UNIT 1: THE NATURAL NUMBERS SET.
1.- The natural numbers set..
Natural numbers are used for counting things.
They are 1, 2, 3, 4, 5, 6,....., 100, 101,....... and so on untill infinity.
Natural numbers have two main purposes: you can use them for counting or for
ordering.
Counting: There are 6 cars in the street.
Ordering: Jerez is the 2nd most important city in the province.
2.- Place value and ordering.
Our number system was invented in India by mathematicians about 1400 years ago and arrived in
Europe with the Arabs. As you know, there are only ten symbols:
zero
one
two
three
four
five
six
seven
eight
nine
Numbers are read from left to right, each digit expresses the order in the number.
24,013,210 = (is read as)= Twenty four million, thrirteen thousand, two hundred and ten.
Eight million, eight thousand and eight = (is written)= 8,008,008.
The value of a digit depends on its place in a number.
In this place value diagram, the digit 6 means...
1000
100
10
Thousands
Hundreds
Tens
2
7
1
Ones
1
6
6 ones
6
5
6 tens
2
4
6 hundreds
4
1
6 thousands
Two thousand, seven hundred an sixteen
8
3
eight thousand, three hundred and sixty five
5
6
five thousand, six hundred and twenty four
6
7
six thousand, seven hundred and fourteen one
452,945
400,000
50,000
2,000
900
40
5
Hundred thousands
Ten thousands
Thousands
Hundreds
Tens
Ones
100,000s
10,000s
1,000s
100s
10s
1s
You say:
Four hundred and fifty two thousand, nine hundred and fourty five.
Or:
Four hundred thousands, five ten thousands, two thousands, nine hundreds, four
tens and five ones.
452,945 = 4 x 100,000 + 5 x 10,000 + 2 x 1000 + 9 x 100 + 4 x 10 + 5 x1
The natural numbers can be represented on a half line (semirrecta) (line with a fixed beginning and
with no fixed ending) that begins with zero and which is divided in equal segments.
0
1
2
3 4
5
6
7
8 ....
To compare two numbers, we can use three symbols:
>
(greater than: mayor que)
=
(equal to: igual a)
<
(less than: menor que).
Solve the exercise:
1.- Put the corresponding symbols between the following numbers:
5 ___ 7
1003 ____1030
10020 ___10200
2.- Put these number in order of size, smallest first:
719, 642, 711, 317, 306, 207, 159
9898____ 9799
12,045,235,003,134
Billion
Thousand
Million
Thousand
Twelve billion forty five thousand two hundred and thirty five million three thousand one hundred and
thirty four.
(Note: In English, numbers are written using commas instead of dots. Example: 12,045,235,003,134)
Solve the exercise:
3.- Describe the following numbers in words, in Spanish and in English:
a) 34.000.340.02
b) 3.004.000.123.004
c) 12.005.000.012.300
d) 373.005.000.000.345
Let’s see: (Writing an amount in figures)
Ten billion one hundred thirteen million two thousand twenty three.
10,000,113,002,023
Billion
Thousand
Million
Thousand
Solve the exercise:
4- Write the following amounts in figures:
a) Twelve billion two hundred and forty thousand million three hundred thousand and five
b) Four billion twenty seven thousand two million five thousand three hundred and fifteen
b) Fifteen billion forty five thousand three hundred and twenty four, six million and two hundred
3- Rounding to a given order of units .
To round a number to the nearest ten, look at the digit in the units column:
If it is less than 5, round down.
If it is 5 or more, round up.
78 would round to 80, to the nearest ten
78
70
75
80
Rounding a number to the nearest hundred is similar to roundind to the nearest ten, except you look at
the digit in the tens column:
If it is less than 5, round down.
If it is 5 or more, round up.
742 would round to 700, to the nearest hundred
742
700
750
800
To round a number to the nearest thousand, look at the digit in the hundreds column:
If it is less than 5, round down.
If it is 5 or more, round up.
6500 rounds to 7000 to the nearest thousand
6,500
6000
6500
7000
Rounding numbers can help when estimating the answers to calculate.
Estimating can be useful when shopping so that you have a rough idea of how much your shopping
bill will be.
It’s also a goodway of checking caculations.
When estimating...
round the numbers to easy numbers, usually to the
nearest ten, hundred or thousand
use these easy numbers to work out the estimate
when multiplying or dividing, never round a number to zero.
Example:
9+21 is approximately 10+20 = 30
12x402 is approximately 10x400 = 4000
5- Round each distance to the nearest 100 Km:
a)
b)
c)
d)
e)
f)
g)
h)
i)
Edimburgh to London 656 Km
Paris to Madrid 1265 Km
Rome to Vienna 1168 Km
Prague to Copenhagen 1033 Km
Amsterdam to Paris 577 Km
Lisbon to Madrid 635 Km
Milan to Rome 689 Km
Paris to Munich 827 Km
Copenhagen to Paris 1329 Km
4.- Operations: Addition, subtraction, multiplication and division.
We read 3 + 5 = 8 like: “Three plus five is equal to eight” or “Three plus five equals eight” or “Three
plus five is eight”. Terms in the addition are called addends and the result is called the sum. In Spanish
the addends are the sumandos.
Example:
The library has lent 45 books last Monday, 50 books on Tuesday and 73 books on Wednesday. How
many books have they lent?
45 + 50 + 73 = 168 books. Answer: They have lent 168 books
We read 13 – 7 = 6 like: “Thirteen minus seven equals six” (sometimes you can see “thirteen take
away seven equals six” but it is better to use the first expression.
The terms of subtraction are called minuend and subtrahend, the outcome is called the difference..
The minuend is the first number, it is the number from which you take something and it must be the
larger number. In Spanish it is called minuendo.
The subtrahend is the number that is subtracted and it must be the smaller number. In Spanish it is
called sustraendo.
The difference is the result of the subtraction. In Spanish it is called diferencia. To check if the
subtraction is correct we add up the subtrahend and the difference. The outcome must be the
minuend.
Minuend = Subtrahend + Difference
Example:
We have saved 3520 euros but we have spent € 745 on a computer. How much money is left?
3520 – 745 = 2775.
Answer: 2775 euros is left.
Addition and substracction are related to each other:
42 + 28 = 70
70 - 28 = 42
70 – 42 = 28
Adding is putting together, combining and calculating a total.
Subtraction is reducing, calculating the difference and calculating what is missing or what is left.
Total = 70 (you have to add)
42
28
To know how much is missing from 42 to reach 70, you have to subtract.
To multiply it is essential that you know the times tables up to 10 x 10.
x
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
2
2
4
6
8
10
12
14
16
18
3
3
6
9
12
15
18
21
24
27
4
4
8
12
16
20
24
28
32
36
5
5
10
15
20
25
30
35
40
45
6
6
12
18
24
30
36
42
48
54
7
7
14
21
28
35
42
49
56
63
8
8
16
24
32
40
48
56
64
72
9
9
18
27
36
45
54
63
72
81
When the numbers you want to add up are the same, the problem can be solved by multiplication:
34 + 34 + ... + 34 = 20 x 34 = 680 (Twenty times thirty four is six hundred and eighty)
(20 times)
To multiply a number by a unit followed by zeros, you add the zeros to the right of the number:
67 x 10 = 670 (sixty seven times ten equals six hundred and seventy)
19 x 1,000 = 19,000 (nineteen times a thousand equals nineteen thousand)
Example:
In my living-room I have a bookcase (estantería) with three shelves (estantes). If there are five books
on each shelf, how many books are there?
5 x 3 = 15
Answer: I have 15 books in my bookcase.
Dividing is to share a quantity into equal groups.
It is the inverse of multiplication. In Spanish we write 6 : 2 , but in English it is always 6 ÷ 2 and never
with the colon (:).
We read 15 ÷ 5 = 3 like: “Fifteen divided by five equals three” .
Dividing is splitting into equal parts:
20
2
6
6
6
Divisor
Dividend
2 0
-18
3
6
02
Quotient
Remainder
As the divisor is 3, we look for answers in the 3 times table:
3 x 5 = 15
3 x 6 = 18
The closest result that is less than or equal to dividend.
3 x 7 = 21
To check if the division is correct we do the division algorithm:
D
Dividend
Divisor
r
Remainder Quotient
d
q
D=dxq+r
20 = 3 x 6 + 2
Example:
Find out the outcome of the division 237 : 13 and then check the result with the division algorithm:
237 : 13 = 18 Remainder = 3
Dividend = Divisor x Quotient + Remainder
13 x 18 + 3 = 237 so it is correct.
The division can be:
a) Exact:: has zero remainder.
b) Whole: has nonzero remainder.
5.- Expressions with brackets and combined operations. Priority of the
operations.
BIDMAS is a made-up word that helps you to remember the order in which calculations take place
B I D M A S
Brackets
Division
Index or powers
Addition
Multiplication
Subtraction
This means that brackets are worked out first, the division and multiplication are done before addition
and subtraction.
Examples:
(5+3) x 6 =
8 x6=
48
Carry out the addition first
5+3x6=
5 + 18 =
23
Carry out the multiplication first
Solve
3 + 6 x (5 + 4) 3 - 7
Step 1:
3 + 6 x (5 + 4) ÷ 3 - 7 = 3 + 6 x 9 ÷ 3 - 7
Brackets
Step 2:
3+6 x
÷ 3 - 7 = 3 + 54 ÷ 3 - 7
Multiplication
Step 3:
3+
÷ 3 - 7 = 3 + 18 - 7
Division
Step 4:
3+
- 7 = 21 - 7
Addition
- 7 = 14
Subtraction
Step 5:
9
54
18
21
using the order of operations.
Evaluate
9 - 5 ÷ (8 - 3) x 2 + 6
using the order of operations.
Step 1:
9 - 5 ÷ (8 - 3) x 2 + 6 = 9 - 5 ÷ 5 x 2 + 6
Brackets
Step 2:
9-5÷5x2+6=9-1x2+6
Division
Step 3:
9-1x2+6=9-2+6
Multiplication
Step 4:
9-2+6=7+6
Subtraction
Step 5:
7 + 6 = 13
Addition
8.- Solve the exercise using the order of operations.
a) 5 + 2 x (10 – 2 x 5 + 1) – 3
b) 10 – 3 x 2 + 35 : (5 – 4 + 3 x 2)
12.- Solving arithmetic problems with natural numbers: usefulness of natural
numbers. Critical analysis of the solutions to a problem.
In order to solve problems you must follow the rules below:
1. Start with a first reading of the problem to know what it is about.
2. Then you do a second reading more slowly, in order to understand the problem and find out
that data provided.
3. Write down the data of the problem clearly. If it is a geometric problem, then you can make a
drawing. You must also check that the units are all the same. If they are not, then you will have
to change them to the appropriate ones.
4. Now you can solve the problem. In this step you do all the necessary operations to solve the
problem.
5. Finally, answering: Read the question of the problem again and answer it with a
sentence.Don't forget to mention the correct unit and check that the answer makes sense.
Example:
John has saved 350 euros in his bank account. He received 37euros as a birthday present and then,
he bought 4 DVDs which cost 15€ each. How much money does he have now?
DATA
SOLVE
Saved: 350euros
350 + 37 = 387 euros
Gift: 37euros
4 x 15 = 60 euros A:
Spent: 4 x 15euros
387 – 60 = 327euros.
ANSWER
Now, he has 327euros