Rational and Real Numbers
- Natural Numbers
- 6 Write Five Rational Numbers Greater Than 20
- 6 Write Five Rational Numbers Greater Than 2/3
Learning Objective(s)
Every rational number can be written as a fraction a/b, where a and b are integers. For example, 3 can be written as 3/1, -0.175 can be written as -7/40, and 1 1/6 can be written as 7/6. All natural numbers, whole numbers, and integers are rationals, but not all rational numbers are natural numbers, whole numbers, or integers. Evolution of rational numbers,difference between fraction and rational numb. Ask Doubt Queries asked on Sunday & after 7pm from Monday to Saturday will. C) 4 9 25 = 2 3 5 2 3 5 = 30 30. Rational and irrational numbers. A rational number is simply a number of arithmetic: a whole number, fraction, mixed number, or decimal; together with its negative images. A rational number has the same ratio to 1 as two natural numbers. That is what a rational number is.
·Identify the subset(s) of the real numbers that a given number belongs to.
·Locate points on a number line.
·Compare rational numbers.
·Identify rational and irrational numbers.
You've worked with fractions and decimals, like 3.8 and . These numbers can be found between the integer numbers on a number line. There are other numbers that can be found on a number line, too. When you include all the numbers that can be put on a number line, you have the real number line. Let's dig deeper into the number line and see what those numbers look like. Let's take a closer look to see where these numbers fall on the number line.
Rational Numbers
The fraction , mixed number , and decimal 5.33… (or ) all represent the same number. This number belongs to a set of numbers that mathematicians call rational numbers. Rational numbers are numbers that can be written as a ratio of two integers. Regardless of the form used, is rational because this number can be written as the ratio of 16 over 3, or .
Examples of rational numbers include the following.
0.5, as it can be written as
, as it can be written as
−1.6, as it can be written as
4, as it can be written as
-10, as it can be written as Cloudmounter 2 2 – mount web servers as local disks.
All of these numbers can be written as the ratio of two integers.
You can locate these points on the number line.
In the following illustration, points are shown for 0.5 or , and for 2.75 or .
As you have seen, rational numbers can be negative. Each positive rational number has an opposite. The opposite of is , for example.
Be careful when placing negative numbers on a number line. The negative sign means the number is to the left of 0, and the absolute value of the number is the distance from 0. So to place −1.6 on a number line, you would find a point that is |−1.6| or 1.6 units to the left of 0. This is more than 1 unit away, but less than 2.
Example | |
Problem | Place on a number line. |
It's helpful to first write this improper fraction as a mixed number: 23 divided by 5 is 4 with a remainder of 3, so is . | |
Since the number is negative, you can think of it as moving units to the left of 0. will be between −4 and −5. | |
Answer |
Which of the following points represents ? |
Comparing Rational Numbers
When two whole numbers are graphed on a number line, the number to the right on the number line is always greater than the number on the left.
The same is true when comparing two integers or rational numbers. The number to the right on the number line is always greater than the one on the left.
Here are some examples.
Numbers to Compare | Comparison | Symbolic Expression |
−2 and −3 | −2 is greater than −3 because −2 is to the right of −3 | −2 > −3 or −3 < −2 |
2 and 3 | 3 is greater than 2 because 3 is to the right of 2 | 3 > 2 or 2 < 3 |
−3.5 and −3.1 | −3.1 is greater than −3.5 because −3.1 is to the right of −3.5 (see below) | −3.1 > −3.5 or −3.5 < −3.1 |
Natural Numbers
Which of the following are true? i. −4.1 > 3.2 ii. −3.2 > −4.1 iii. 3.2 > 4.1 Winx dvd ripper 6 2 0 download free. iv. −4.6 < −4.1 A) i and iv B) i and ii C) ii and iii D) ii and iv E) i, ii, and iii |
Irrational and Real Numbers
There are also numbers that are not rational. Irrational numbers cannot be written as the ratio of two integers.
Any square root of a number that is not a perfect square, for example , is irrational. Irrational numbers are most commonly written in one of three ways: as a root (such as a square root), using a special symbol (such as ), or as a nonrepeating, nonterminating decimal.
The fraction , mixed number , and decimal 5.33… (or ) all represent the same number. This number belongs to a set of numbers that mathematicians call rational numbers. Rational numbers are numbers that can be written as a ratio of two integers. Regardless of the form used, is rational because this number can be written as the ratio of 16 over 3, or .
Examples of rational numbers include the following.
0.5, as it can be written as
, as it can be written as
−1.6, as it can be written as
4, as it can be written as
-10, as it can be written as Cloudmounter 2 2 – mount web servers as local disks.
All of these numbers can be written as the ratio of two integers.
You can locate these points on the number line.
In the following illustration, points are shown for 0.5 or , and for 2.75 or .
As you have seen, rational numbers can be negative. Each positive rational number has an opposite. The opposite of is , for example.
Be careful when placing negative numbers on a number line. The negative sign means the number is to the left of 0, and the absolute value of the number is the distance from 0. So to place −1.6 on a number line, you would find a point that is |−1.6| or 1.6 units to the left of 0. This is more than 1 unit away, but less than 2.
Example | |
Problem | Place on a number line. |
It's helpful to first write this improper fraction as a mixed number: 23 divided by 5 is 4 with a remainder of 3, so is . | |
Since the number is negative, you can think of it as moving units to the left of 0. will be between −4 and −5. | |
Answer |
Which of the following points represents ? |
Comparing Rational Numbers
When two whole numbers are graphed on a number line, the number to the right on the number line is always greater than the number on the left.
The same is true when comparing two integers or rational numbers. The number to the right on the number line is always greater than the one on the left.
Here are some examples.
Numbers to Compare | Comparison | Symbolic Expression |
−2 and −3 | −2 is greater than −3 because −2 is to the right of −3 | −2 > −3 or −3 < −2 |
2 and 3 | 3 is greater than 2 because 3 is to the right of 2 | 3 > 2 or 2 < 3 |
−3.5 and −3.1 | −3.1 is greater than −3.5 because −3.1 is to the right of −3.5 (see below) | −3.1 > −3.5 or −3.5 < −3.1 |
Natural Numbers
Which of the following are true? i. −4.1 > 3.2 ii. −3.2 > −4.1 iii. 3.2 > 4.1 Winx dvd ripper 6 2 0 download free. iv. −4.6 < −4.1 A) i and iv B) i and ii C) ii and iii D) ii and iv E) i, ii, and iii |
Irrational and Real Numbers
There are also numbers that are not rational. Irrational numbers cannot be written as the ratio of two integers.
Any square root of a number that is not a perfect square, for example , is irrational. Irrational numbers are most commonly written in one of three ways: as a root (such as a square root), using a special symbol (such as ), or as a nonrepeating, nonterminating decimal.
Numbers with a decimal part can either be terminating decimals or nonterminating decimals. Terminating means the digits stop eventually (although you can always write 0s at the end). For example, 1.3 is terminating, because there's a last digit. The decimal form of is 0.25. Terminating decimals are always rational.
Nonterminating decimals have digits (other than 0) that continue forever. For example, consider the decimal form of , which is 0.3333…. The 3s continue indefinitely. Or the decimal form of , which is 0.090909…: the sequence '09' continues forever.
In addition to being nonterminating, these two numbers are also repeating decimals. Their decimal parts are made of a number or sequence of numbers that repeats again and again. A nonrepeating decimal has digits that never form a repeating pattern. The value of, for example, is 1.414213562…. No matter how far you carry out the numbers, the digits will never repeat a previous sequence.
If a number is terminating or repeating, it must be rational; if it is both nonterminating and nonrepeating, the number is irrational.
Type of Decimal | Rational or Irrational | Examples |
Terminating | Rational | 0.25 (or ) 1.3 (or ) |
Nonterminating and Repeating | Rational | 0.66… (or ) 3.242424… (or) |
Nonterminating and Nonrepeating | Irrational | (or 3.14159…) (or 2.6457…) |
Example | ||
Problem | Is −82.91 rational or irrational? | |
Answer | −82.91 is rational. | The number is rational, because it is a terminating decimal. |
The set of real numbers is made by combining the set of rational numbers and the set of irrational numbers. The real numbers include natural numbers or counting numbers, whole numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers. The set of real numbers is all the numbers that have a location on the number line.
Sets of Numbers Natural numbers 1, 2, 3, … Whole numbers 0, 1, 2, 3, … Integers …, −3, −2, −1, 0, 1, 2, 3, … Rational numbers numbers that can be written as a ratio of two integers—rational numbers are terminating or repeating when written in decimal form Irrational numbers numbers than cannot be written as a ratio of two integers—irrational numbers are nonterminating and nonrepeating when written in decimal form Real numbers any number that is rational or irrational |
Example | ||
Problem | What sets of numbers does 32 belong to? | |
Answer | The number 32 belongs to all these sets of numbers: Natural numbers Whole numbersIntegers Rational numbers Outline 3 21 2 – view onenote notebooks for dummies. Real numbers | Every natural or counting number belongs to all of these sets! |
Example | ||
Problem | What sets of numbers does belong to? | |
Answer | Superduper 2 9 download free. belongs to these sets of numbers: Rational numbers Real numbers | The number is rational because it's a repeating decimal. It's equal to or or . |
Example | ||
Problem | What sets of numbers does belong to? | |
Answer | belongs to these sets of numbers: Irrational numbers Real numbers | The number is irrational because it can't be written as a ratio of two integers. Square roots that aren't perfect squares are always irrational. |
Which of the following sets does belong to? whole numbers integers rational numbers irrational numbers real numbers A) rational numbers only B) irrational numbers only C) rational and real numbers D) irrational and real numbers E) integers, rational numbers, and real numbers F) whole numbers, integers, rational numbers, and real numbers |
6 Write Five Rational Numbers Greater Than 20
Summary
6 Write Five Rational Numbers Greater Than 2/3
The set of real numbers is all numbers that can be shown on a number line. This includes natural or counting numbers, whole numbers, and integers. It also includes rational numbers, which are numbers that can be written as a ratio of two integers, and irrational numbers, which cannot be written as a the ratio of two integers. When comparing two numbers, the one with the greater value would appear on the number line to the right of the other one.
