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Chapter 3:Rational Numbers
3.1 What Is a Rational Number?Pg 94-105
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Quick Review of What You SHOULD Know
• If we are looking for the SUM of two numbers, that means two numbers added together.
• Ex. The sum of 2 and 3 is?– Aka. 2+3=?
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Quick Review of What You SHOULD Know
• The DIFFERENCE between two numbers is one number subtracted by another number.
• Ex. What is the difference of 5 and 2?– Aka. 5-2=?
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Quick Review of What You SHOULD Know
• The PRODUCT of two numbers is one number multiplied by the other.
• Ex. The product of 12 and 5 is?.– Aka. 12 x 5 = ?
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Quick Review of What You SHOULD Know
• 4 5 = = 0.8
• This would be called finding the quotient
• What are these quotients?a. ¾b. 6/2 c. -11/2
4
5
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Quickly what are the different types of numbers we know?
?????
Integers
Whole
Natural
Natural numbers: 1, 2, 3, 4, 5, 6, …..Whole Numbers: 0, 1, 2, 3, 4, 5, 6, ……Integers: ……, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, ...
Anything bigger?
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But what about……?
• Where does 0.5 fit in?• Or what about ?
• Are these numbers? Do they fit in the sets we have?
3
4
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• We need a new set to fit these numbers
• They are called rational numbers.
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Rational Numbers
• Rigorous Definition:– A rational number is of the form m/n where m and
n are any integer and n ≠ 0
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What does that actually mean?
• Basically if you can write a number as a fraction, it is rational.
• SO! All integers are rational because we can write them like this.2=2/1 , 439=439/1 , 7993857667=7993857667/1
• And obviously all fractions are rational.
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Do all decimals work?
• 0.25? 0.5? Why?
• What about something like this?– 0.123456789123456789123456789…….
– 3.1415962535897……..
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• Not all decimals will be rational. They will be rational if they are not infinite, or they have a finite repeatable pattern.
• So if something is not rational it must then be…
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IRRATIONAL!
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• So everything that isn’t rational is then irrational.
• The example I gave was Pi – Part of your homework is to find two other
irrational numbers.
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How to write a rational number on a number line.
• First we need to review what a zero pair is. • ZERO PAIR is two numbers that have the same
value but opposite sign.
• Exs. -2, +2 -20.5, +20.5 -926, +926– Take any of these pairs and add them, you will get
zero every time. Thus zero pair.• Zero pairs have the same distance from zero
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How to write a rational number on a number line.
Let’s try placing these values onto a number line: 2, 3, -1, -4, 0.5, -0.75
0 1 2 3 4 5-1-2-3-4-5
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What do we do if they are fractions?
• How do we put numbers like or onto a number line?
4
5
2
3
0 1 2 3 4 5-1-2-3-4-5
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• Easy. Turn them into decimals and place (to a reasonable estimation) onto the number line.4
5
2
3= -0.8 = 0.6
0 1 2 3 4 5-1-2-3-4-5
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Ordering Rational #’s
• List in order from least to greatest.• Same strategy; place them on a number line
and list them off in order.• -3/5, 1.1, 13/12, -0.5, 12/12
0 1 2 3 4 5-1-2-3-4-5
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• How do you write a rational number between two given numbers?
• How many can we find between -5 and 5?
0 1 2 3 4 5-1-2-3-4-5
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Rational
Integer
Whole
Natural
Irrational
Everything is contained in the Real Numbers