Further Mathematics, with Calculator – Unit 2

ROUNDING NUMBERS

Calculations, especially when using a calculator, result in a ridiculous (implied) degree of accuracy or precision. For example £70 shared between 42 people gives an impossible amount of £1.666666667. Rounding these figures means expressing them with less precision.

ROUNDING TO A NUMBER OF DECIMAL PLACES

The number of decimal places is the number of digits to the RIGHT of the decimal point.

EXAMPLES

3.1416 is written to 4 decimal places ( 4 d.p )

2.718 is written to 3 decimal places

21.08 is written to 2 decimal places

0.029 is written to 3 decimal places

13 is written to 0 decimal places (13) is a whole number and the decimal point after the digit 3 is not normally shown.

To round a number, draw an imaginary line where you wish the precision to stop. Add one to this last digit if the digit to the RIGHT of the line is 5 or more, otherwise leave it as it is.

EXAMPLES

407.28|67 = 407.29 to 2 d.p.

74.1|256 = 74.1 to 1 d.p.

0.2944|82 = 0.2945 to 4 d.p.

9.245|5 = 9.246 to 3 d.p.

Note that 14.98 to one decimal place is 15.0 as the “8” turns the preceding “9” into a “0” with a carry of “1” which turns the “4” in the units column into a “5” The zero after the decimal point here indicates that the rounded number is accurate to the first decimal place.

Practice Questions 1:

Write the following numbers correct to 1 d.p.

1 (a) 4.29 (b) 49.62 (c) 0.0724 (d) 0.834 (e) 18.996

Write the following numbers correct to 3 d.p.

2 (a) 8.7656 (b) 0.15524 (c) 8.9998 (d) 0.08555 (e) 5.010101

Practice Questions 1 (Answers):

1 (a) 4.3 (b) 49.6 (c) 0.1 (d) 0.8 (e) 19.0

2 (a) 8.766 (b) 0.155 (c) 9.000 (d) 0.086 (e) 5.010

ROUNDING TO A NUMBER OF SIGNIFICANT FIGURES

If we were measuring the width of a window with a steel tape measure, we might find it to be 85.7 cm. However, if we used a linen tape we could not rely on the seven in the first decimal place and would normally quote 86 cm as the width.

We say that 85.7 has 3 SIGNIFICANT FIGURES and the width is 86 cm quoted to 2 significant figures.

Significant figures start with the first NON ZERO on the left of the measurement

The following numbers have been rounded to a number of significant figures.

7.08 has 3 significant figures

12.49 has 4 significant figures

0.049 has 2 significant figures (zeros to the left of 4 are not counted)

28.00 has 4 significant figures (zeros to the right of the decimal point are counted because their presence implies an accuracy to 4 significant figures otherwise they would not be there)

28 has 2 significant figures.

3900 has 2 significant figures (zeros on right give size of number ie thousands)

Practice Questions 2:

State the number of significant figures in the following rounded numbers.

(a) 14.6 (b) 509 (c) 0.004 (d) 498.8

(e) 19.01 (f) 3240 (g) 400 (h) 8

Practice Questions 2 (Answers):

(a) 14.6 (b) 509 (c) 0.004 (d) 498.8

(e) 19.01 (f) 3240 (g) 400 (h) 8

Note that when rounding numbers to a given number of significant figures, the same rule applies as when rounding to a given number of decimal places.

EXAMPLES

68.64 = 69 to 2 significant figures

Counting from the left, “6” and “8” are the first two significant figures but the following digit “6” changes the “8” into a “9” and the resulting figure into “69” (Using the rule : 5 or more add 1 to preceding digit ) This is the same as rounding 68.64 to the nearest unit

9843 = 9800 to 2 significant figures

The first two significant figures are “9” and “8”. The following digit is”4" which leaves the 98 unchanged. However, the zeros must be included to maintain the size of the number ie thousands (98 would not be a good rounded approximation to 9843 !) Note that this is the same as rounding 9843 to the nearest hundred.

0.0588 = 0.059 to 2 significant figures

Starting with the first non-zero digit on the left ie “5” and writing down the two digit numbers we have 0.058. However, the following “8” will change the preceding “8” into a “9” giving the answer 0.059

Practice Questions 3:

Round to the number of significant figures indicated in brackets :

(a) 9.125 (2) (b) 8.008 (3) (c) 47718 (2)

(d) 4.999 (3) (e) 2109 (1) (f) 11.004 (4)

(g) 84.76 (2) (h) 8.45 (2) (i) 94 (1)

(j) 1.355 (3) (k) 0.0984 (2) (l) 0.0078 (1)

Practice Questions 3 (Answers):

(a) 9.1 (b) 8.01 (c) 4800 (d) 5.00

(e) 2000 (f) 11.00 (g) 85 (h) 8.5

(i) 90 (j) 1.36 (k) 0.098 (l) 0.008

USE OF CALCULATOR

Photo of CASIO fx -85 MS scientific calculator.

This unit has been developed using a CASIO fx - 85 MS calculator. It is important to use this one or any equivalent scientific calculator. Refer to the manufacturer’s instructions as necessary. This calculator is solar powered with a back up battery. It also displays the calculation as well as the answer which is a useful feature. Here are a number of examples to illustrate its use in calculations you may meet.

Example (1)

73 + 48

Press ON key - to turn calculator ON then MODE followed by 1 to select calculation mode (you will see a small number 1 top right of display ) Press 7 then 3 keys Press + key Press 4 then 8 keys Press = key Read off result (121) from display

In future these instructions will be abbreviated to:

ON , MODE , 1 , 7 , 3 , + , 4 , 8 , =

where there are commas between each press of a button.

The instruction to turn on should be unnecessary in future !

Note that the calculator turns itself off automatically after a time if not in use, but you can turn it off by pressing SHIFT followed by AC button with OFF above it.

As you would expect, subtraction is similar with - replacing + . A sequence of additions and subtractions can be carried out as follows :

Example (2)

58.35 - 39.2 + 7.1

5 , 8 , . , 3 , 5 , - , 3 , 9 , . , 2 , + , 7 , . , 1 , =

and the result in the display should be 26.25. (You should be able to check it without a calculator as in the introductory unit ?)

Note that the AC button clears the display for the next calculation.

Example (3)

8 , 0 , 5 , x , 9 , 8 , 4 , 1 , = , (product displayed) ÷ , 7 , 2 , =

(product displayed 7922005 and result = 110027.8472 before rounding ) Using brackets this example could be done by :

( , 8 , 0 , 5 , x , 9 , 8 , 4 , 1 , ) , ÷ , 7 , 2 , =

Note that brackets should be used when dealing with negative numbers :

Example (4)

127 x - 81

1 , 2 , 7 , x , ( , - , 8 , 1 , ) , =

Common fraction calculations can be carried out on the calculator using the a button:

Example (5)

The calculator method is :

4 , a , 5 , x , 3 , a , 4 , =

Note that if you multiply these fractions you obtain so the calculator has given the answer in

its simplest form of

Note also that the answer can be converted to a decimal display by pressing

= , a to give answer 0.6

Example (6)

is processed as

4 , a , 3 , a , 4 , ÷ , 1 , a , 1 , a , 2 , =

giving an answer of

(Convert this to the decimal answer by pressing which key?)

Finally some examples of squares, cubes, powers in general, square roots, cube roots and reciprocals

Example (7)

√, ( , 1 , 2 , 3 , x² , + , 9 , 8 , x² , ) , =

and the answer displayed before rounding should be 157.2672884

(What would this be rounded to 2 significant figures ?)

Example (8)

48³

4 , 8 , x³ , =

should display 110592.

To check this find the cube root of 110592 which is displayed and the process is:

SHIFT , x³ , =

Any root can be found using the SHIFT button followed by ^ button because the SHIFT operates the root function written in little gold figures above the ^ button.

Example (9)

5th root of 16807 (that is 168071/5 as you will see when indices are introduced)

5 , SHIFT , ^ , 16807 , =

How might you check your answer using the calculator ? ( ^, 5, = )

Example (10)

Resistors in parallel provide a good example of the use of the reciprocal button

Wh

ere =

If the total resistance in a parallel circuit is given by:

Calculate where = 20Ω and = 40Ω

Using the calculator :

20 , , + , 40 , , =, ( now displayed) , =

The displayed should be 0.075 then pressing the reciprocal button again and = gives the result for of 13.33333....... which you may recognise as (Press a )

The manufacturers instruction booklet will give many more applications of the calculator usually in small print and several languages

Practice Questions 4:

Use the calculator to evaluate the following giving your answers correct to 4 significant figures except for questions 8 - 11 which should be given as fractions:

1. 398.75 - 2.18 + 0.0018

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14. 5 22

15.

16. 9.7³

17.

18.

19.

20.

Practice Questions 4 (Answers):

1. 396.6 2. 18.58 3. 0.8639

4. 5.405 x in standard form as in calculator display, 0.005405 in normal form.

5. 16.34 6. 4.347 7. 5.166

8. 9. 10.

11. 12. 0.1103 13. 0.1379

14. - 0.3814 15. 29.09 16. 912.7

17. 0.001096 or 1.096 x

18. 4.518 19. 2.441 x or 0.0002441

20. 0.8251

INDICES

A convenient (or lazy) way of writing 3 x 3 x 3 x 3 is . (= 81) is read as “3 to the power 4”. 3 is called the base, 4 is called the index and 81 is 3 to the power 4.

In general, if a is a real number and n is an integer, then = a x a x a ........ x a with n factors.

LAW 1 – MULTIPLICATION

Example Simplify y² x y³

y² x y³ = (y x y) x (y x y x y)

= y x y x y x y x y

=

Example Simplify x k²

x k² = (k x k x k x k) x (k x k)

= k x k x k x k x k x k

=

Look at the results of the above two examples.

y² x y³ =

x K² =

NOTE that when MULTIPLYING powers we ADD the INDICES.

LAW 1

x = +

Example

Simplify t² x x

t² x x = t² + + (LAW I)

=

(NOTE that LAW 1 may be applied to more than two terms)

Example

Simplify 4y³ x 9y²

4y³ x 9y² = 36y³ + ² (LAW I)

=

NOTE we MULTIPLY the COEFFICIENTS but ADD the INDICES.

Example:

Simplify 7p x 3p (LAW I) (Remember that p = p1

7p x 3p = 21p + 1

= 21p

Practice Questions 5:

Simplify:

1. a² x a³ 2. p² x p² 3. y x y

4. 5y² x 7y 5. 3a³ x 2a 6. 2f x f³ x 8f²

7. 3b³ x 4b 8. 12m² x 2m 9. 10a³ x 2a² x a

Practice Questions 5 (Answers):

1. a 2. p 3. y

4. 35y 5. 6a 6. 16f

7. 12b 8. 24m³ 9. 20a

LAW 2 - DIVISION Example

Simplify

=

= y

Example

Simplify

=

= m

Look at the results of the above examples.

NOTE that when DIVIDING powers we SUBTRACT the INDICES

LAW II

Example

Simplify

= (LAW II)

=

Example

Simplify

= W (LAW II)

= W ³

Example

Simplify

= 9r³ (LAW II)

NOTE: We DIVIDE the COEFFICIENTS but SUBTRACT the INDICES.

Practice Questions 6:

1. a ÷ a² 2. b ÷ b 3. a² ÷ a

4. 4m ÷ 2m 5. 10m ÷ 5m 6. 2 ÷ 2

Practice Questions 6 (Answer):

1. a 2. b 3. a

4. 2m 5. 2m 6. 2 = 4

LAW 3 - RAISING to POWERS

Example

Simplify (y)

(y) = y x y x y

= y + + (LAW I)

= y

Example

Simplify (k)

(k) = k x k x k x k

= k + + + (LAW I)

= k

Look at the results of the above examples.

(y) = y

(k)= k

NOTE that when a POWER is raised to a POWER we MULTIPLY the INDICES.

LAW III

(a) = a

Example

Simplify (k)

(k) = k

Example

Simplify (4x)

(4x) = x

= 16x

NOTE EVERYTHING inside the brackets must be raised to the power 2

Example

Simplify (2p)

(2p) = 2 p

= 16p

LAW IV

(ab) = ab

Practice Questions 7:

1. (a²)³ 2. (b³) 3. (c)²

4. (2a²)² 5. (3b³)² 6. (2m³)³

Practice Questions 7 (Answer):

1. a 2. b 3. c

4. 4a 5. 9b 6. 8m

Example

Simplify

= [LAW I]

= 3 [LAW II]

Practice Questions 8:

1. 2. 3.

Practice Questions 8 (Answers):

1. 6y 2. 6w² 3. 27d

THE ZERO INDEX

In all the examples to date the indices have all belonged to the set of NATURAL NUMBERS.

y³ = y x y x y

y² = y x y

y¹ = y

What does mean?

Consider x y3 = + 3 (LAW I)

x y3 = y3

Now 1 x y3 =y 3

Hence =1

Similarly x m6 = + 6 (LAW I)

x m6 = m6

But 1 x m6 = m6

Hence =1

Taking a number as base

x = + (LAW I)

x =

But 1 x =

Hence = 1

From three underlined results it is obvious that (ANYTHING) = 1

LAW V = 1

NEGATIVE INDICES

Having included zero we have now discussed the indices belonging to W, the Set of Whole Numbers. Remember this set DOES include 0.

We shall now investigate the NEGATIVE INDICES.

What does really mean?

Consider x

x = + 2 (LAW I)

=

x = 1

Divide both sides by

=

IN GENERAL = This is LAW VI

LAW VI =

Examples

= = =

= (2 =

Practice Questions 9:

Simplify:

1. 2. 3. 4.

5. 6. 7. (3a) 8. (2b)

Practice Questions 9 (Answers):

1. 1 2. 1 3. 4.

5. 6. 7. 8.

NOTE we must NEVER have a negative index in our final answer.

Hence, if we have p-7, this would become

Let us consider the situation where we have

You will note that the negative index is in the denominator.

= 1 x

=

Hence =

And =

NOTE if the negative index is in numerator take it down to the denominator and change the sign of index. If the negative index is in the denominator bring it up to the numerator and change sign of index.

Example

Simplify

= = =

ANOTHER METHOD FOR THE LAST EXAMPLE

=

=

= t²

= t²

= t

I think the first method is better, having less confusion with negative signs. The main point in showing the TWO methods is that there is no particular “order” for using the laws. Many examples, as above, can be tackled in more than one way but as long as you use the Laws correctly you will always arrive at the same answer.

Practice Questions 10:

Simplify:

1. 2. 3.

Practice Questions 10 (Answers):

1. 2. 3.

There is one point where you MUST TAKE CARE. Look at the following example and see if YOU can spot the mistake.

Example

Simplify

=

Have you spotted it?

What is the index of the 4? No, it is NOT -2.

If it was we would have been given (4y)

The index of the 4 is POSITIVE 1.

If we bring down to the denominator, it would become

As the 4 has a positive index it will REMAIN on top, hence

4y =

Example

Simplify x

x = x

=

= (LAW II)

=

=

The above example may require several readings.

Note that we cannot leave our answer as .

Example

Simplify and find the value of this expression when p = 2.

=

=

=

=

When p = 2

= = =

Practice Questions 11:

Simplify:

1. x

2. x 3a

3. x

4. x a

5. x

6. x

Practice Questions 11 (Answers):

1. 2. 3.4. 18a²

5. 6.

STANDARD FORM / SCIENTIFIC NOTATION

VERY LARGE NUMBERS

Television signals travel at about 30 000 000 000 cm/s.

The number of seconds in a century is 3 153 600 000.

Scientists and others have to work with large numbers such as these.

They use a shorthand method of writing these numbers called STANDARD FORM or SCIENTIFIC NOTATION.

Look at this table showing integer (ie whole number) powers of ten.

10 000 = 10x10x10x10 = (read as ten to the power 4)

1000 = 10x10x10 = (read as ten to the power 3)

100 = 10x10 =

10 = 10 =

1 = =

The last result is a special case - any number raised to power zero is 1.

Using this table any large number can be written in shorthand by expressing it as a number between one and ten multiplied by a power of 10.

For example:

5400 = 54 x 100

= 5.4 x 10 x 100

= 5.4 x 1000

= 5.4 x

This way of writing numbers is called STANDARD FORM or SCIENTIFIC NOTATION.

Notice that the first number must lie between 1 and 10. i.e the decimal point is always after the first digit.

Each answer is in the form a x where a is between 1 and 10 and n is 1, 2, 3 ..... i.e. the index tells you how many places to move the decimal point to get the normal form of the number (by normal we mean the usual way of writing numbers)

Thus the speed of television signals is 3 x kg and the number of seconds in a century is 3.153 x in standard form.

CHANGING NUMBERS FROM STANDARD FORM

To change numbers from standard form into the format of ordinary numbers, the decimal point must be moved by a given number of places and zeros inserted as required.

For example

3.5 x is a number in standard form.

Since = 10 x 10 x 10 = 1000 (a thousand)

3.5 x = 3.5 thousands

and 3.5 x = 3500.

moving point 3 places to the RIGHT (inserting zeros as required)

= 3500 (the decimal point is NOT shown in a whole number).

EXAMPLES

4.72 x = 472000 move point 5 places to the RIGHT

1.79 x = 17.9 move point 1 place to the RIGHT

9.3 x = 9300000 move point 6 places to the RIGHT

1.92 x = 1.92 move point 0 places to the RIGHT