<< Chapter < Page Chapter >> Page >

The set of rational numbers    is written as { m n | m  and  n  are integers and  n 0 } . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

  1. a terminating decimal: 15 8 = 1.875 , or
  2. a repeating decimal: 4 11 = 0.36363636 = 0. 36 ¯

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing integers as rational numbers

Write each of the following as a rational number.

  1. 7
  2. 0
  3. –8

Write a fraction with the integer in the numerator and 1 in the denominator.

  1. 7 = 7 1
  2. 0 = 0 1
  3. −8 = 8 1
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write each of the following as a rational number.

  1. 11
  2. 3
  3. –4
  1. 11 1
  2. 3 1
  3. 4 1
Got questions? Get instant answers now!

Identifying rational numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

  1. 5 7
  2. 15 5
  3. 13 25

Write each fraction as a decimal by dividing the numerator by the denominator.

  1. 5 7 = −0. 714285 ——— , a repeating decimal
  2. 15 5 = 3 (or 3.0), a terminating decimal
  3. 13 25 = 0.52 , a terminating decimal
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write each of the following rational numbers as either a terminating or repeating decimal.

  1. 68 17
  2. 8 13
  3. 17 20
  1. 4 (or 4.0), terminating;
  2. 0. 615384 ¯ , repeating;
  3. –0.85, terminating
Got questions? Get instant answers now!

Irrational numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 3 2 , but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers    . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

{ h | h  is not a rational number }

Differentiating rational and irrational numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

  1. 25
  2. 33 9
  3. 11
  4. 17 34
  5. 0.3033033303333
  1. 25 : This can be simplified as 25 = 5. Therefore, 25 is rational.
  2. 33 9 : Because it is a fraction, 33 9 is a rational number. Next, simplify and divide.
    33 9 = 33 11 9 3 = 11 3 = 3. 6 ¯

    So, 33 9 is rational and a repeating decimal.

  3. 11 : This cannot be simplified any further. Therefore, 11 is an irrational number.
  4. 17 34 : Because it is a fraction, 17 34 is a rational number. Simplify and divide.
    17 34 = 17 1 34 2 = 1 2 = 0.5

    So, 17 34 is rational and a terminating decimal.

  5. 0.3033033303333 is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

how to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
what is isotopes
Nangun Reply
nuclei having the same Z and different N s
AI-Robot
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask