applying the distributive law

7/31/2019 Applying the Distributive Law

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Applying the Distributive LawTo cleverly calculateA = 57 968 + 43 968, we can apply the distributive law, which allows us to write:A = (57 + 43) 968 = 100 968. We thenget the answer 96,800.What are the different distributive laws, and when should we use them?

I. Distributive laws

In the following two paragraphs, a, b, and kdesignate any numbers.

A. Distribution of multiplication with respect to addition

This gives us the rule: k (a + b) = k a + k bMore simply, we can write: k(a + b) = ka + kb

Example: 2 (3 + 4) = 2 3 + 2 4We can check that: 2 (3 + 4) = 2 7 = 14and that: 2 3 + 2 4 = 6 + 8 = 14.

Note: We also have: (a + b) k= a k+ b kand therefore:(a + b)k= ak+ bk= ka + kb = k(a + b).

B. Distribution of multiplication with respect to subtraction

This gives us the rule: k (ab) = k ak bMore simply, we can write: k(ab) = kakb

Example: 3 (52) = 3 5 - 3 2Note: We also have: (ab) k= a kb kand therefore:(ab)k= akbk= kakb = k(ab).

C. Generalization

The preceding formulas can be generalized to any number of terms inside the parentheses.Example: 2 (3 + 45) = 2 3 + 2 42 5

II. Examples of application

A. Mental calculation

Example 1: We want to calculate 25 11; 25 21; and 25 31.We can proceed like this:25 11 = 25 (10 + 1) = 25 10 + 25 1 = 250 + 25 = 275In the same way:25 21 = 25 20 + 25 = 500 + 25 = 52525 31 = 25 30 + 25 = 750 + 25 = 775

Example 2: We want to calculate 24 9; 24 19; and 24 29.We can proceed like this:24 9 = 24 (101) = 24 1024 1 = 24024 = 216In the same way:24 19 = 24 (201) = 24 2024 = 48024 = 45624 29 = 24 (301) = 24 3024 = 72024 = 696

B. Calculating the lateral surface area of a right prism


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Figure 1 shows the net of the lateral surface of a right prism. The unit of length is centimeters.To calculate the area of this surface, we can carry out the following calculation:24 14 + 24 19 + 24 12This would amount to adding together the areas of the three rectangles.It is, however, simpler to carry out this calculation:24 (14 + 19 + 12) = 24 45 = 1,080So we find that the area of this surface is equal to 1,080 cm2.

Note: (14 + 19 + 12) is the size in centimeters of the perimeter of a base of the prism.

3. Calculating the area of a circular ring

We want to calculate the area of the circular ring shaded in figure 2. The unit of length is centimeters. The unit of area will be centimeters squared.

The area of this ring will be equal to the difference between the area of the large circle with a radius of 3 cm and the area of the small circle with aradius of 2 cm. We have:

area of the large circle: 32 = 3 3 = 9 ;

area of the small circle: 22 = 2 2 = 4 ;

area of the ring: 9 - 4 = (94) = 5 .

The area of the ring is therefore equal to 5 cm2.

By taking 3.14, we find the area is around 15.7 cm2.

Note: Generally, the area of a circular ring bordered by two concentric circles with respective radii ofR and ris equal to R r, which is (Rr).

Copyright 2006 Ruedesecoles, translated and reprinted by permission. Translation Copyright of Microsoft Corporation. All rights reserved.

1993-2008 Microsoft Corporation. All rights reserved.

Calculating a Numerical Expression (1)

A teacher was giving an oral test. She read out numerical expressionsA,B, C, andD in the following way:A = (93) 2 + 1 Subtract 3 from 9, double your answer, and then add 1.B = 93 2 + 1 Multiply 3 by 2, subtract from 9, and then add 1.C= 9(3 2 + 1) Multiply 3 by 2, add 1, and subtract your answer from 9.

D = 93 (2 + 1) Add 2 to 1, triple your answer, then subtract it from 9.Although they all consist of the same numbers and the same operation symbols, these numerical expressions are not calculated in the same way andeach one gives a different result (A = 13,B = 4, C= 2,D = 0).What are the rules for calculating such expressions? The clues are in the way the teacher read them out.

I. Order of operations


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To be sure to get the correct answer, conventions have been agreed upon. So, to calculate a numerical expression without parentheses or brackets, wecomplete the calculations from left to right, starting with the multiplication and the division, which take priority over add ition and subtraction.

Note: If there is only addition and subtraction (or only multiplication and division), we complete the calculations from left to right.Example:B = 93 2 + 1.First, do the multiplication, which takes priority over the addition and the subtraction:B = 96 + 1.Then do the subtraction and the addition, from left to right:B = 3 + 1 = 4.

II. Calculating with parentheses

If there are parentheses, start by doing the calculations inside the parentheses. Complete the calculations, following the pr iorities defined in section I.Example 1:C= 9(3 2 + 1).Start by calculating 3 2 + 1, which is in parentheses. To do this, first work out the multiplication, which takes priority: C= 9(6 + 1).Then complete the calculation inside the parentheses: C= 97 = 2.

Example 2: .This expression can be written in the following form:G = 3 + (6 + 4) (72).Start by working out 6 + 4 and 72: G = 3 + 10 5.Then do the division, which takes priority over the addition: G = 3 + 2 = 5.



Calculating a Numerical Expression (2)How do we calculate an expression involving addition and subtraction?

I. Calculating an algebraic sum

A. From left to right

In general, we calculate from left to right.Example: To work out (-2) + (-1.5) - (-0.3), first calculate (-2) + (-1.5), then subtract (-0.3) from the result.We can present the calculations in the following manner, showing one stage of the calculation on each line:

A = (-2) + (-1.5) - (-0.3)A = (-3.5) - (-0.3)A = (-3.5) + (+0.3)

A = -3.2

B. By changing the order of the terms

It is possible to change the order of the terms in order to make calculations easier.Example with two terms:

B = (-24.8) - (-32.5) + (+24.8)B = (-24.8) + (+24.8) - (-32.5)B = 0 + (+32.5)B = +32.5

II. Giving a simplified notation of an algebraic sum

A. Rules

1. For positive numbers, writing the + sign and brackets is optional.2. In a sum, if the first term is negative, the brackets are optional.3. Adding a number is the same as subtracting its opposite (and subtracting a number is the same as adding its opposite).

B. Examples with two terms


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(+7) - (+2) can be written 7 - 2, according to rule 1.We write: (+7) - (+2) = 7 - 2 = 5.(-3) + (+1) can be written (-3) + 1, according to rule 1, then -3 + 1, according to rule 2.We write: (-3) + (+1) = -3 + 1 = -2.(+1) + (- 4) can be written (+1) - (+4), according to rule 3, then 1 - 4, according to rule 1.We write: (+1) + (- 4) = 1 - 4 = -3.

C. Generalization

We want to simplify how the following expression is written:E= (-2) + (+3) + (-4) - (+5) - (-6).This gives us:

E= (-2) + (+3) - (+4) - (+5) + (+6), according to rule 3.

E= -2 + 3 - 4 - 5 + 6, according to rule 2.In practice and for speed, we can use the following rules for rewriting notation:

III. Calculating an expression with more than one set of brackets or parentheses

Start by completing the calculations inside the innermost brackets or parentheses.Examples with two terms:C= (- 5) - ((-2) + (+7))C= (-5) - (+5) = (-5) + (-5) = -10

D = [(-1.2) - (-2)] - [3 - (7.1 - (-8.5) + (-3)]Using simplified notations:

D = [-1.2 + 2] - [3 - (7.1 + 8.5 - 3)]D = 0.8 - [3 - 12.6]D = 0.8 - [-9.6]D = 0.8 + 9.6D = 10.4



Describing Different Types of NumbersChildren use natural numbers to learn to count. However, this type of number is too limited to solve certain problems, such as in geometry. Newsystems are needed and so we add symbols to the numbers: the decimal point, the fraction bar, root signs, etc.What are the different types of numbers?

I. A brief history of numbers

A. Natural numbers

As the adjective natural suggests, these are the first numbers we use: Just as a child learns to count using his fingers, early people began to countobjects or animals. Naturally, we count: 1, 2, 3, etc.

B. Whole numbers

The whole numbers consist of the natural numbers combined with the number zero.


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C. Rational numbers

Problems involving division and measurement of lengths gave rise to fractions of whole numbers, which are called rational numbers in mathematics,

such as , for example. The Greeks knew about natural numbers and rational numbers.

D. Decimal numbers: Decimal numbers come from fractions: They correspond to particular fractions. Nowadays, a decimal number indicates a

number written with a decimal point and followed by a number of figures, 23.45 for example.

E. Negative numbers:There is evidence that negative numbers were being used in India in the 7th century. It is important to note that Indians

used zero, a necessary precondition to conceiving negative numbers. Negative numbers were called debt numbers for commercial reasons, just as

today's bank statements have two columns entitled credit(for receipts) and debit(for expenditure).

The use of negative numbers in the West came much later. Italian Renaissance mathematicians, who were specialists in algebra (the science of solvinequations), understood that without negative numbers they could not solve certain equations (x + 7 = 0, for example). However, they were unsurewhether to call these solutions proper numbers. And even in the 17th century, French mathematician Descartes described negative numbers asfalsenumbers.It was not until the 19th century that negative numbers were finally treated as true numbers.

F. Irrational numbers The followers of Pythagoras proved that the length of a diagonal of a square of side 1 is not a rational number. Today, we

write this number as and call this kind of number irrational because it cannot be expressed as the ratio of two whole numbers.

II. The different types of numbers in mathematics:Relationships exist between the different types of numbers.

A. Whole numbers:Every number used in mathematics is derived from a whole number. The letter Wis used to denote this set.

Examples: 12, 5, and zero are whole numbers.

B. Integers

Integers are whole numbers to which we add '+' for positive numbers or '-' for negative numbers. The letterZ(from the GermanZahl, meaningnumber) is used to denote this set of numbers.

Examples: +3, zero, and - 72 are integers.We know that positive numbers can be written without the '+' sign, for example: +7 = 7.

Consequently, every whole number is an integer. In mathematics, we say that the set W is included in the set Z. This is written: .

C. Rational numbers

These are fractions of the type , where a and b are integers and .The letter Q is used to denote the set of rational numbers.

Examples: and are rational numbers.

D. Real numbers

As we said before, there are numbers, such as for example, that are not rational. We also acknowledge a set of numbers called real numberscontaining all the numbers mentioned previously. Real numbers are denoted by the letterR. The real numbers usually suffice as a basis for themathematics studied in school, though still other types of numbers exist, such as complex numbers and quaternions.

Examples: 5;29;49.21; ; , and are real numbers.


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So at school we work with four sets of numbers contained in one another: .



Determining the Common Factors of Two IntegersHow do you find the common factors of two whole numbers?

I. Factors and multiples

A. Definitions

a and b are two whole numbers.b is a factor ofa if a whole number, q, exists, such that a = b q.a is also referred to as a multiple ofb, i.e., a is divisible by b.

Notes:Ifb is a factor ofa, then the division ofa by b gives a remainder of 0. Thus, we can write a = b q, where q is the quotient ofa divided by b;Some calculators have a key for division where sometimes both the quotient and remainder are displayed on the screen.

Example: Are 13 and 7 factors of 221?Carry out division of 221 by 13, then 221 by 7:221 = 13 17, therefore 13 is a factor of 221.221 = (7 31) + 4, therefore 7 is not a factor of 221.

B. Reminder of tests of divisibility

It is not always necessary to perform a division to know if a whole number is divisible by another; the following rules serve as a reminder:A whole number is divisible by 2 if its last digit is zero, 2, 4, 6, or 8.A whole number is divisible by 3 if the sum of its digits is divisible by 3.A whole number is divisible by 5 if its last digit is zero or 5.A whole number is divisible by 9 if the sum of its figures is divisible by 9.A whole number is divisible by 10 if its last digit is zero.

Example: According to these criteria, we can say that 975 is divisible by 3 and 5, but is not divisible by 2, 9, or 10.

II. The factors of a whole number

A. Some rules

Consider a whole number a other than zero or 1. This number has at least two factors: 1 and itself.It is always true that a = a 1.The number 1 only has one factor: itself.The number zero is divisible by every non-zero whole number.Often, the factors of a whole number can be paired: For example, 8 and 9 are a pair of factors of 72, because 72 is divisible by 8 and 9, and 72 = 8 9

B. Method


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The method of finding all the factors of a whole number can be explained using an example: Find all the factors of 72. This can be done by dividing 7by every successive whole number: 1, 2, 3, etc.Where the remainder is zero, write the corresponding equation and the pair of factors obtained.

The process stops because the following equation, 72 = 9 8, gives the factors 8 and 9, which have already been obtained.The factors of 72 are therefore: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

III. The common factors of two whole numbers

A. Example

Find all the common factors of 72 and 54. To do this, find the factors of each of these numbers using the method explained above, then take thenumbers that appear in both lists:The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, and 54.The common factors of 72 and 54 are therefore: 1, 2, 3, 6, 9, and 18.

B. Definition of the HCF

The highest common factor (or greatest common divisor) of two whole numbers is, in abbreviated form, the HCF of these two whole numbers.Example: The HCF of 72 and 54 is 18 (from the example given above).This is written: HCF(72, 54) = 18.Returning to the list of common factors of 72 and 54: 1, 2, 3, 6, 9, and 18, note that these are all factors of 18.Property: The common factors of two whole numbers are the factors of their HCF. If we know the HCF of two whole numbers, we can just find all ofits factors to find the common factors of these two whole numbers.

Copyright 2006 Ruedesecoles, translated and reprinted by permission. Translation Copyright of Microsoft Corporation. All ri ghts reserved.


Recognizing a Proportional RelationshipConsider the following expressions:At the supermarket, the price of a steak is proportional to its weight;at constant speed, the fuel consumption of a car is proportional to the distance traveled.Each of these describes a proportional relationship.What does that mean? Under what conditions can we say that a situation displays a proportional relationship?

I. Recognizing a proportional table

Example:

The above table is a proportional table; the numbers on the second line are each equal to four times the numbers on the first line:2 4 = 87 4 = 280.8 4 = 3.2

Definition: A proportional table is a table (formed from two lines of numbers) for which there is a number kso that the numbers on the second line areach equal to ktimes the numbers on the first line.The number kis known as the coefficient of proportionality.


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II. Applications

A. Example 1: the price of steak

The weights (in pounds) and prices (in dollars) of four packages of steak in a supermarket were recorded, and the results are shown in the table below

Lets carry out the following calculations:$6.25 1.250 lb = $5/lb, so 1.250 lb $5/lb = $6.251.700 lb $5/lb = $8.502.300 lb $5/lb = $11.502.520 lb $5/lb = $12.60These calculations demonstrate that the table above is a proportional table. This is how we should understand the phrase, th e price of steak is

proportional to its weight.Note: The coefficient of proportionality is equal to $5/lb; $5 will be the price of a one-pound steak. This coefficient appears on the package labels inthe form price per pound: $5.

B. Example 2: the fuel consumption of a car

The graph in this figure shows the fuel consumption of a car as a function of the distance traveled at a constant speed. For example, we can read fromit that the car consumes 5 gallons of gas for 100 miles traveled.

All of the points on the graph are aligned with the point of origin. This alignment is characteristic of a proportional situation.

We can highlight this proportional relationship by setting up the table shown below using the values on the graph:

Note: The coefficient of proportionality is equal to 0.05 gallons/mile; that is the gas consumption in gallons for one mile traveled.

C. Example 3: the annual expenditure on a video-rental club

In a video club, you pay $18 for a one-year subscription, and then $2 each time you rent a film.Is the annual expenditure proportional to the number of films rented?Look at the table below.

You spend $20 if you rent one film ($18 subscription + $2 for the film).You spend $22 if you rent two films ($18 subscription + $4 for the two films).

By examining just the first two columns in the table, we can see that it is not a proportional table, since: 20 = 1 20 and 22 2 20.We can conclude from this that annual expenditure is not proportional to the number of films rented. However, if the yearly subscription fee were $0(that is, if membership were free), the annual expenditure would be proportional to the number of films rented.




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Rounding to the Nearest UnitPeter owes Jack $5.60, but he only has $1 bills and Jack does not have any change. Peter decides to give Jack $6, which is $5 .60 rounded up.What rule allows us to round decimal numbers? And what is the difference between rounding and truncating?

I. Rounding to the nearest unit

Example: In figure 1, we have placed the numbers 8, 8.36, 8.5, 8.74, and 9 on an axis.As 8.36 is closer to 8 than to 9, when we round 8.36 to the nearest unit we get the number 8.

For a similar reason, when we round 8.74 to the nearest unit, we get the number 9.For the number 8.5, we must make a choice, because it is positioned at the same distance from 8 and from 9 . We will decide to choose 9 when weround 8.5 to the nearest unit.

Definition:a is a decimal number:When the decimal part ofa is 0.5, rounding to the nearest unit gives us the integer just greater than a;in other cases, rounding a to the nearest unit gives us the whole number that is closest to a.

II. Truncating to the unit

Example:Figure 2 shows a way of truncating the number 31.42 to the unit. This truncation gives us 31. We just need to cut off the decimal part ofthe number.

Definition: Truncating a decimal number dto the unit gives the largest integer less than or equal to d.Therefore there are two possibilities:The truncation is equal to rounding to the nearest unit (this is the case for 8.36);the truncation is equal to 1 less than rounding to the nearest unit (this is the case for 8.74).



The Effect of Addition and Multiplication on the Order of Numbers

Peters schoolbag is heavier than Maries. Deciding that their bags are too heavy, each of them takes out the third -year mathematics book. It is obviouthat Peters bag will still be the heavier of the two.What is the mathematical translation of this situation?

I. Addition and order

1. Property

Example: We want to comparex = 23.4 + 7.986 andy = 19.74 + 7.986.Carrying out the two additions is one way to find the answer, but there is another way as well. Effectively we are adding the same number, 7.986, toboth 23.4 and 19.74. Since 23.4 > 19.74, we see thatx >y.

Rule: Ifa, b, and c are integers, the integers a + c and b + c come in the same order as a and b.In other words, if we take the same number and add it to (or subtract it from) each member of an inequality, we obtain an inequality in the same

direction, that is equivalent to the original.

2. Applications

Example 1: We want to compare and .

We note that we are subtracting the same number, , from both and . We know that , since two numbers with the same positivenumerator are arranged in the inverse order of their denominators.


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Therefore we know that .Example 2:x is an integer such that . What does this tell us aboutx?If we subtract the same number from each member of an inequality, we find an inequality in the same direction.Therefore we know that , that is, . So the numberx is less than or equal to -10.

II. Multiplication and order

1. Property

Examples: We want to comparex = 4.7 2.93 andy = 7.9 2.93.Effectively we have multiplied 4.7 and 7.9 by the same positive number, 2.93. Since 4.7 < 7.9, we know thatx (-3) 7 (that is, -12 > -21).This example shows the importance of the strictly positive condition.

2. Applications

Example 1: We want to compare and .

We note that we are multiplying both and by the same positive number, .

Since , we can see that .

Example 2:x is an integer such that . What does this tell us aboutx?If we divide each member of an inequality by the same positive real number, this gives an equivalent inequality in the same direction. Therefore

, that is, .

So the numberx is less than or equal to .



Writing a Numerical Expression That Corresponds to a Sequence of OperationsAt the market, you buy three cans of beans for $0.85 each and two bags of apples for $1.95 each.If you pay with a $10 bill, how much change will you get?Here is a detailed solution given by a student:

Is it possible to write a numerical expression that summarizes all the calculations carried out above? In other words, is it possible to write a numericalexpression that contains all the data from the statement and that gives the result 3.55?

I. First example: total to pay and change

Use the example above. The total to pay is calculated by working out 2.55 + 3.9 or 0.85 3 + 1.95 2 (respecting the rules of the order of operations


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The change is the difference between 10 dollars and the total price to pay; this corresponds to the expression:10(0.85 3 + 1.95 2). This expression summarizes the calculations carried out step by step in the introduction.

II. Second example: calculating an area

Figure 1 shows the plan of a house. We want to write an expression that will allow us to calculate the area of the backyard.

The backyard is rectangular. Its dimensions (in meters) are:length: 14 m5 m;width: 10 m6 m.The area of the backyard (in square meters) is therefore equal to: (14 m5 m) (10 m6 m).This expression summarizes the stages of the calculation and allows us to calculate the area of the backyard (equal to 36 m).



Writing Numbers and the Properties of Numbers'God made whole numbers. The rest is the work of man.'Leopold Kronecker (1823-1891), German mathematician.During certain periods of history, whole numbers were considered to be an innate knowledge or a gift of the gods. Since then, other families ofnumbers have been constructed to solve new problems: decimal numbers to improve operational techniques, integers to take account of commercialexchanges, rational and irrational numbers to measure size, etc.We will study these different groups of numbers below.

1. How do we decide the group(s) to which a number belongs?

We can distinguish several groups of numbers.

denotes the set of whole numbers. = .

denotes the set of integers.

denotes the set of rational numbers. These are numbers that can be written in the form where and * (therefore they arewhole quotients).

For example, and are rational numbers, but is not a rational number.

denotes the set of real numbers. This set comprises all the numbers that we use. They can be shown on a coordinate number line:

Every real number is represented by a point and each point represents a real number.This set includes irrational numbers, i.e., real numbers that are not rational.

These sets of numbers are subsets of each other as follows: .This means that a whole number is also an integer; an integer is also a decimal, etc.To recognize a number's type:Simplify how it is written as much as possible.

In the case of an irreducible quotient , carry out the division. If it is finite (i.e., if the remainder is zero) then is a decimal; if it is infinite, it is


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referred to as recurring and is a rational number that is not a decimal.If the number cannot be written as a quotient of integers, then it is irrational.

2. How do we write a decimal number?

In the science subjects, we often use powers of 10 to write a decimal number, so we immediately have an idea of the size of t he number.

Example:Take the decimal number 28,642.357.

We can write this as a whole number and a power of 10: .

The scientific notation of this number comprises a decimal between 1 and 10, excluding 10, and a power of 10: .By rounding the decimal number in scientific notation to the nearest whole number, we find an approximation of the number:

.

3. How do we work out an approximate value?

If we want to write a real number as a decimal where it is not a decimal number, we have to use an approximate value.

For example . This is not equal as the number 3 repeats an infinite number of times (or place values).

An approximate value can be defined by rounding up or down. We refer to a value approximated to the nearest , wherep is a whole number,

when the difference between the number and its approximate value is less than .To determine the approximate value of a positive real number to n decimal places:When rounding down, we shorten to n decimal places of this number (this means deleting the decimal places that come after the first n places).

When rounding up, we take the approximate value obtained by rounding down and add 1 to the last decimal place (this means adding to theapproximation from rounding down).To determine the approximate value of a negative real number to n decimal places:When rounding up, we shorten to n decimal places of this number.When rounding down, we take the approximate value obtained by rounding up and add 1 to the last decimal place.To calculate the rounded value of a real number to n decimal places, we shorten the number to n decimal places, then:

The shortened number is the rounded value if the decimal place is 0, 1, 2, 3, or 4;

The rounded value is found by adding 1 to the final decimal place of the shortened number if the decimal place of the real number is 5, 6, 7, or 9.

4. How do we know whether a whole number is divisible or not?

We say that number b can be divided by number a when a number q exists, such that .We can then say that b is a factor ofa and that a is a multiple ofb.To find particular divisors, we need only know certain criteria for divisibility:A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8;A number is divisible by 3 if the sum of its digits is a multiple of 3;A number is divisible by 4 if the number formed by taking its last two digits is divisible by four;A number is divisible by 5 if its last digit is either 0 or 5;A number is divisible by 9 if the sum of its digits is a multiple of 9;A number is divisible by 10 if its last digit is 0.A prime number is a whole number with only two distinct factors: 1 and itself.

It is important to know the smallest prime numbers. These numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,83, 89, 97, 101, ...To find out if a number is a prime number:First check that the divisibility criteria do not apply;Then divide the number by the prime numbers 7, 11, 13, 17,... Each time, check that the remainder from the division is not zero (the result of thedivision is not exact). When the quotient becomes less than the divisor, there is no need to continue.

5. How and why do we break a number down into its prime factors?


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252 is not a prime number, as . So the number 252 is a product of prime numbers. In more generaterms, we can say that any whole number that is greater than or equal to 2 is prime or the product of prime numbers.When we break a number down into the product of primes, we say that this number is broken down into its prime factors.To break a number down into its prime factors, for example 72, we can either:

Use multiplication tables: ;Or successively divide by the prime numbers 2, 3, 5, 7, etc. For our example, this gives the following table:

From this, we find that .We use the process of breaking down into prime factors to simplify the way of writing quantities with fractions and radicals. We also use it to workout the greatest common divisor of a group of numbers or the least common multiple.

Reminder:

.Writing a decimal number in scientific notation means writing it as the product of a decimal number between 1 and 10, excluding 10, and a power of10.A prime number is a whole number with only two distinct factors: 1 and itself.



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Taking a Fraction of a NumberWhen we say that a runner missed out on first place by a fraction of a second, we mean that he was several tenths or hundredths of a second behind thwinner. In mathematics, taking a fraction of a number has a very precise meaning. What is it?

I. General method

A. Example

In a group of 120 people, three quarters are wearing pants. To find the number of people wearing pants, all we need to know is how to work out threequarters of 120.

Three quarters is the same as three times one quarter .If we know how to find a quarter of 120, then we know how to find three-quarters of 120, i.e., by multiplying by 3.

As one quarter of 120 is , finding three-quarters of 120 is the same as calculating .

Since , we find that .Therefore, 90 people in this group are wearing pants.

Note: . From this we can see that .

B. Generalization

We have just seen that three-quarters of 120 is the same as and that . So, we can make sense of the product , which means three-quarters of 120.


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In more general terms, given a number n, three quarters ofn will be .

We can generalize further by replacing the fraction with any other fraction. So, if we take a fraction written as , where b 0, then

ofn is: .Special case for decimal fractions:

Lets say that a is a natural number. If we want to take a fraction , , or of a number, we simply multiply by a, then move thedecimal point one, two, or three places to the left respectively in the result obtained.

For example, of 12 (thirty-seven hundredths of 12) is 4.44, since 37 12 = 444.

In the same way, of 5.4 (twenty-three tenths of five point four) is 12.42, since 23 5.4 = 124.2.



Multiplying Two FractionsWe cut a pie into four equal parts, then we split each of these parts in two.What fraction of the pie does each of these small parts represent, and how does this example allow us to illustrate the multiplication of two numbersthat are written as fractions?

I. Starting example

We cut a pie into four equal parts.

Colored fraction of the pie: .Next we split each of these parts in two.

Colored fraction of the pie: half of ; therefore .What fraction of the pie does each of these small parts represent?


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Answer: . From this we can deduce that .

II. Rules of calculation

A. General rule

To multiply two numbers that are written as fractions, we multiply the numerators together and the denominators together:

, with b 0 and d 0.Examples:

B. Particular case

If one of the factors is not written as a fraction, we can apply the following rule: , with d 0. (We can find this rule by thinking that

and then applying the rule above.)

Example:

C. Generalization

The rule stated above can be generalized for more than two factors.

Example:

D. Simplification

Before working out the products of the numerators and denominators it can help to simplify.

Examples:

(we have simplified by canceling the common factor 27)

(we have simplified by canceling the common factors 7, 5, 2, and 13)

III. Example of application


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In a class of 30 students, three-fifths of the students are girls, and five-sixths of the girls study French. What fraction of the students in the class isrepresented by girls who study French?1st method:

; there are 18 girls in the class.

; there are 15 girls who study French.

; the girls who study French therefore represent half of the class.2nd method:

; the girls who study French represent half of the class.This second method is quicker than the first; also, it can be used even when we do not know how many students there are in the class.



Simplifying Fractions (A)The simplest form of a fraction is the form of that fraction where the numerator and denominator do not have any common factors apart from 1. If weare given a fraction, how do we find the simplest form?

I. Simplifying a fraction

To simplify a fraction is to divide the numerator and the denominator by the same positive integer. This integer must be a common factor of thenumerator and the denominator.

Example: Can be written in a simplified form?

We can see that 150 and 400 are multiples of 10. Therefore, we can write: .

We say that we have simplified the fraction by a factor of 10.

Now we can see that 15 and 40 are multiples of 5. Therefore we have: .

We say that we have simplified the fraction by a factor of 5.

Finally, we can write: . Now we can say that we have simplified the fraction .

Note: If we had started by simplifying by 5 and then by 10, we would still have obtained (and also if we had started with 2, then 5, then 5 again).

II. Simplifying to obtain the simplest form of a fraction

Among all the different fraction forms of a number, there is one that cannot be simplified. We say that it is the simplest form of the number. How dowe find it?There are several methods:Perhaps, as is the previous example, we could just make obvious simplifications, however we might stop simplifying too soon;instead, we could look for the greatest common factor of the numerator and the denominator, and then divide the numerator and denominator of thefraction by this number.

Example: How do we find the simplest form of the fraction ?The factors of 24 are: 1; 2; 3; 4; 6; 8; 12; and 24.The factors of 42 are: 1; 2; 3; 6; 7; 14; 21; and 42.1; 2; 3; and 6 are the common factors of 24 and 42. 6 is the greatest.

We can write:

is the simplest form of . (4 and 7 do not have any common factors apart from the number 1.)Note:


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Some calculators have a button (Simp or d/c) that allows you to simplify fractions.For example, the input sequences:3 2 3 / 1 5 3 = Simp3 2 3 ab/c 1 5 3 d/crespectively return 19/9 and 19_|9. (The calculator has simplified by 17.)

III. Applications of fraction calculations

Before launching into a complicated calculation, it might be wise to first simplify any fractions.

Example: Calculate .If we look for a common multiple for 77 and 48, we find (at best): 3,696.

Whereas: and .

Finally: .Subtracting or adding the simplified forms of the fractions is easier.



Adding and Subtracting FractionsWhen we want to add or subtract numbers that are written as fractions, we distinguish between two different cases: one where the fractions have thesame denominator and one where they have different denominators.

I. The denominator is the same

In figure 1, we see that (read three-sevenths) of the circle is colored red and is green.

Without specifying the colors, we can say that of the circle is colored.

This is written .Rule:To add two numbers written as fractions with the same denominator:keep the common denominator;add the numerators.To subtract two numbers written as fractions with the same denominator:keep the common denominator;subtract the numerators.


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In other words, using letters (a, b, and drepresenting signed numbers; d 0): and .Examples:

(which can also be written as )

II. The denominators are different

A. Establishing that two fractions are equal

Rule: The value of a quotient is not changed if we multiply its numerator and its denominator by the same number other than zero.

In other words, using letters (a, c, and krepresenting numbers; c 0 and k 0):

Examples: and specifically:

B. Adding or subtracting

Rule: To add (or subtract) numbers that do not have the same denominator, first replace them with quotients that have the same denominator (andrepresent the same numbers); then apply the rule from Section I.

Examples:

Specific cases:If the denominators are powers of 10 (1, 10, 100 ), these numbers can be written in decimal form before adding or subtractin g them.

If the sum includes numbers written in decimal form, we can convert them into fractions with the denominators being powers of 10.



Taking a Fraction of a NumberWhen we say that a runner missed out on first place by a fraction of a second, we mean that he was several tenths or hundredths of a second behind thwinner. In mathematics, taking a fraction of a number has a very precise meaning. What is it?

I. General method

A. Example

In a group of 120 people, three quarters are wearing pants. To find the number of people wearing pants, all we need to know is how to work out threequarters of 120.


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Three quarters is the same as three times one quarter .If we know how to find a quarter of 120, then we know how to find three-quarters of 120, i.e., by multiplying by 3.

As one quarter of 120 is , finding three-quarters of 120 is the same as calculating .

Since , we find that .Therefore, 90 people in this group are wearing pants.

Note: . From this we can see that .

B. Generalization

We have just seen that three-quarters of 120 is the same as and that . So, we can make sense of the product which means three-quarters of 120.

In more general terms, given a number n, three quarters ofn will be .

We can generalize further by replacing the fraction with any other fraction. So, if we take a fraction written as , where b 0, then ofn is:

.Special case for decimal fractions:

Lets say that a is a natural number. If we want to take a fraction , , or of a number, we simply multiply by a, then move thedecimal point one, two, or three places to the left respectively in the result obtained.

For example, of 12 (thirty-seven hundredths of 12) is 4.44, since 37 12 = 444.

In the same way, of 5.4 (twenty-three tenths of five point four) is 12.42, since 23 5.4 = 124.2.



Checking Whether Two Fractions Are EqualWe want to distribute 36 marbles from one packet equally among four people and 27 marbles from another packet among three people. It is easy to se

that each person will receive 9 marbles in both of these distributions, as and . The two fractions and represent the samenumber, so we say that they are equal. We can see this by working out the divisions 36 4 and 27 3.

I. Using a calculator

If we have a calculator (e.g., the one on a computer), we can see whether two fractions are equal by looking at the quotients obtained.

For example, the fractions and are equal, since102 6 = 17 and 119 7 = 17.

II. Without using a calculator

In this case, recognizing two equal fractions assumes that we know how to establish that a fraction is equal to a given fraction.To do this, we use the following rule: To establish that a fraction is equal to a given fraction, simply multiply or divide its numerator and itsdenominator by the same natural number (not zero).


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For example, the fractions and are equal, since:

and .



Solving Speed Problems

I. Calculating an average speed

To calculate the average speed Vof a vehicle, when we know the distance dtraveled and the duration tof the journey, we use the formula . Thformula also allows us to calculate the distance traveled if we know the duration and average speed of the journey (d= Vt) or the duration if we know

the distance traveled and the average speed .When carrying out this sort of calculation, be careful to always use matching units!

A. Example 1

Problem: A boat makes a journey of 28 km in 2 hours 20 minutes. What is its average speed?Solution: The journey time is given in hours and minutes.

We begin by converting this time into hours: 60 minutes is equal to one hour, so 20 minutes is equal to h, and 2 h 20 min is equal to h (because

).

Next we apply the formula (dis in km and tis in h, so Vwill be in km/h).

We replace dand twith their values, and get: .The journey is made at an average speed of 12 km/h.

B. Example 2

Problem: While cycling in the mountains, Martin travels 8 km up a hill, at an average speed of 8 km/h. He descends along the same route at anaverage speed of 32 km/h.So what is his average speed for the round trip?

Solution: From the problem we can quickly work out that the ascent took an hour and the descent took a quarter of an hour ; sothe round trip lasted an hour and a quarter, or 1.25 h (it is easier to use tenths or hundredths of an hour than minutes).The length of the round trip is 16 km (2 8 = 16).

To calculate the average speed of the total journey, we use the formula (dis in km and tis in h, so Vwill be in km/h). We replace dand twith

their values, and get: .The average speed of the round trip is 12.8 km/h.

Important Note: The average speed of the round trip is not the average of the two average speeds (that would be equal to , or 20 km/h).

II. Calculating a distance

Problem: The speed of light is about 300,000 km/second. It takes about 8 minutes for light to get from the Sun to Earth. What is the distance of theplanet Earth from the Sun?


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Solution: We should convert the time of 8 minutes into seconds: 1 minute is equal to 60 seconds, so 8 minutes is equal to 480 seconds (because 8 min 60 sec/min = 480 sec);use the formula d= Vt(Vis in km/s and tis in s, so dwill be in km). We replace Vand twith their values, and get: d= 300,000 km/sec 480 sec= 144,000,000 km.The distance of Earth from the Sun is about 144 million kilometers.

III. Calculating a time

Problem: Emily is going for a 15 km walk. She wants to get back at 4:30 PM and plans to walk at an average speed of 4 km/h. At what time should shset out?

Solution: First we can work out the journey time using the formula (dis in km and Vis in km/h, so twill be in h); we replace Vand dwith

their values, and get: , so the walk will last three and three-quarter hours.Now we can calculate Emilys departure time. For ease of subtraction, we will convert the clock time 4:30 PM into the following notation: 16 hours 30

minutes. Since h is equal to 45 min, we must carry out the following subtraction: 16 h 30 min3 h 45 min, which is the same as the 15 h 90 min3 h 45 min. We get the result 12 h 45 min.

Emily must leave at 12:45 PM (or a quarter to one) to get back at 4:30 PM.


1993-2008 Microsoft Corporation. All rights reserved.Using Compound UnitsIf a motorist covers 720 kilometers in six hours, his average speed is equal to 120 kilometers (km) per hour (h) (720 km 6 h = 120 km/h).The length of the journey (720 km) and the duration (6 h) are simple measures.The average speed of the motorist (120 km/h) is a compound measure; more precisely, it is a quotient measure.How are product measures and quotient measures used?

I. Product measures

Product measures are products of simple measures. The unit of the product measure depends upon the units in which the simple measures areexpressed. We must always take care to use corresponding units, as shown in the following examples.

A. Areas and volumes

Areas and volumes are product measures: An area is the product of two lengths (expressed in the same unit), and a volume is the product of threelengths (expressed in the same unit).

Example: We want to calculate the altitude of a pyramid where the volume is equal to 500 centimeters3 (cm3) and the area of the base is 2 decimeters(dm).

The volume of a pyramid is given by the formula , whereB and h respectively denote the area of the base and the altitude of the pyramid,expressed in corresponding units. We can convert the area B into cm, so that 2 dm = 200 cm.

We have: , hence .The altitude of the pyramid is therefore equal to 7.5 cm.

B. Electrical energy

The power of an electrical appliance is expressed in watts (W) or in kilowatts (kW); we have: 1 kW = 1,000 W.If we use an electrical appliance whose power is equal to P for a length of time t, the formula:E= P tallows us to calculate the energyEconsumedby this appliance. Energy is therefore measured by a product measure. The unit used to measureEdepends on the units chosen to express P and t.The most common unit of energy is the kilowatt-hour (symbol: kWh): electric meters in homes use this unit to measure the energy consumed. So that

Eis expressed in kilowatt-hours, we must express P in kilowatts and tin hours (h).Example: You use a steam iron with a power of 1,500 W for two hours. We want to compare the energy consumed by this with that consumed byseven 75 W light bulbs lit for five hours. These energies will be expressed in kilowatt-hours.The power of the steam iron is equal to 1,500 W, which is 1.5 kW. The energy consumed by the steam iron is therefore:E1 = 1.5 kW 2 h, which is3 kWh.


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The power of a light bulb is equal to 75 W, which is 0.075 kW. The energy consumed by the seven light bulbs is therefore:E2 = 7 bulbs (0.075 kW 5 h), which is 2.625 kWh.Using the iron for two hours therefore consumes more energy than leaving the seven light bulbs lit for five hours.

II. Quotient measures

Quotient measures are quotients of simple measures.The unit of the quotient measure depends on the units in which the simple measures are expressed. We must always take care to use corresponding

units, as shown in the following examples.

A. Speed

Take the formula: , where the speed v, the distance traveled d, and the duration of the journey tare expressed in corresponding units.Example: We want to calculate the average speed of a sprinter who runs 100 meters (m) in 10 seconds (s); the speed will be expressed in m/s, then inkilometers per hour (km/h).d= 100 m and t= 10 s; therefore the average speed of the sprinter in m/s is equal to 100 m 10 s, which gives 10 m/s.The speed expressed in km/h represents the distance that would be covered by the sprinter in one hour: we have just seen that in one second he covers10 m, therefore, in one hour, he covers 3,600 s/h 10 m/s (because 1 h = 3,600 s), which is 36,000 m. However, 36,000 m = 36 km.The average speed of the sprinter is therefore equal to 36 km/h.

B. Flow

If you turn on a tap and let the water run, the flow dof the tap is given by the formula: , where v denotes the volume of water and tthe timeelapsed; d, v, and tmust be expressed in corresponding units: If the volume is in liters (L) and the time in minutes (min), then the flow will beexpressed in L/min.

Example: We want to calculate the time, in hours (h) and minutes, necessary to fill a 300-liter bathtub where the flow of the bathtubs tap is 4 L/min.

Using the formula , we have: .

We therefore obtain the time tin minutes using the calculation , so t= 75 min.75 min = 1 h 15 min, therefore the bathtub is filled in 1 h 15 min.

Copyright 2006 Ruedesecoles, translated and reprinted by permission. Translation Copyright of Microsoft Corporation. All rights reserved. 1993-2008 Microsoft Corporation. All rights reserved.

Solving Percentage ProblemsYouve probably heard phrases such as 20% off everything in the store, GDP rose by 3.7% in one year, Inflation this month was 0.3%, andThe birth rate is 16%(read: 16 per thousand). We can see that percentages are often used in everyday life.How do we calculate and interpret percentages?

I. Calculating and applying percentages

A. An example

Problem: The population of the county of Essex rose from 1,078,145 to 1,192,932 between surveys carried out in 1990 and 1999.1. What was the percentage increase between those two dates?

2. If the growth continues at the same rate, what can we estimate that the population of the county will be in 2008?Solution:1. To calculate the percentage increase, we think of the increase in the number of inhabitants as proportional to the initial population (in 1990). Weimagine an initial population of 100 and try to find out its increase n.

Using cross products, we can write: 1,078,145 n = 100 114,787.


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n 10.6Between the two surveys, the population of Essex increased by approximately 10.6%.2. From 1999 to 2008 spans the same amount of time (nine years) as between 1990 and 1999. If the rate of increase stays the same, the population witherefore increase by the same percentage, 10.6%, between these two dates.

So the population will increase by about 126,450 inhabitants ( ).We can estimate that the population of Essex in 2008 will be between 1,319,000 and 1,320,000 inhabitants (1,192,932 + 126,450 = 1,319,382).

Note:When we make forecasts, we should not be too precise. For example, in the problem we have just looked at, we should not give the answer 1,319,382inhabitants, or even 1,319,400; the answer should be to the nearest 1,000.

We could have multiplied 1,192,932 directly by 1.106 ( ), because

B. Rules

To increase an amount by t%, we can just multiply by: .

To decrease an amount by t%, we can just multiply by: .

Example: If prices increase by 200%, that means prices have tripled; the new price Q is equal to the old price P multiplied by , so

.Note: Unlike a rate of increase, a rate of reduction will be less than 100%. If prices dropped by 100%, everything would be free!

II. Other types of percentage calculations

A. Calculating a percentage of a percentage

Problem:On a container of cream cheese, the label says: 45% fat and also 82% water content. So what is the percentage of fat c ontent in the

cream cheese?Solution: The problem might seem not to make sense at first: How can a cheese contain more than 127% (45% + 82% = 127%) of ingredients? In facthe fat content is calculated on the dry product, that is, what would be left if all the water were taken out.Imagine 100 g of the product: There will be 18 g of dry product (because 100 g82 g = 18 g). It is from this 18 g that we calculate the amount of fatcontent.So there is 8.1 g of fat content (because 18 0.45 = 8.1) for 100 g of cream cheese.The fat content represents 8.1% of the weight of cream cheese.

B. Calculating a merged percentage

Problem: In 1998 car sales in France included 225,200 Citrons and 322,340 Peugeots. In 1999, the sales of Citrons increased by about 4.5% andPeugeot sales increased by about 21%. What was the percentage of increase in the sales of the Peugeot-Citron group?Solution: We want to find the sales of the Peugeot-Citron group between 1998 and 1999.In 1998, the group sold 547,540 cars (225,200 + 322,340 = 547,540).In 1999, Citron sold 235,334 cars (225,200 1.045 = 235,334) and Peugeot sold 390,031 cars (322,340 1.21 = 390,031), which gives a total of625,365 cars for the group (235,334 + 390,031 = 625,365).The total increase in sales was 77,825 cars (625,365547,540 = 77,825).

Call the percentage we are looking forx. We have: , so , which givesx 14.2.The sales of the Peugeot-Citron group increased in 1999 by approximately 14.2%.

Note: We can see that there is not a simple relation between the percentages in the problem (4.5% and 21%) and the percentage result (14.2%); note

that: .


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Dividing Positive and Negative Integers

In a physics book, we are asked to calculate the resistanceR using the formula , whereR1 andR2 also denote resistances. We are

givenR1 = 3 andR2 = 2 .

We carry out the following calculation: .

Using the (or ) button on the calculator, we can findR = 1.2 (we input the sequence: 5 6 = ).What does this button do, and how do we find the result without using a calculator?

I. The reciprocal of an integer

A. Definition

Two numbers are reciprocals of one another if their product is equal to 1.Examples:2 and0.5 are reciprocals. -2 (0.5) = 1

is the reciprocal of .

7 has as its reciprocal.Note:All non-zero numbers have a reciprocal;

x is a number that is not 0. We write its reciprocal as (read one overx) orx1(read as the reciprocal ofx or xto the power of minus 1);

calculators often have a button ( or ) that will give the result.

B. Properties

a and b are two non-zero integers; the reciprocal of is .A non-zero number has a reciprocal with the same sign.The reciprocal of the opposite number is the opposite of the reciprocal.

Example: (the reciprocal of -3 is the opposite of the reciprocal of 3).

II. Carrying out a division of two integers

A. Definition

x is an integer andy is a non-zero integer; dividingx byy is the same as multiplyingx by the reciprocal ofy. In other words:

B. First examples

With numbers:

With algebraic notation:


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a and b are two numbers (and b 0 ), we have:

a, b, c, and dare numbers (and b, c, and dare not zero), we have:Note: We can tell in advance what the sign of the quotient ofa b will be.Ifa and b have the same sign, then the quotient will be positive.Ifa and b have different signs, then the quotient will be negative

C. Complex fractions

In the following examples, we will be calculating quotients of quotients, or complex fractions.

Example 1:

Example 2:

Example 3:

Note: The placement of the = sign determines the significance ofA,B, and C.Generalization: a is an integer; b, c, and dare non-zero integers.We have the following equalities:



Multiplying Positive and Negative NumbersWe already know how to multiply positive numbers. But how does it work for negative numbers?

I. Multiplying two negative numbers

A. General rules

The product of two numbers is a number such that:the absolute value of the result is the product of the absolute values of the two numbers;if the two numbers have the same sign, then their product is positive;if the two numbers have different signs, then their product is negative;if one or both of the two numbers is zero, then their product is zero.


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Examples:(7) (3) = 21

3.2 6.5 =20.8

, because the two numbers have the same sign.

B. With a calculator

First we should note that there are two buttons on the calculator keypad that should not be confused:

the button, which is used for subtraction;

the (or ) button, which allows us to enter negative numbers or to turn a number into its opposite.Examples:

if we input the sequence: 7 4 , 3 will be returned on the screen;

if we input the sequence: 5 , -5 will be returned;

if we input the sequence: 7 , first -7 and then 7 will be returned;

if we input the sequence: 5 3 ,15 will be returned;

if we input the sequence: 5 3 , 15 will be returned.The results returned by the calculator verify the rules set out in the previous paragraph.

II. Multiplying several positive and negative numbers

A. Property of multiplication of positive and negative numbers

We can note that:(4) (2.7) = 10.8 and (2.7) (4) = 10.8((3) 5) (2) = (15) (2) = 30 and (3) (5 (2)) = (3) (10) = 30In a series of multiplications of positive and negative numbers, we can rearrange the factors as we wish.

Example: calculateA = (1.25) 6.28 8.A = (1.25) 6.28 8 = (1.25) 8 6.25 = (10) 6.28 =62.8

This is easier than first calculating (

1.25) 6.28.

B. The sign of a product of several factors

The sign of the product of several factors depends only on the number of negative factors:if there is an even number of negative factors, then the product is positive;if there is an odd number of negative factors then the product is negative.

Examples:(3) (5) 7 (2) is negative, because there are three negative factors and three is odd.(1) (2) (3) (4) (5) (6) 7 is positive, because there are six negative factors and six is even.(1)2001 = -1, because there are 2,001 negative factors, all equal to -1, and 2001 is odd.(1)2000 = 1, because there are 2,000 negative factors, all equal to -1, and 2000 is even.

III. Multiplying positive and negative fractions

A. The sign of a quotient

The quotient of two non-zero numbers with the same sign is positive.The quotient of two non-zero numbers with different signs is negative.

Examples:

and are positive. In general, we would replace with .


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and are negative.

Note: we have:

B. Multiplication of quotients

To multiply two fractions, we apply the rule above then we multiply the numerators together and the denominators together; and remember tosimplify, if possible, before carrying out the calculation.

Example:



Using ExponentsThe area of a square with sides of length a is a (read asquared, or ato the power of 2).In the same way, the volume of a cube with edges of length a is a3(read acubed or ato the power of 3).But what is the meaning of similar notations such as 75, (2)7, or 53?

I. Definition

a is a non-zero number and n is a positive integer.

Ifn 2, then an is the product ofn factors all equal to a: .Ifn = 1, then a1 = a.

Also, .

Finally, ifa 0 we say that a0 = 1. Therefore we consider 00 to be undefined.Vocabulary:an and an are called the powers ofa.n (orn) is called the exponent.

For an, we read a to the power ofn. 107 is read as 10 to the power of 7, but is read as 3 over 7, all to the power of 5.Examples:

Notes:The number 210 (which is equal to 1,024) is often used in data processing. We often take 1,000 as an approximation for this, for example when we talabout kilobytes (with a capital K to show that this is different from the kilo- that is used as a prefix to indicate a multiplication by 1,000);

to calculate, using a calculator, the power of a number, we can use the (or or or ) key. So, to calculate 2.34, we input the

sequence: 2 . 3 4 , which gives us: 27.9841.

II. Properties

a and b are non-zero numbers, and n andp are integers; we have:


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an ap = an + p; ;

an bn = (ab)n ; ;

(an)p

= an p

.Examples:34 37 = 34 + 7 = 311 = 177,147

26 56 = (2 5) 6 = 106 = 1,000,000

III. Applications

A. Writing a number in exponent form

Lets consider the numberA equal to 3 3 3 3 3 3.

It has 6 factors all equal to 3, thereforeA = 3

6

.

In the same way, if , then .Finally, ifC= (3) (3) (3) (3) 0.7 0.7 0.7 0.7, then:C= (3)4 0.74C= ((3) 0.7)4C= (2.1)4

Note: In the notation for C, there are four negative factors and so the result is positive. Therefore we also have: C= 2.14.This note can be generalized. Ifa is a negative number and n is a positive integer:ifn is even then an is positive;ifn is odd then an is negative.

B. Using powers of 10

This is a consequence of the definition of the power of a non-zero number.n is a positive integer.10n is written as: 1 followed by n zeros.10n is written as: 0.1 with n zeros in total (of which there are n1 zeros after the decimal point).

Examples:103 = 1,000 (three zeros); 106 = 1,000,000 (six zeros).102 = 0.01 (two zeros); 106 = 0.000001 (six zeros).Of course, 100 = 1.This allows us to write decimal numbers in scientific notation.

Examples:1,500,000 = 1.5 1,000,000 = 1.5 1060.0000000547 = 5.47 0.00000001 = 5.47 108



Calculating and Using Map ScalesA house is 12 meters (m) long; it is represented on an architectural plan by a rectangle 48 centimeters (cm) long. What is the scale of the plan?

While walking, Peter sees that the distance that he still has to cover is represented by 5 cm on a map with a scale of . What real distance doethis represent?What calculation methods can we use to answer these questions?

I. Applying a scale


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Example: Bernard wants to make a plan of his bedroom; it is rectangular and is 5 m long and 2.50 m wide.He decides to divide the real dimensions by 20:5 m = 500 cm and 500 cm 20 = 25 cm; 2.5 m = 250 cm and 250 20 = 12.5 cm.So he draws a rectangle with a length of 25 cm and a width of 12.5 cm.

This rectangle is a plan of his room to the scale of .

Note: The dimensions of the plan are the real dimensions multiplied by the scale factor of ; in fact: and

;

the dimensions of the plan are proportional to the real dimensions; the scale factor is .Definition: On a map (or a plan), the dimensions are equal to the real dimensions multiplied by the same number e. The number e is called the mapscale.IfD is a real distance that is represented on the map by a distance d, then

D e = d(the distances must be expressed in the same unit).

II. Calculating a scale

Example 1: What is the scale e of the architectural plan mentioned in the introduction (12 meters represented by 48 centimeters)?So:D = 12 m = 1,200 cm and d= 48 cm.

So: 1,200 e = 48, or (simplifying by 48).

The scale of the plan is equal to .

Note: ; we can also say that the scale factor is equal to 0.04, but it is usual, where possible, to write the scale as a fraction with a numeratoof 1 when the scale is less than 1.

Example 2: On a road map, a straight road 1 kilometer (km) long is represented by 1 cm. What is the scale of this map?So:D = 1 km = 100,000 cm and d= 1 cm. We can call the scale of the map e.

So: 100,000 e = 1, or .

The scale of the map is equal to .Example 3: Using a microscope, you photograph a paramecium that is 0.2 millimeters (mm) long. On the photograph, the paramecium is 10 cm long.What is the scale of this photograph?So:D = 10 cm = 100 mm and d= 0.2 mm. We can call the scale of the photograph e.So: 0.2 e = 100, or e = 100 0.2 = 500.The scale of the photograph is equal to 500.

Note: In this example, the photograph is an enlargement; this is because the scale is greater than 1.

III Using a scale

A. Example 1: calculating a real distance

Look again at the second example given in the introduction. What is the distance that Peter must cover (the distance represented by 5 cm on a map

with a scale of ?

We apply the formulaD e = d, with and d= 5 cm.

So: , orD = 5 25,000 = 125,000.Therefore D = 125,000 cm = 1.25 km.Peter must cover 1.25 km.

B. Example 2: calculating a reduced length


30/30

On the same map, how is a path 750 m long represented?

We apply the formulaD e = d, with and D = 750 m.

So: , therefore d = 0.03 m = 3 cm.

On the map with a scale of , a path 750 m long is represented by 3 cm.

Copyright 2006 Ruedesecoles, translated and reprinted by permission. Translation Copyright of Microsoft Corporation. All ri ghts reserved. 1993-2008 Microsoft Corporation. All rights reserved.

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