unit 1: integers · 1/9/2019 · unit 1: integers. mathematics 2nd e.s.o. opposite of an integer:...

Unit 1: Integers. Mathematics 2nd E.S.O.

UNIT 1: INTEGERS

Integers:

The first set of number we knew was the set of Natural Numbers (also called whole numbers):

ℕ = {0, 1, 2, 3, 4, 5, 6, 7, …}

There are many situations in which you need to use numbers below zero, one of these is

temperature, others are money that you can deposit (positive) or withdraw (negative) in a bank,

steps that you can take forwards (positive) or backwards (negative).

Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5, …

Negative Integers are all the opposites of these whole numbers: -1, -2, -3, -4, -5, …

Integers allow us to count and order below and above zero. The set of all Integers is represented by

the letter ℤ = { … -4, -3, -2, -1, 0, 1, 2, 3, 4, … }

The natural numbers are included in the set of Integers. This fact is represented by the symbol

“ ⊂ “ .

The Number Line:

ℕ⊂ℤ is read ℕ is a subset of ℤ .

The number line is a line labelled with the integers in increasing order from left to right, that

extends in both directions:

For any two different places on the number line, the integer on the right is greater than the integer

on the left.

Examples:

4>-1 is read: “four is greater than negative one”

-3<2 is read: “negative three is less than two”


Opposite of an integer:

The opposite of an integer is the same number with the other sign. The distance from a number to

zero is the same as the distance from its opposite to zero.

The opposite of +5 is -5 The opposite of -7 is +7

Absolute value of an integer:

The absolute value of an integer is the number of units is from zero on the number line. If the

number is positive, the absolute value is the same number.

If the number is negative, the absolute value is the opposite.

The absolute value of an integer is always a positive number (or zero). We specify the absolute

value of a number n in between two vertical bars: ∣n∣ .

Examples:

∣+3∣=3

∣−5∣=5

∣+4∣=4

∣−7∣=7

1. Plot on the number line and after order them from less to great.

-2 +8 0 -5 3

Write the opposite and the absolute value of all these numbers.


Adding and Subtracting Integers:

Rules for Addition:

When adding integers with the same sign: We add their absolute values, and give the result with

the same sign.

(+4)+ (+6)= +10

(−3)+(−6)=−9

(−5)+(−2)=−7

When adding integers with the opposite signs: We subtract their absolute values (we subtract the

smaller absolute value from the larger), and give the result with the sign of the integer with the

larger absolute value.

(+7)+(−9)= −2

(+8)+(−5 ) = +3

(−6) + (+1) = −5

Rules for Subtraction:

Subtracting an integer is the same as adding the opposite.

We convert the subtracted integer to its opposite, and add the two integers: The result of subtracting

two integers could be positive or negative.

(+3) − (+7) = (+3) + (−7) = −4

(−2) − (−8) = (−2) + (+8) = +6

Unit 1: Integers. Mathematics 2 nd E.S.O. Teacher: Miguel Angel Hernández

You can use a number line to help you to add or subtract integers:

Calculate 4-6:

Start at 4 and subtract 6 (move 6 units to the left):

The answer is -2.

Calculate -3+7:

Start at -3 and add 7 (move 7 units to the right):

The answer is 4.

1. Calculate:

a) −5−2 b) −913 c) 15−12d) 7−13 e) −2−10 f) 4−−5g) −9−−6 h) −8−3 i) 12−3

2. Calculate using one of the methods of the example:

Example: -3 + 8 – 4 + 2 – 5 =

1st Method (Doing the operations in order) = 5 – 4 + 2 – 5 = 1 + 2 – 5 = 3 – 5 = - 22nd Method (Grouping positive and negative) = (8+2)+( - 3 – 4 – 5 ) = 10 – 12 = - 2

a) −107−5−68b) 4−913−16−211−5c) 14−138−20−125d) −2−3−54−1e) −71030−5015

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3. Calculate:

(-6) + (5-2) - (-6+1) =

1st Method (Removing first brackets) = - 6 + 5 – 2 + 6 – 1 = 22nd Method ( Operating first the expressions into brackets) = - 6 + 3 - (-5) = -6 + 3 +5 = 2

a) 8−3−25b) 2−7−5−4−1−6c) 7−4−923−1d) 24−1−7−4−9e) −1−23−73−4−2−3−5f) −5−104 12−9−8−16g) 2−36−8−−182−9−1h) −9−4−5−7−104−6

4. Calculate:

a) 3−5−7−410−2−−2b) 5−31−−2−1c) 8−5−4 −2−3d) 43−4 −2−6−8e) −713−6−8

5. Calculate:

a) 6[3−25]b) 4−[7−51]c) 5−[2−6−9]d) 3−[2−4−27 ]−5−1e) 5−[53−2−9]−3−10f) 1−2[4−6−−3−4]1g) 1−[2−5−3−9−1]−2−3

6. Calculate:

a) 7−2[4−3−1]b) −4−3−26−5−34c) −2−[−4−15]d) −4−13−[2−4−−17]e) −31−4−[1−2−3]f) [4−35]−[4−35]

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Multiplying and Dividing Integers:

Rules for Multiplication.

To multiply a pair of integers:

– If both numbers have the same sign (positive or negative), their product is the product of their absolute values (their product is positive).

– If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative).

– If a number is 0, the product is 0.

Look at the chart below:PRODUCT + -

+ + -- - +

5· 4=20 5· −4=−20

−5· 4=−20 −5· −4=20

To multiply any numbers of integers:

1. Count the number of negative integers in the product. If this numbers is even, the product is positive, but if the number is odd, the product is negative.

2. Take the product of their absolute values.

( If any of the integers in the product is 0, the product is 0).

−4·6· −2=48 −2·−3·−5=−30

0 ·−5.7=0 −5· 2·−2·4 =80

−1· 3· −2 ·−10=−60 −1· −2 ·1 ·−3· −5=30

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Rules for Division:

To divide a pair of integers the rules are the same than for the product:

– If both numbers have the same sign (positive or negative), divide the absolute values of the first integer by the absolute value of the second integer (the result is positive).

– If the number have opposite signs, divide the absolute value of the first integer by the absolute value of the second integer, and give the result a negative sign.

Look at the chart below:DIVISION + -

+ + -- - +

12: 3=4 12: −3=−4

−12: 3=−4 −12: −3=4

1. Calculate:

a) −3· −4b) −5·4c) 10· −3d) −15: 3e) 40: −8f) −56: −7

2. Calculate:

a) −5·−2 ·−1b) 2·−1 ·4c) −18: −2: 3d) −20:2 :−1e) 9 ·−2·−1 ·2f) −5·−2 ·1· 8g) −36 :9·−2h) −15·−3: −5

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Powers of Integers:

Powers are products of equal factors:

an=a ·a ·...... · a , n times

where a is the base and n is the exponent or index.

Examples:

42=4·4=1643=4·4· 4=64−34=−3·−3·−3 ·−3=81−33=−3· −3· −3=−27

Sign of the power of an integer:

• If the base is positive, the sign will always be positive.• If the base is negative, the sign will be positive if the exponent is even, and negative if it is

odd.

Example:

−21=−2−22=−2 ·−2=4−23=−2· −2·−2=−8−24=−2·−2·−2 ·−2=16−25=−2· −2 ·−2·−2· −2=−32−26=−2· −2·−2·−2 ·−2·−2=64 …

Operations with powers:

Multiplying powers: You can multiply powers with the same base by adding the exponents.

am⋅an=amn

Examples: 34⋅37=347=311 −25⋅−24⋅−2=−2541=−210

Dividing powers:You can divide powers with the same base by subtracting the exponents.

am :an=am−n

Examples: −312 : −37=−312−7=−35 25 :2=25−1=24

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Power of a power:You can simplify the power of a power by multiplying the exponents.

amn=am⋅n

Examples: 425=42 ·5=410 [−34]2=−34⋅2=−38

Multiplying powers with the same exponent:You can multiply powers with the same exponent by multiplying the bases.

a⋅bn=an⋅bn

Examples: 34⋅54=3⋅54=154 [−2⋅3]5=−25⋅35

Dividing powers with the same exponent:You can divide powers with the same exponent by dividing the bases.

a : bn=an :bn

Examples: 154 :34=15 :34=54 6 :35=65:35

1. Express as just one power:

a) 23 · 25 · 26

b) 3 ·32 · 37

c) −62 ·−65 ·−64

d) 610 : 65

e) −76 :−72

f) −22·−23· −2g) [−33]5

h) [−26 ]2

2. Express as just one power:

a) [−22 ]3 : −24

b) 24 · 23 : 25

c) 32·35 ·36 : 34·35 d) 84:82 :82

e) 76 3 · 72 3

f) 64 : [ 28 :27 ·3]3

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NEGATIVE EXPONENTS

Simplify. Your answer should contain only positive exponents.

1) 7–2

2) 2–4

3)

4)

5)

6)

7)

8)8

3

6

6

9)5

3

2

2

10)4

2

5

5

11) 3225 4·4

12) ()

13) 25 a

14) 4c –3

15) –18a2 b–3

16) 4

5z

17) 1

23

y

x

18)1

4

3

19)2

5

2

20)3

3

2

c

a


Square root:

The square root of a number a is another number b whose squared is a.

a=b when b2=a

The number a is called radicand, the sign √ is called radical and b is called the squared root of a.

The numbers with an exact square root are called perfect squares.

Examples:

1=1 because 12=14=2 because 22=49=3 because 32=916=4 because 42=1625=5 because 52=2536=6 because 62=3649=7 because 72=4964=8 because 82=6481=9 because 92=81100=10 because 102=100

121=11 because 112=121144=12 because 122=144169=13 because 132=169196=14 because 142=196225=15 because 152=196256=16 because 162=256289=17 because 172=289324=18 because 182=324361=19 because 192=361400=20 because 202=400

But, be careful! If we are working in the set of integers, a number can have two square roots:

Example:

36=±6 , because 62=36 and −62=36100=±10 , because 102=100 and −102=100−4 , it does not exist because any squared integer is negative. −9 , it does not exist.

Integer square root:

If a radicand is not a perfect square, the square root is not exact. In this case, we talk about integer square root.The integer square root of a number a is the greater number b whose squared is less than a.The remainder of the integer square root is the difference between the radicand a and the squared of the integer root b.

Examples:

11≈3 Remainder= 11−32=11−9=229≈5 Remainder= 29−52=29−25=437≈6 Remainder= 37−62=37−36=1

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Radicals

By definition, an nth root of a number is this:

The number c is an nth root of a, so can if acn

EX: The square roots of 81 are 9 and -9, because 8192 and 81)9( 2

The index of a radical symbol nn indicates what root is being

found.

EX: 3 x means the “cube root” of x. Similarly, 7 x means the “7th

root” of x.

ODD ROOTS: Every real number has just one real odd root (index

is odd): odd roots of positive numbers are positive and odd roots of

negative numbers are negative.

EX: 5 32 = 2 and 2325 . It’s not a problem to have a negative

radicand with an odd root.

EVEN ROOTS: Every positive real number has two real even roots

(index is even). Negative numbers do not have real nth roots when n is even.

EX: The 4th roots of 16 are 2 and -2. Negative 16 (-16) does NOT have a 4th root, since no real number multiplied to itself 4 times can

be negative.


Order of operations:

Do all operations in brackets (or square brackets) first.Then, do all the exponents (indexes and roots)Then, do multiplications and divisions in the order they appear.Finally, do additions and subtractions in the order they appear.

Easy way to remember:

Parenthesis Exponents Multiplications Divisions Additions Subtractions

This gives you: PEMDAS: “Please Excuse My Dear Aunt Sally”.

1. Calculate:

a) 5· −8−3·−2

b) 15: −5−22:−2

c) −5·−6−4·−3

d) −14: −2−6: −3

e) −50: [−6−4]

f) 3. [−56: −8]

2. Calculate:

a) 15: −3 ·−1

b) −80: [−8 ·2]

c) [−80 :−8] ·2

d) [9 ·−8] : [−3· −4]

e) 2−5·4: −2

f) −5−40 :−5 ·31

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3. Calculate:

a) 35−828−12−33·−5

b) 17−[2 ·35−24]·3−6 ·−5

c) −22−−32−13

d) 42−42−−42−42

e) 23−3242−17

f) [52·−32]: −15

4. Calculate:

a) 32−6573−1024

b) 2 ·[−315−201]−2

c) −23 ·−23−21: 1−3

d) 3 ·5−82 ·3−12−53 ·2

e) 2−[ 4−3−7]

f) −5−[23−6· 5−3]

5. Calculate:

a) 36−464−10

b) −5 · 49−81

c) −1−23−24

d) −17−1849

e) 25−9 : [2 :−2 ]

f) 144−100 ·9−8

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6. Calculate:

a) [9−6−24]: 3

b) 3: −3−12: 63

c) 54 :[−10−4−4]

d) −3· −4−−24 :615 : −3

e) 10 :−2−−7·−34

f) 1[−30: −52]−35 : −7

1. Mount Everest is 29 028 feet above sea level. The Dead Sea is 1 312 feet below sea level. What is the difference of altitude between these two points?

2. The temperature in Chicago was 4º C at two in the afternoon. If the temperature dropped 12º C at midnight, what is the temperature now?

3. A submarine was situated 21 000 feet below sea level. If it ascends 1 230 feet, what is its new position?

4. Aristotle was born in 384 B.C. and died in 322 B.C. How old was he when he died?

5. A submarine was situated 1 230 feet below sea level. If it descends 125 feet, what is the new position?

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Activities.


6. This is the three-day forecast for Yellowknife (Canada) from the 24th of November 2008.

Today Tue WedNov 24 Nov 25 Nov 26

Snow Cloudy Snow-6º C -7ºC -6ºC-13º C -8ºC -14ºC

What is the difference between the maximum and minimum temperatures each day?

What are the maximum and minimum temperatures during these three days?

7. This is the three day forecast for Birmingham (UK) from the 24th of November 2008.

Today Tue WedNov 24 Nov 25 Nov 26

Rain Sunny Partly Cloudy 13º C 14º C 14º C 2º C -2º C -1º C

What is the difference between the maximum and minimum temperatures each day?

What are the maximum and minimum temperatures during these three days?

8. The Punic wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last?

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9. This a table with the melting and boiling points of some metals

Metal Melting Points ºC Boiling Points ºCAluminium 660,32 2519Iron 1538 2861Gold 1064,18 2856Mercury -38,83 656,73

a) Calculate the difference between the melting and boiling point of each metal.

b) How much warmer is the melting point of mercury than the melting point of iron.

10. On the 2nd of January, the temperature dropped from 3º C at two o'clock in the afternoon to -11º C at 8 a.m. the next day. How many degrees did the temperature fall?

11. A Greek treasure was buried in the year 164 B.C. and found in 1843 A.D. How long was the treasure hidden?

12. On the 1st of December, the level of the water in a reservoir was 130 cm above its average level. On the 1st of July it was 110 cm below its average level. How many cm did the water level drop in this time?

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unit 1: integers · 1/9/2019 · unit 1: integers. mathematics 2nd e.s.o. opposite of an integer:...

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