MATH 9
unit 3.1: what is a rational number?
what do we need to know?
Comparing fractions
Equivalent fractions
How to convert fractions to decimals and vice-versa
Equivalent fractions
How to convert fractions to decimals and vice-versa
More information
In this unit, we will be dealing with rational numbers. Here is a short introduction on the concept of rational numbers, and a short summary on what we will be learning in this unit.
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What is the difference between rational and irrational numbers?
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Take a look at this explanation of negative rational numbers:
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which type of numbers are rational numbers?
Rational numbers are numbers (positive and negative) that can be made into a fraction. They are:
- INTEGERS: All integers have a DENOMINATOR of 1, and therefore considered fractions.
- DECIMALS: As you already know, decimals can be made into fractions.
- MIXED NUMBERS: These numbers are easily converted unto improper fractions.
- SQUARE ROOT OF PERFECT SQUARE numbers.
- INTEGERS: All integers have a DENOMINATOR of 1, and therefore considered fractions.
- DECIMALS: As you already know, decimals can be made into fractions.
- MIXED NUMBERS: These numbers are easily converted unto improper fractions.
- SQUARE ROOT OF PERFECT SQUARE numbers.
the irrational numbers
comparing rational numbers
When comparing rational numbers:
- Make sure all numbers are of the same type. That is, convert them all to fractions OR to decimals.
- Remember that POSITIVE NUMBERS ARE ALWAYS GREATER THAN NEGATIVE NUMBERS.
- The closer a negative number is to zero, the greater it is.
- FRACTIONS WITH SAME DENOMINATOR: The greater the numerator, the greater the value.
- FRACTIONS WITH SAME NUMERATOR: The greater the denominator, the greater the value.
- When comparing decimals, make sure the decimal numbers have the same amount of decimal places (you can always add a 0), and compare them from right to left.
- Make sure all numbers are of the same type. That is, convert them all to fractions OR to decimals.
- Remember that POSITIVE NUMBERS ARE ALWAYS GREATER THAN NEGATIVE NUMBERS.
- The closer a negative number is to zero, the greater it is.
- FRACTIONS WITH SAME DENOMINATOR: The greater the numerator, the greater the value.
- FRACTIONS WITH SAME NUMERATOR: The greater the denominator, the greater the value.
- When comparing decimals, make sure the decimal numbers have the same amount of decimal places (you can always add a 0), and compare them from right to left.
finding numbers between two rational numbers
If m and n be two rational numbers such that m < n then 1/2 (m + n) is a rational number between m and n. This is called the AVERAGE.
Let's do some examples!
1. Find out a rational number lying halfway between 2/7 and 3/4.
- Solution: Find the average: (2/7 + 3/4)/2 = 29/56
- Hence, 29/56 is a rational number lying halfway between 2/7 and 3/4.
Let's prove it: 2/7 = 0.285
3/4 = 0.75
29/56 = 0.517
0.75 > 0.517 > 0.285
2. Find out a rational number lying between -1/3 and 1/2.
- Solution: Find the average: (-1/3 + 1/2)/2 = 1/12
- Hence, 1/12 is a rational number lying between 1/3 and 1/2.
3. Find out ten rational numbers lying between -3/11 and 8/11.
- Solution: Notice that the denominator is the same, and because
-3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8, then,
-2/11, -1/11, 0/11, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11 and 7/11 are the ten rational numbers lying between -3/11 and 8/11.
4. Find out three rational numbers lying between 3 and 4.
- Solution: Find the average: 1/2 (3 + 4) = 7/2. Then, 3 < 7/2 < 4
- Now, find the average of 3 and 7/2: (3 + 7/2)/2 = 13/4
- Do the same to find a rational number between 7/2 and 4: (7/2 + 4)/2 = 15/4
3 < 13/4 < 7/2 < 15/4 < 4
- Hence, 13/4, 7/2 and 15/4 are the three rational numbers lying between 3 and 4.
Let's do some examples!
1. Find out a rational number lying halfway between 2/7 and 3/4.
- Solution: Find the average: (2/7 + 3/4)/2 = 29/56
- Hence, 29/56 is a rational number lying halfway between 2/7 and 3/4.
Let's prove it: 2/7 = 0.285
3/4 = 0.75
29/56 = 0.517
0.75 > 0.517 > 0.285
2. Find out a rational number lying between -1/3 and 1/2.
- Solution: Find the average: (-1/3 + 1/2)/2 = 1/12
- Hence, 1/12 is a rational number lying between 1/3 and 1/2.
3. Find out ten rational numbers lying between -3/11 and 8/11.
- Solution: Notice that the denominator is the same, and because
-3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8, then,
-2/11, -1/11, 0/11, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11 and 7/11 are the ten rational numbers lying between -3/11 and 8/11.
4. Find out three rational numbers lying between 3 and 4.
- Solution: Find the average: 1/2 (3 + 4) = 7/2. Then, 3 < 7/2 < 4
- Now, find the average of 3 and 7/2: (3 + 7/2)/2 = 13/4
- Do the same to find a rational number between 7/2 and 4: (7/2 + 4)/2 = 15/4
3 < 13/4 < 7/2 < 15/4 < 4
- Hence, 13/4, 7/2 and 15/4 are the three rational numbers lying between 3 and 4.
HERE is THE ANSWER TO THE WORKSHEET GIVEN IN CLASS
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videos that may help
how do we represent rational numbers on a number line?
how to find any number between two numbers
comparing fractions
INTERACTIVE ONLINE ACTIVITIES
worksheets
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review - workbook
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