chap

ter 1

Real Numbers

 How to define the real numbers

 About factors, multiples and prime factors

 How to write a whole number as a product of prime factors

 The properties of integers

 The properties of rational numbers

 About rounding and significant figures

 How to geometrically construct √__

2 and √__

3

 About orders of magnitude and scientific notation

Learning Outcomes

In this chapter you will learn:

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–8 –7 –6 –5 –4 –3 –2 –1 876543210

This is not a very rigorous definition of the real numbers. However, it will serve our purposes. The discovery of a proper rigorous definition of the real numbers was one of the most important developments in the mathematics of the nineteenth century. The main contributors to the field were a French mathematician, Augustin-Louis Cauchy (1789–1857), and two German mathematicians, Richard Dedekind (1831–1916) and Karl Weierstrass (1815–1897).

Augustin-Louis Cauchy (1789–1857)

Richard Dedekind (1831–1916)

Karl Weierstrass (1815–1897)

The natural numbers, the integers, the rational numbers and the irrational numbers are all subsets of the real number system.

1.1 FactoRs, multiples aNd pRime FactoRs

Natural NumbersThe natural numbers are the ordinary counting numbers. The set of natural numbers is an infinite set. This means that the set is never-ending. The letter N is used to label the set of natural numbers.

Factors

YOU SHOULD REMEMBER.. .

�

KEY W ORDS

�

�

�

�

�

�

�

�

�

�

�

RealsRationals

Naturals

Integers

Irrationals

For example, all the factors of 24 are {1, 2, 3, 4, 6, 8, 12, 24}.

As you can see, 1 is a factor of 24 and 24 is a factor of 24.

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smultiples

multiple

The multiples of 6 are {6, 12, 18, 24, 30, 36 ...}.

As you can see, the set of multiples is an infinite set, i.e. it goes on forever.

prime Numbers

Prime numbers

� 7 is a prime number as it has two factors only: 1 and 7.

� 2 is the only even prime number. Its two factors are 1 and 2.

� 11 is the first two-digit prime. Its two factors are 1 and 11.

� 1 is not a prime as it has one factor only, itself.

� 0 is not a prime as it has an infinite number of factors, and it is not a natural number.

Euclid

the Fundamental theorem of arithmetic

The Fundamental Theorem of Arithmetic is an important result that shows that the primes are the building blocks of the natural numbers.

composite numbers

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Worked example 1.2RFind (i) the HCF and (ii) the LCM of 512 and 280.

SolutionExpress both numbers as a product of primes.

2 512

2 256

2 128

2 64

2 32

2 16

2 8

2 4

2 2

1

2 280

2 140

2 70

5 35

7 7

1

512 = 2 9 and 280 = 2 3 × 5 × 7

(i) HCF (512, 280) = 2 3 = 8

(ii) LCM (512, 280) = 2 9 × 5 × 7 = 17,920

The highest common factor of 12 and 20 is 4, as 4 is the largest natural number that divides evenly into both 12 and 20.

The lowest common multiple of 3 and 4 is 12, as 12 is the smallest number that both 3 and 4 divide evenly into. ACTIVITIes 1.1, 1.2

In Activities 1.1 and 1.2 you discovered how to find the HCF and LCM of two natural numbers, by writing each number as a product of primes.

Worked example 1.1RExpress 240 as a product of prime numbers.

Solution

2 240

2 120

2 60

2 30

3 15

5 5

1

Start with the lowest prime that is a factor.

240 = 2 × 2 × 2 × 2 × 3 × 5

∴ 240 = 2 4 × 3 × 5

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Worked example 1.4RPeriodical cicadas are insects with very long larval periods and brief adult lives. For each species of periodical cicada with larval period of 17 years, there is a similar species with larval period of 13 years. If both the 17-year and 13-year species emerged in a particular location in 2011, when will they next both emerge in that location?

SolutionHere we are looking for the LCM of 17 and 13. As both numbers are prime, the LCM of 13 and 17 is 17 × 13 = 221.

Therefore, both species will next appear together in that location in the year 2232.

Worked example 1.3RShow that if p is a prime number and p divides evenly into r 2 , then p divides evenly into r.

SolutionLet r = r 1

a 1 r 2 a 2 r 3

a 3 ... r n a n , where each rî is prime and each aî ∈ N.

Therefore, using the rules of indices:

r 2 = ( r 1 a 1 r 2

a 2 r 3 a 3 ... r n a n ) 2

= r 1 2a 1 r 2

2a 2 r 3 2a 3 ... r n 2a n

If p | r2 (p divides r2), then p | one of rî2aî.

Since each rî is prime, ⇒ p | one of rî.

Hence, p divides r 1 a 1 r 2

a 2 r 3 a 3 ... r n a n = r.

Worked example 1.5REvaluate (i) 5! and (ii) 5 × 4!

Solution (i) 5! = 5 × 4 × 3 × 2 × 1

∴ 5! = 120

(ii) 5 × 4! = 5 × (4 × 3 × 2 × 1)

= 120

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exercise 1.1

1. Express the following numbers as a product of prime factors:

(i) 160 (v) 1,155 (viii) 102

(ii) 273 (vi) 1,870 (ix) 1,224

(iii) 128 (vii) 10,500 (x) 38,016

(iv) 368

2. (a) Express each of the following pairs of numbers as the product of prime factors.

(b) Hence, find the LCM and HCF for each set of numbers.

(i) 102 and 170 (vi) 123 and 615

(ii) 117 and 130 (vii) 69 and 123

(iii) 368 and 621 (viii) 20, 30 and 60

(iv) 58 and 174 (ix) 8, 10 and 20

(v) 60 and 765 (x) 294, 252 and 210

3. Let n be a natural number. What is the HCF of:

(i) n and 2n (ii) n and n 2

4. If 3 divides 334,6112, explain why 3 must also divide 334,611.

5. r is an even natural number and s is an odd natural number. The prime factorisations of r and s are:

r = r 1 a 1 r 2

a 2 r 3 a 3 ... r n a n

s = s 1 q 1 s 2

q 2 s 3 q 3 ... s m q m

(i) Explain why one of the primes, r 1 , r 2 , r 3 , ... r n must be 2.

(ii) Explain why none of the primes, s 1 , s 2 , s 3 , ... s m is 2.

6. Let u be any natural number with prime factorisation,

u = u 1 f 1 u 2

f 2 u 3 f 3 ... u n f n

(i) Explain why none of the primes, u 1 , u 2 , u 3 , ... u n divide u + 1.

(ii) Hence, write down the HCF of u and u + 1.

7. Let u be any natural number with prime factorisation:

u = u 1 f 1 u 2

f 2 u 3 f 3 ... u n f n

(i) If u is even, then find the HCF of u and u + 2.

(ii) If u is odd, then what is the HCF of u and u + 2.

8. Kate has two pieces of material. One piece is 72 cm wide and the other piece is 90 cm wide. She wants to cut both pieces into strips of equal width that are as wide as possible.

How wide should she cut the strips?

9. Tom exercises every 14 days and Katie every nine days. Tom and Katie both exercised on March 12. On what date will they both exercise together again?

10. Ms Hoover has 160 crayons and 30 colouring books to give to her students. If each student gets an equal number of crayons and an equal number of colouring books what is the largest number of students she can have in her class?

11. Bart is making a board game with dimensions of 16 cm by 25 cm. He wants to use square tiles. What are the dimensions of the largest tile he can use?

infinitude of primesThere are infinitely many prime numbers. In the next activity you will discover a proof, by contradiction, that this statement is correct.

ACTIVITy 1.3

Proof by contradiction

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running until Friday, The Breakfast Show gives away €100 to every 10 0 th caller who gets through to the show. During the week before a Saturday night concert, the show offers two free tickets to the concert for every 7 0 th caller. How many callers must get through before one wins the tickets and the €100?

13. The Ulam numbers, u n , n ∈ N, are defined as follows:

u 1 = 1, u 2 = 2

and each successive natural number, m, where m > 2, u m is an Ulam number if and only if it can be written uniquely as the sum of two distinct Ulam numbers.

(i) Explain the words in bold.

(ii) Why is 5 not an Ulam number?

(iii) Find the first 10 Ulam numbers.

14. Let n be a natural number. Define the function, F(n) as follows:

F(n) = n __ 2 if n is even.

F(n) = 3n + 1 _______ 2 if n is odd.

(i) Find F(1) and F(2).

(ii) Explain why F(n) is always a natural number.

(iii) Consider the sequence:

n, F(n), F(F(n), F(F(F(n))) ....

Construct the first 15 terms of the sequence for each of the following values of n:

6, 10, 15, 32 and 17.

(iv) Based on your results make a conjecture about the sequence.

(v) Test your conjecture for n = 39.

15. Consider the finite sequence:

10! + 2, 10! + 3, 10! + 4, ... 10! + 9, 10! + 10

(i) How many terms are in the sequence?

(ii) Explain why none of the terms are prime.

(iii) Construct a sequence of 20 consecutive natural numbers none of which are prime.

(iv) Is it possible to have a set of consecutive natural numbers of any given size, that does not contain any prime numbers? Explain.

1.2 iNtegeRs aNd RatioNal NumbeRsThe integers are made up of zero and all the positive and negative whole numbers. Mathematicians use the letter Z to represent the set of integers.

properties of integers � a + b and a × b are integers whenever a and b are integers. (Closure property)

� a + b = b + a and a × b = b × a (Commutative properties)

� (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c) (Associative properties)

� a × (b + c) = (a × b) + (a × c) (Distributive property)

� a + 0 = a and a × 1 = a for all integers a. (Identity elements)

� For every integer a, there exists an integer –a, such that a + (–a) = 0. We say –a is the additive inverse of a. (Additive inverse)

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Rational NumbersThe letter Q is used to represent the set of rational numbers.

� Rational numbers are also called fractions.

� All integers can be written in the form a _ b and hence are rational numbers. For example, 3 = 3 _ 1 .

� The properties listed for the integers also carry over into the rationals.

� Two fractions are equivalent if they have the same value. For example, 1 _ 2 = 3 _ 6 .

� The reciprocal of a fraction is found by turning the fraction upside down (inverting).

For example, the reciprocal of 11 __ 12 is 12 __ 11 .

� Every fraction, with the exception of zero, has a multiplicative inverse. The product of a number

and its multiplicative inverse is always 1. The multiplicative inverse of a fraction is its reciprocal.

For example, the multiplicative inverse of 3 _ 4 is 4 _ 3 .

Worked example 1.6RUse the properties of the integers to prove that for any integer a × 0 = 0.

Solution 0 + 0 = 0 (Identity)

⇒ a × (0 + 0) = a × 0 (Multiplying both sides by a)

⇒ a × 0 + a × 0 = a × 0 (Distributive property)

a × 0 + a × 0 + (–a × 0) = a × 0 + (–a × 0) ( Adding –a × 0 to both sides)

a × 0 + [a × 0 + (–a × 0)] = a × 0 + (–a × 0) (Associative property)

a × 0 + 0 = 0 (Additive inverse)

∴ a × 0 = 0 (Identity)

Worked example 1.7RUse the properties of the integers to prove that for all integers a and b, (a)(–b) = –ab.

Solution (a)(–b) + (a)(b) = a(–b + b) (Distributive

property)

= a(0) (Additive inverse)

∴ (a)(–b) + (a)(b) = 0

Therefore, (a)(–b) is the additive inverse of (a)(b).

But (a)(b) = ab and its additive inverse is –ab.

∴ (a)(–b) = –ab

Worked example 1.8RUse the properties of the integers to prove that for all integers a and b, (–a)(–b) = ab.

Solution (–a)(–b) + (–a)(b) = –a(–b + b) (Distributive

property)

= –a(0) (Additive inverse)

⇒ (–a)(–b) + (–a)(b) = 0

Therefore, (–a)(–b) is the additive inverse of (–a)(b).

But (–a)(b) = –ab and its additive inverse is ab.

∴ (–a)(–b) = ab

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exercise 1.2

1. Evaluate each of the following:

(i) –8 – 5 + 13 (iii) (–5)(–8)

(ii) (2)(–3) (iv) 3(4 ) 2 + 2(4) – 56

(v) 3(5 – 2)2 – 3(4 – 2)2 + 5(3)3

________________________ (5 – 2)2

2. Evaluate each of the following, leaving your answers in their simplest form:

(i) 14 ___ 15 × 3 __ 8

(vi) 1 3 __ 4 × 1 2 __ 3 + 2 __ 3 × 5 __ 6

________________ 7 1 __ 2 – 4

(ii) 2 1 __ 2 × 5 (vii) 8.4(19.6 – 12.2)2

_______________ (14.4 – 12.2)3

(iii) 1 3 __ 4 × 1 2 __ 3 (viii) 3 __ 5 – 13 ___ 20

______ 5 __ 8

(iv) 1 __ 2 – 1 __ 8 (ix) 3(2.5 – 1.2 ) 2

(v) 2 __ 3 + 5 __ 6

3. Use the properties of the integers to prove that 4 × –5 = –(4 × 5).

4. Use the properties of the integers to prove that –3 × –6 = (3 × 6).

5. Use the properties of the integers to prove that –5 × 8 = –(5 × 8).

6. A box contains oranges and grapes. An equal number of the oranges and grapes are rotten. 2 _ 3 of all the oranges are rotten and 3 _ 4 of all the grapes are rotten. What fraction of the total number of pieces of fruit in the box are rotten?

7. Alice and Bob share an allotment. The ratio of the area of Alice’s portion to the area of Bob’s portion is 3 : 2. They each grow vegetables and fruit on the allotment.The entire allotment is covered by vegetables and fruit in the ratio 7 : 3. On Alice’s portion of the allotment, the ratio of vegetables to fruit is 4 : 1. What is the ratio of vegetables to fruit in Bob’s portion?

8. Three cans of juice fill 2 _ 3 of a 1 litre jug. How many cans of juice are are needed to completely fill eight 1 litre jugs?

9. A rectangle has a length of 3 _ 5 units and an area of 1 _ 3 units2. What is the width of the rectangle?

10. In the diagram, the number line is marked at consecutive integers, but the numbers themselves are not shown. The four large red dots represent two numbers that are multiples of 3 and two numbers that are multiples of 5. Which of the black points represents a number that is a multiple of 15? Give an explanation for your choice.

P J Q G R C DA B

+ + + + + + + + + + + + + + +

Worked example 1.9R (i) Find the sum of the squares of the four consecutive integers, –2, –1, 0 and 1.

(ii) Show that the sum of the squares of any four consecutive integers is always an even number.

Solution (i) (–2 ) 2 + (–1 ) 2 + (0 ) 2 + (1 ) 2 = 4 + 1 + 0 + 1

= 6

(ii) n, n + 1, n + 2, and n + 3, n ∈ Z, are four consecutive integers.

∴ (n ) 2 +(n + 1 ) 2 +(n + 2 ) 2 +(n + 3 ) 2 = n 2 + n 2 + 2n +1 + n 2 + 4n + 4 + n 2 + 6n + 9

= 4 n 2 + 12n + 14

= 2(2 n 2 + 6n + 7) [an even number as 2 is a factor]

∴ The sum of any four consecutive integers is an even number.

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11. The integers from 1 to 9 are listed on a whiteboard. If an additional m 8s and n 9s are added to the list, then the average of all the numbers is 7.3. Find m + n.

12. Five square tiles are shown. Each tile has a side of integer length. The side lengths can be arranged as consecutive integers. The sum of the areas of the five squares is 1,815.

13. Seán has a pile of tiles, each measuring 1 cm by 1 cm. He tries to put these small tiles together to form a larger square of length n cm, but finds that he has 92 tiles left over. If he had increased the side length to (n + 2) cm, then he would have been 100 tiles short. How many tiles does Seán have?

14. A palindromic number is a positive integer that is the same when read forwards or backwards. For example, 31213 and 1237321 are palindromic numbers.

(i) Find the total number of three digit palindromic numbers.

(ii) Determine the total number of palindromic numbers between 1 0 6 and 1 0 7 .

(iii) If the palindromic numbers in part (ii) are written in order, find the 2,12 5 th number on the list.

(i) Show that the sum of the squares of five consecutive integers is divisible by 5.

(ii) Find the dimensions of the largest square.

1.3 iRRatioNal NumbeRsIn the right-angled triangle shown, the value for x can be found using the theorem of Pythagoras. Here is the solution:

x 2 = 1 2 + 1 2

x 2 = 1 + 1

x

1

1

x 2 = 2

x = √__

2 (as x > 0)

Can √__

2 be written as a fraction? This problem preoccupied the ancient Greek mathematicians for many years. Around 500 bc, Hippasus, a follower of Pythagoras, proved that √

__ 2 could not be written as a fraction. Pythagoras,

who did not believe in the existence of irrational numbers, was so enraged by this proof that he had Hippasus thrown overboard from a ship and Hippasus subsequently drowned. Numbers that cannot be written as fractions are called irrational numbers. √

__ 2 was the first known irrational number.

Hippasus, a follower of Pythagoras

While √__

2 cannot be written as a fraction, it is possible to find an approximation for √

__ 2 . A calculator gives the

approximation √__

2 = 1.414213562, but this decimal goes on forever with no pattern or repetition.

Real Number System

1

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sproof that √

__ 3 is irrational

To prove: √__

3 is irrational.

The proof of this result is another example of proof by contradiction.

Proof: Assume that √__

3 is rational and can therefore be written in the form a _ b , a, b ∈ Z, b ≠ 0.

Also, assume that the fraction a _ b is written in simplest terms, i.e. HCF(a, b) = 1.

√__

3 = a _ b ,

⇒ 3 = a 2 __ b 2

(squaring both sides)

∴ a 2 = 3 b 2 (*)

As b 2 is an integer, a 2 has to be a multiple of 3, which means that 3 divides a 2 .

If 3 divides a 2 , then 3 divides a. (Worked Example 1.3)

∴ a = 3k, for some integer k. Substituting 3k for a in (*) gives,

(3k ) 2 = 3 b 2

9k2 = 3 b 2

⇒ b 2 = 3 k 2

As k 2 is an integer, b 2 has to be a multiple of 3, which means that 3 divides b 2 .

Therefore, 3 divides b. If 3 divides a and 3 divides b, then this contradicts the assumption that HCF(a, b) = 1. This completes the proof.

constructing √__

2 and √__

3 √

__ 2 and √

__ 3 cannot be written as fractions, but can be constructed.

Construct √__

2

1. Let the line segment AB be of length 1 unit.

BA 2. Construct a line m perpendicular to [AB] at B.

B

m

A

3. Construct a circle with centre B and radius [AB].

B

m

A

4. Mark the intersection, C, of the circle and m.

B

Cm

A

5. Draw the line segment CA. |AC| = √

__ 2

B

Cm

A

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Proof: |AB| = |BC| = 1 (radii of circle)

|AB | 2 + |BC | 2 = |AC | 2 (Theorem of Pythagoras)

1 2 + 1 2 = |AC | 2

|AC | 2 = 2

∴ |AC| = √__

2

Construct √__

3

1. Let the line segment AB be of length 1 unit.

BA

2. Construct a circle with centre A and radius length |AB|.

BA

3. Construct a circle with centre B and radius length |AB|.

BA

4. Mark the intersection of the two circles as C and D.

B

C

D

A

5. Draw the line segment [CD]. |CD| = √

__ 3

BE

C

D

A

Proof: CD is the perpendicular bisector of [AB] (Construction).

∴ |AE| = |EB| = 1 __ 2

|AC| = |BC| = 1 (Construction)

|AE | 2 + |EC | 2 = |AC | 2 (Theorem of Pythagoras)

( 1 __ 2 ) 2 + |EC | 2 = 1 2

|EC | 2 = 1 – 1 __ 4 = 3 __ 4

∴ |EC| = √__

3 __ 4 = √__

3 ___ 2

|CD| = 2 |EC|

= 2 ( √__

3 ___ 2 ) ⇒ |CD| = √

__ 3

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sexercise 1.3

1. Copy the diagram and use it to show √__

5 on the numberline.

–0.5

–0.5

0.5

1

190º 90º

1

1

1

1.5

2

0.5 1 1.5 2 2.50

0

–1–1.5

90º

90º

2. Using a diagram similar to that used in Question 1, show √__

7 on the numberline.

3. Prove by contradiction that √__

2 is irrational.

4. p, the ratio of the circumference of a circle to its diameter, is also an irrational number. The British mathematician John Wallis (1616–1703) discovered the following formula for approximating p:

p __ 2 = 22 _____ (1)(3) × 42

_____ (3)(5)

× 62 _____

(5)(7) × ... × (2n)2 ______________

(2n – 1)(2n + 1)

(i) Use the first seven terms of the product to approximate p to four decimal places.

(ii) Use your calculator to approximate p to four decimal places.

(iii) Assuming that your calculator has given the true approximation to four decimal places, calculate the percentage error in using seven terms of Wallis’ formula (answer to two decimal places).

5. Show that x = √________

3 + 2 √__

2 – √_______

3 – 2 √__

2 is a rational number. ( Hint: find x 2 . )

6. Two quantities are in the golden ratio, if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. In the rectangle below a > b and

a + b _____ a = a __ b ; therefore, the ratio a : b is the golden ratio.

a + b

a

a b This ratio is a constant and also irrational. It can be shown that the ratio is the positive solution to the equation:

r = 1 + 1 __ r

(i) Solve the equation to find the golden ratio.

(ii) Use your calculator to approximate the ratio to four decimal places.

1.4 RouNdiNg aNd sigNiFicaNt FiguRes

Rounding to decimal placesIn geometry, the number of times the diameter of a circle divides into the circumference is called p. We normally substitute 3.14 for p in these calculations. However, 3.14 is just an approximation:

� There are infinitely many decimal places in p.

� p is 3.141592654 to nine decimal places.

� For simplicity, we often write p as 3.14, i.e. to two decimal places.

� Engineers use 3.1416 as an approximation for p.

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(i) 2.57 (ii) 39.32

Solution (i) 2.57

When rounding to one decimal place, we look at the second number after the decimal point. If this number is 5 or greater, we round up to 2.6. Otherwise, the corrected answer is 2.5. As 7 is the second number after the decimal point, we round up to 2.6.

Answer = 2.6

(ii) 39.32

Here, the second number after the decimal point is 2, which is less than 5. Therefore, the number rounded to one decimal place is 39.3.

Answer = 39.3

R Worked example 1.10

significant Figures

Correct the following numbers to two significant figures:

(i) 3.67765 (ii) 61,343 (iii) 0.00356

Solution (i) 3.67765

The first significant figure in a number is the first non-zero digit in the number. In this number, 3 is the first significant figure in the number. We need to correct to two significant figures, so we look at the third significant digit. If this number is 5 or greater, we round up the second digit. The third digit is 7, so the corrected number is 3.7.

R Worked example 1.11

3.6|7765

1st sig. figure 2nd sig. figure

Answer = 3.7

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Worked example 1.12RWrite the following numbers in scientific notation:

(i) 725,000,000,000 (iii) 0.0000056

(ii) 980,000 (iv) 0.000000034

Solution (i) First, note that dividing a number N by 10n, where n ∈ N, moves the decimal point n places to

the left.

For example, 144.25 _______ 102 = 1.4425 (Decimal point moves two places to the left)

725,000,000,000 = 725,000,000,000

_______________ 1 0 11 × 1 0 11

= 7.25 × 1 0 11

(ii) 61,343

Here, the third digit is 3, which is less than 5. Therefore, the rounded number is 61,000.

61,|343

1st sig. figure 2nd sig. figure

Answer = 61,000 Note that all other digits after the rounded digit change to zero.

(iii) 0.00356

The third significant digit is 6. Therefore, the rounded number is 0.0036.

0.0035|6

1st sig. figure 2nd sig. figure

Answer = 0.0036 �

�

1.5 oRdeRs oF magNitude aNd scieNtiFic NotatioN

When doing calculations, scientists often use very large numbers or very small numbers. For example, the speed of light is about 300,000,000 metres per second.

Very large or very small numbers can be awkward to write down. So, scientists use scientific notation to write down these numbers.

scientific notation

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Worked example 1.13RBy how many orders of magnitude does 345,632 differ from 567,123,423?

SolutionWrite both numbers in scientific notation:

345,632 = 3.45632 × 105 567,123,423 = 5.67123423 × 108

�

�

≈1×105 ≈10×108

= 100 × 105 = 101 × 108

= 105 = 109

109 ____

105 = 109 – 5 = 104

Therefore, both numbers differ by four orders of magnitude.

orders of magnitudeOrders of magnitude are generally used to make very approximate comparisons. If two numbers differ by one order of magnitude, one is about ten times larger than the other.

order of magnitude

(ii) 980,000 = 980,000

________ 1 0 5

× 1 0 5

= 9.8 × 1 0 5

(iii) Note that dividing a number N by 10n, where n is a negative integer, moves the decimal n places to the right.

For example, 0.00146 ________ 10–3 = 0.00146 × 1 ____

1 0 –3

= 0.00146 × 103 (Rules of indices)

= 1.46 (Decimal point moves three places to the right)

0.0000056 = 0.0000056 __________ 1 0 –6 × 1 0 –6

= 5.6 × 1 0 –6

(iv) 0.000000034 = 0.000000034 ____________ 10–8 × 1 0 –8

= 3.4 × 1 0 –8

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sexercise 1.4

1. (a) Write these numbers correct to three decimal places:

(i) 5.1456 (iv) 62.1235321

(ii) 7.2983 (v) 23.7654

(iii) 17.8943 (vi) 0.07893

(b) Write these numbers correct to two decimal places:

(i) 1.263 (iv) 21.3

(ii) 5.9876 (v) 22

(iii) 21.456 (vi) 0.00391

2. (a) Write these numbers correct to two significant figures:

(i) 0.00985 (vi) 0.000849

(ii) 0.00234 (vii) 0.238

(iii) 0.0125 (viii) 52.00285

(iv) 0.000000785 (ix) 52.487

(v) 1.000034 (x) 967,333

(b) Write these numbers correct to one significant figure:

(i) 32.14 (iv) 1,698

(ii) 3.857 (v) 5,965

(iii) 19,345 (vi) 999

3. Write these numbers in scientific notation:

(i) 34,000,000 (vi) 0.000032

(ii) 0.25 (vii) 5,000,000

(iii) 4,570 (viii) 0.6464

(iv) 0.0001258 (ix) 532,600

(v) 7,206 (x) 5,000

4. Write these as decimal numbers:

(i) 2.65 × 1 0 2 (vi) 4 × 10 –2

(ii) 4.53 × 10 –3 (vii) 2.64 × 1 0 7

(iii) 7.2 × 1 0 6 (viii) 7.612 × 1 0 3

(iv) 1.7 × 10 –5 (ix) 2.76 × 1 0 8

(v) 3 × 10 2 (x) 3.02 × 10 –9

5. Calculate each of the following, giving your answers as decimal numbers:

(i) 3.4 × 103 + 2.8 × 10 3

(ii) 5.2 × 109 + 3.5 × 109

6. The following numbers are written in scientific notation. Rewrite the numbers in ordinary form.

(i) 2 × 1 0 6 (iv) 6.47 × 1 0 5

(ii) 1.69 × 1 0 4 (v) 6.12 × 1 0 1

(iii) 2.48 × 1 0 3 (vi) 9.43 × 105

7. Write the following in the form a × 1 0 n , where 1 ≤ a < 10, a ∈ R, n ∈ Z:

(i) 0.000036 (iv) 0.00063

(ii) 0.0005613 (v) 0.0078

(iii) 0.0345 (vi) 0.0011

8. Write the following numbers in the form a × 1 0 n , where 1 ≤ a < 10:

(i) 0.00068 (iv) 0.0000000097

(ii) 0.0000328 (v) 0.00000056

(iii) 0.0657 (vi) 0.0030307

9. The following numbers are written in scientific notation. Rewrite the numbers in ordinary form.

(i) 1.5 × 10 –3 (iii) 3.5 × 10 –5

(ii) 2.54 × 10 –4 (iv) 6.67 × 10 –6

10. By how many orders of magnitude do the following numbers differ:

(i) 868, 932, 145 and 284

(ii) 453, 987, 312 and 3548

(iii) 767, 894, 567,000 and 23,000,000

(iv) 0.1 and 0.00042

(v) 1.8 and 234

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Revision exercises

1. The German mathematician Christian Goldbach conjectured that every odd positive integer greater than 5 is the sum of three primes. Verify this conjecture for each of the following odd integers:

(i) 11 (ii) 33 (iii) 97 (iv) 199 (v) 17

2. Two bikers are riding in a circular path. The first rider completes a circuit in 12 minutes. The second rider completes a circuit in 18 minutes. They both started at the same place and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?

3. Use the properties of integers to prove the following:

(i) 5 × –3 = –15 (ii) –5 × –3 = 15

4. Show that no integer of the form n 3 + 1 is prime, other than 2 = 1 3 + 1.

5. (i) List the smallest five consecutive integers that are composite.

(ii) Find 10,000 consecutive integers that are composite.

6. Express each of the following numbers as the product of prime factors, and hence, find the LCM and HCF of each pair:

(i) 68 and 102 (iii) 104 and 351

(ii) 69 and 123 (iv) 123 and 615

7. (a) What fraction when added to 1 _ 4 gives 1 _ 3 ?

(b) A mathematician states that her children’s ages are all prime numbers that multiply together to give 7,429. She also says that two of her children are teenagers.

(i) How many children does she have?

(ii) What are their ages?

8. A palindromic number is a number that reads the same forwards and backwards. For example, 52,325 is a palindromic number. All four-digit palindromic numbers have 11 as a prime factor.

(i) Find the prime factorisations of the palindromic numbers 2,332 and 6,776.

(ii) Hence, find the HCF and LCM of 2,332 and 6,776.

9. Write the following numbers correct to two significant places:

(i) 852,233 (iv) 0.000054

(ii) 0.134 (v) 652,494

(iii) 2.00062 (vi) 0.000814

10. Given a line segment of length one unit, show clearly how to construct a line segment of length √

__ 2 units.

11. Given a line segment of length one unit, show clearly how to construct a line segment of length √

__ 3 units.

12. (a) (i) Find the values of the primes p and q, if p 3 × 13 × q = 1,768.

(ii) Find the values of the primes m and n, if 2 4 × m × n = 3,192.

(iii) Using your answers to parts (i) and (ii), find the HCF and LCM of 1,768 and 3,192.

(b) Evaluate the following, giving your answer in scientific notation:

3 3 __ 7 × ( 2 2 __ 5 + 1 1 ___ 10 )

_______________ 3 × 104

13. (i) P represents a pointer on a gauge. What is the decimal value shown by the pointer?

P

0.010

(ii) A computer shop buys a batch of iPod nanos for €99 each and marks the price up by 1 _ 3 . The goods fail to sell so they are included in the next sale where all prices are reduced by 1 _ 4 .

What price would you pay for an iPod nano in the sale?

(iii) John calculates correctly 85 × 142 = 12,070. Using this information, what should his answer be for 12,070 ÷ 850?

14. Prove that √__

5 is irrational.

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s 15. (i) A single-celled protozoa is about one-

tenth of a millimetre in diameter. Write down its diameter in metres, giving your answer in scientific notation.

(ii) A bag contains 350 disks. Seán takes four-fifths of the disks out of the bag and divides them into seven equal groups. How many disks are in each group?

16. A man died, leaving some money to be divided among his children in the following manner:

� €x to the first child plus 1 __ 16 of what remains.

� €2x to the second child plus 1 __ 16 of what then remains.

� €3x to the third child plus 1 __ 16 of what then remains and so on.

When all the money was distributed, each child received the same amount of money and no money was left over.

How many children did the man have?

17. The digits 1, 2, 3, 4 and 5 are each used once to create a five-digit number vwxyz, which satisfies the following conditions:

� The three-digit number vwx is odd.

� The three-digit number wxy is divisible by 5.

� The three-digit number xyz is divisible by 3.

Determine the six five-digit numbers that satisfy all three conditions.

18. A well known series for the irrational p is:

p __ 2 = 1 + 1 __ 3 + 1 × 2 ______ 3 × 5 + 1 × 2 × 3 _________ 3 × 5 × 7 + 1 × 2 × 3 × 4 _____________ 3 × 5 × 7 × 9 + ...

By summing the first 10 terms of the series, find an approximation for p to two decimal places.

ACTIVE MATHS – ACTIVITIES 1

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Real Numbers 1Activity 1.1R

1. If p × q = r, then we say that p is a factor of r and q is a factor of r. For example, 2 3 × 2 7 = 2 10 implies that 2 3 is a factor of 2 10 and 2 7 is a factor of 2 10 .

20 × 25 25 21 × 24 25 22 × 23 25

Using the table above, write down the factors of 2 5 . Write the factors in the form 2 n .

2. Find one prime factor of each of the following numbers:

Number Prime factor12356251

221203

Example: Write 12 as a product of prime factors.

Solution

Step 1 From Note 1 above, 12 has a prime factor. 2 is a prime factor of 12.

Therefore, 12 = 2 × 6.

Step 2 The number 6 has a prime factor 2.

Therefore, 6 = 2 × 3.

Step 3 12 = 2 × 2 × 3 = 2 2 × 3

The work can be summarised more neatly as follows:

Repeated division by primes

2 12

2 6

3 3

1

OR

12 = 2 2 × 3

Factor trees

12

2

2 3

6

12 = 22 × 3

ACTIVE MATHS – ACTIVITIES

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3. Write the following as a product of their prime factors:

24 60 36 27

Activity 1.2R1. The highest common factor (HCF) of two natural numbers is the largest natural number that divides evenly into

both numbers. In this activity you will find the HCF of 36 and 48.

(i) Write 36 and 48 as a product of primes. Write your answers in the form 2 m × 3 n .

36 48

(ii) What is the highest power of 2 that divides into both 36 and 48?

(iii) What is the highest power of 3 that divides into both 36 and 48?

(iv) Using your answers from parts (ii) and (iii), write down the HCF of 36 and 48.

2. The lowest common multiple (LCM) of two natural numbers is the smallest natural number into which both numbers divide evenly.

(i) What is the LCM of 2 9 and 2 3 ? Give your answer in the form 2 n .

(ii) What is the LCM of 5 8 and 5 2 ? Give your answer in the form 5 n .

(iii) Using your results from (i) and (ii), write down the LCM of 2 9 × 5 2 and 2 3 × 5 8 .

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3. Write 50 and 160 as a product of prime factors, and hence, find the LCM of 50 and 160.

50 160

LCM of 50 and 160 =

Activity 1.3RIn this Activity you will prove that there are infinitely many prime numbers.

1. How many 6’s are there in 20? There are three 6’s in 20, with 2 left over. We call ‘3’ the quotient and ‘2’ the remainder. Complete the table.

Quotient Remainder12 ÷ 931 ÷ 5

500 ÷ 5041 ÷ 10

2. Write down the quotient and remainder in each of the following divisions:

Quotient Remainder(2 × 3 + 1) ÷ 3

(2 × 3 × 5 + 1) ÷ 2(2 × 3 × 5 × 7 + 1) ÷ 7

(2 × 3 × 5 × 7 × 11 + 1) ÷ 11

3. In the following table each p n is a prime number. Complete the table.

Quotient Remainder( p1 × p 3 + 1) ÷ p1

( p 2 × p 3 × p 4 + 1) ÷ p 2 ( p1 × p 3 × p 6 × p 7 + 1) ÷ p 6

( p 6 × p 7 × p 8 + 1) ÷ p 7

4. Does p 1 divide evenly into the number N = p 1 × p2 × p 3 + 1, where each of p 1 , p 2 and p 3 are primes? Explain your answer.

ACTIVE MATHS – ACTIVITIES

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5. Complete the following sentence:

Every natural number greater than 1 has a prime .

6. Consider the number N = p 1 × p 2 × p 3 ... × pn + 1. Explain why none of the primes p 1 to p n divide N.

7. In this section we will prove by contradiction that there are infinitely many primes.

Theorem: There are infinitely many prime numbers.

Proof: Assume that there are finitely many primes p 1 , p 2 , p 3 , ..., p n , where pn is the largest prime.

Consider the number N = p 1 × p 2 × p 3 ... × p n + 1.

Either N is prime or it is not prime.

If N is prime, we have just written a prime greater than p, which is a contradiction to our supposition.

If N is not prime, it must be divisible by some prime of p1, p2, p3, ... pn.

None of the numbers p 1 , p 2 , p 3 , ..., p n divide evenly into N. Why?

However, every number has a prime divisor and we have just produced a number that does not have a prime divisor. Therefore, our original assumption that there are finitely many primes must be false.