 0we have: x:y = \frac{a}{b}:\frac{c}{d} = \frac{a}{b}.\frac{d}{c} = \frac{ad}{bc}

For example:

Multiply rational numbers: \frac{2}{5}\times\frac{7}{3}\times\frac{2}{3}=\frac{28}{45}

Divide rational numbers: \frac{4}{5}\div\frac{3}{7}=\frac{4\times7}{5\times3}=\frac{28}{15}

ii) Properties

– Rational number multiplication also has the same properties as fraction multiplication: commutation, combination, multiplication by 1 and distributive properties of multiplication over addition.

– Every rational number other than 0 has an inverse.

– With a,b,c\mathbb{\in Q} we have:

Absolute value of a rational number

– The absolute value of a rational number a, denoted by |a|is the distance from point a to point 0 on the number line.

|a| = \left\{ \begin{matrix} a\ \ \ \ \ \ \when\ \a \geq 0\ \ \\ - a\ \ \ \when\ \a < 0 \\ \end{matrix} \right .

For example:

x = \frac{1}{5} \Rightarrow |x| = \left| \frac{1}{5} \right| = \frac{1}{5} (Because \frac{1}{5} > 0″ width=”45″ height=”37″ data-type=”2″ data-latex=”\frac{1}{5} > 0″ class=”lazy” data-src=”https://tex.vdoc.vn?tex=%5Cfrac%7B1%7D%7B5%7D%20%3E%200″/></span>)</p>
<p><span class=x = - 1\frac{3}{4} \Rightarrow |x| = \left| - 1\frac{3}{4} \right| = - \left( - 1\frac{3}{4} \right) = 1\frac{3}{4} (Because - 1\frac{3}{4} < 0)

Compare two rational numbers

– With any two rational numbers x, y we always have or x = y or x < y or x > y” width=”40″ height=”15″ data-type=”0″ data-latex=”x > y” class=”lazy” data-src=”https://tex.vdoc.vn?tex=x%20%3E%20y”/>.</p>
<p>– To compare two rational numbers <img loading= we do the following:

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