- Natural Numbers or “Counting Numbers” N
1, 2, 3, 4, 5, . . .
- Whole Numbers: Natural Numbers together with “zero”
0, 1, 2, 3, 4, 5, . . .
- Integers Z (Z is from German Zahl, "number".) Whole numbers plus negatives
. . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .
- Rational Numbers Q (Q is from quotient.) All numbers of the form a/b , where a and b are integers (but b cannot be zero). Rational numbers include what we usually call fractions
Types of fractions:
- PROPER FRACTIONS: Smaller than 1, like 1/2 or 3/4
- IMPROPER FRACTIONS: Bigger than 1 , like two-and-a-half, which we could also write as 5/2
- The bottom of the fraction is called the denominator. Think of it as the denomination—it tells you what size fraction we are talking about: fourths, fifths, etc.
- The top of the fraction is called the numerator. It tells you how many fourths, fifths, or whatever
RESTRICTION: The denominator cannot be zero!
Remember: READING FRACTIONS
When the
denominator is 2 we use the word HALF (plural HALVES)…
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1 / 2
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One half
/ Three halves.
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When the
denominator is 3 we use the word THIRD…………...........
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1/3
|
One
third.
|
When the
denominator is 4 we use the word QUARTER or
FOURTH…
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3/$
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Three quarters
/ three fourths.
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When the
denominator is 5 we use the word FIFTH
………...............
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2/5
|
Two
fifths.
|
When the
denominator is 10 we use the word TENTH ………………
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7/10
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Seven
tenths.
|
And so on. |
||
First
read the numerator.
Then add -th or –th to the denominator.
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1/100
11/100
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One
hundredth.
Eleven
hundredths.
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You can read also the fraction a/b as “a over b”.
|
For example,11/100 is “eleven over a
hundred”.
|
|
Some questions:
- What are equivalent fractions? Give some situation in which they are used
- When is a fraction in its lowest terms?
Classifying decimal numbers
There are three types of decimal numbers:
a) Terminating decimals stop at some point. n1.5 is an
example of a terminating decimal. So is 4.75, and 0.33265985. No matter how
many decimal places there are, provided it stops, it
is a terminating decimal.
b) Recurring decimals repeat forever, but there is a pattern to their repetition. These are
presented either by dots after the number, showing that it continues, or using
proper mathematical notation, a dot above the number that repeats.
E.g. Put 1/3 into a calculator,
and it will either show 0.33333333333… .
If a pattern of numbers repeats, rather than just a single
number, then two dots show where the repeating pattern begins and ends, e.g.
1/7 = 0.1 ̇42857 ̇ = 0.142857142857142857…
c) Non-recurring decimals are decimals
that go on forever, but there is no
pattern. These are also called ‘irrational’ numbers.
Examples include π and √2.
IS IT RATIONAL?
For each of the numbers below, decide whether it is rational or irrational. Explain your reasoning in detail
Work in small groups:
You will make a poster classifying rational and irrational numbers. You are going to get some cards with numbers on them. You have to decide whether the number is rational or irrational and where it fits on your poster:
NUMBERS' CARD 1
NUMBERS' CARD 2
ROUNDING DECIMALS
Remember (rounding whole numbers)
To round to the nearest ten (or hundred, thousand…):
1º Look at the digit right after the tens position (or hundreds, thousand..)
4 3 7 look at this digit
2º If it is 5 or more, round up the digit in the tens position (or hundreds, thousands…). If not, the digit remains the same:
4 4 0
We can round:
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How to round decimal numbers?
Look at the examples:
27.17469 rounded to the nearest whole number is 27
36.74691 rounded to the nearest whole number is 37
12.34690 rounded to the nearest tenth is 12.3
89.46917 rounded to the nearest tenth is 89.5
50.02139 rounded to the nearest hundredth is 50.02
72.63539 rounded to the nearest hundredth is 72.64
46.83531 rounded to the nearest thousandth is 46.835
9.63967 rounded to the nearest thousandth is 9.640
RULE TO ROUND DECIMAL NUMBERS:
1. Retain the correct number of decimal places (e.g. 3 for thousandths, 0 for whole numbers)
2. If the next decimal place value is 5 or more, increase the value in the last retained decimal place by 1.
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