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SLIDE 1
Approximation by decimal places
When approximating by decimal places we count the digits following the decimal point. The approximation is given to the number of decimal places (dp).
When approximating we use the rule that any digit 5 or greater is rounded up, and any digit 4 or less is rounded down.
Write down the following correct to the number of decimal places stated:
? 45.482 to (a) 2 dp, (b) 1 dp
? 0.008518 to (a) 5 dp, (b) 3 dp, (c) 2 dp
? 5.9702 to (a) 3 dp, (b) 2 dp
? 164.2618 to (a) 3 dp, (b) 2 dp, (c) 1 dp
SLIDE 2
? 45.482 = 45.48 (2 dp) = 45.5 (1 dp)
? 0.008518 = 0.00852 (5 dp) = 0.009 (3 dp) = 0.01 (2 dp)
? 5.9702 = 5.970 (3 dp) = 5.97 (2 dp)
? 164.2618 = 164.262 (3 dp) = 164.26 (2 dp) = 164.3 (1 dp)
SLIDE 3
Significant figures
Another way of approximating is to use significant figures.
In the number 3178, 3 is the most significant figure because it has the largest value of 3000. We count down from the most significant figure.
Example
Write 259.935 to (a) 5 sf, (b) 4 sf, (c) 2 sf.
Solution.
259.935 = (a) 259.94 (5 sf), (b) 259.9 (4 sf), (c) 260 (3 sf).
In the final answer here we write 260 and not 26. Clearly, the number 259.935 is not approximately 26, which gets the size of the number all wrong; it is approximately 260. We must introduce a zero to preserve the right size.
SLIDE 4
Write the following correct to the number of significant figures stated.
? 22.945 to (a) 4 sf, (b) 2 sf
? 0.008327 to (a) 3 sf, (b) 2 sf, (c) 1 sf
? 36.602 to (a) 4 sf, (b) 3 sf
? 45783 to (a) 4 sf, (b) 3 sf, (c) 2 sf
? 24,361,288 to (a) 4 sf, (b) 3 sf, (c) 2 sf
SLIDE 5
? 22.945 = 22.95 (4 sf) = 23 (2 sf)
? 0.008327 = 0.00833 (3 sf) = 0.0083 (2 sf) = 0.008 (1 sf)
? 36.602 = 36.60 (4 sf) = 36.6 (3 sf)
? 45783 = 45780 (4 sf) = 45800 (3 sf) = 46000 (2 sf)
? 24,361,288 = 24,360,000 (4 sf) = 24,400,000 (3 sf) = 24,000,000 (2 sf)
SLIDE 6
Error bound
A ruler is used to measure the length of a piece of paper, which is found to be 8.6 cm to the nearest tenth of a centimetre (millimetre). What is the error interval for the length of paper?
Solution
The ruler is only accurate to the nearest 1/10 of a centimetre.
The actual length of the paper could be anywhere between 8.55 and 8.65 cm; it is 8.6 cm to the nearest 1/10 of a centimetre. The error bound may be written using inequalities as
Since in rounding and approximation a 5 rounds up, the lower bound is 8.55 length; the upper bound is length 8.65, because if it is equal to 8.65 it rounds up instead to 8.7.
SLIDE 7
A number, n, is rounded to 2 decimal places. The result is 5.77. Using inequalities write down the error interval for n.
SLIDE 8
SLIDE 9
Jam is being prepared in a large pot containing 6 litres to the nearest litre. The jam is poured out into jars holding litre to the nearest of a litre. What is the greatest number of jars that could possibly be filled with the jam?
SLIDE 10
Largest amount of jam in the large pot = 6.5 L.
Smallest amount of jam in a jar = 0.45 L.
Greatest number of jars = 14.
SLIDE 11
To make a bracelet two pieces of chain must be cut from a roll of chain. The first piece is 12 cm to the nearest cm; the second piece is 15 cm to the nearest cm. If the roll of chain is exactly 52 metres long, what is the greatest number of bracelets that could possibly be made from the roll?
SLIDE 12
Smallest first piece = 11.5 cm
Smallest second piece 14.5 cm
11.5 + 14.5 = 26 cm
Greatest number = 200
SLIDE 13
The metric and customary systems of measurement
Europe and most of the world use the metric system, whose principle units are metre, kilogram and litre.
The USA has continued to use the older customary system, whose units include the mile, pound and pint. Britain retains vestiges of the customary system, which is called imperial system by the British. For example, the British continue to measure distances on motorways in miles.
The metric system
Length
1 kilometre = 1000 metres
1 meter = 1000 millimetres
Volume
1 litre= 1000 millilitres
Mass
1 metric tonne = 1000 kilograms
1 kilogram = 1000 grams
SLIDE 14
Converting from US customary to metric and vice-versa
We use approximate conversions here.
1 inch = 25 millimetres.
1 mile = 8/5 kilometres. Multiply by 8 and divide by 5.
1 US pint = 1/2 litre
1 pound = 4/9 kilograms. Multiply by 4 and divide by 9.
Exercise
Use the approximate conversions given above.
? Convert 11 inches to millimetres.
? There are 12 inches (in) in one foot (ft). Convert 3.52 meters to feet to 4 sf.
? Convert 22 ¾ lb to kilograms.
? There are 2000 lb in one US ton. Convert 420 kilograms to US tons.
? There are 8 pints to 1 gallon. Convert 1 gallon to litres.
? Convert 60 litres to gallons.
SLIDE 15
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?
?
?
?
?
SLIDE 16
Precise conversions
The conversions used in the previous exercise are approximate. Precise conversions given below are, where appropriate, to 4 significant figures.
1 inch = 25.4 millimetres
1 mile = 1.609 kilometre
1 pint = 473.2 millilitre
1 pound = 0.4536 kilograms
1 metre = 39.37 inches
1 kilometre = 0.6214 miles
1 litre = 2.113 pints
1 kilogram = 2.205 pounds
SLIDE 17
Standard form
In business and science, we sometimes have to represent large and small numbers. For this purpose, we use standard form.
Examples
? The number 2,000,000 (two million) is made of a 2 followed by 6 zeros, so we can write it as .
? The number 0.003 could be represented by the fraction , so it can be written as .
The superscript is called an index or exponent.
When we are multiplying by 10 the index is a positive number (+1), and when we are dividing by 10 the index is a negative number (–1, minus sign).
SLIDE 18
Write the following in standard form
? 458 ? 8390
? 29,500 ? 7,034,000
? 0.05 ? 0.0067
? 0.000008 ? 0.000825
SLIDE 19
? 458 ?
? ?
? ?
? ?
SLIDE 20
Write the following as ordinary numbers
? ?
? ?
? ?
?
SLIDE 21
?
?
?
?
?
?
?
SLIDE 22
Operations in standard form
To add and subtract: convert to ordinary numbers, add or subtract, then convert back to standard form.
To multiply or divide: multiply (divide) the numbers and add (subtract) the indices.
Examples
SLIDE 23
? Add and
? Subtract
? Multiply
? Divide
SLIDE 24
?
?
?
?
SLIDE 25
? Add
? Subtract
? Multiply
? Divide
SLIDE 26
?
?
?
? Divide
SLIDE 27
? Divide 8100 million by 300 giving your answer in standard form.
? Work out . Give your answer in standard form.
? The probability of A is . The probability of B is How many times more likely is A than B?
? Work out .
SLIDE 28
?
?
? The probability of A is . The probability of B is . How many times more likely is A than B?
15 times more likely.
? .
SLIDE 29
Clearing the memory of a calculator
Begin by clearing the memory of your calculator
Try the following
SLIDE 30
This should clear the memory, producing a series of displays
SLIDE 31
Modern calculators are too complicated for general use
You are not likely to need even a small fraction of all the functions that your calculator can perform.
Setting the mode
For the coming period you are most likely to need to use the calculator in computation mode with angles measured in degrees.
As your calculator may accidentally be not in these modes, you need to be able to reset the mode.
Setting to computation mode
You select the COMP mode. Your calculator may do it differently.
SLIDE 32
Setting to computation mode
SLIDE 33
Resetting to degree mode
You press CLR twice.
You select the Deg mode. Your calculator may do it differently.
Angles are also measured in radians and gradians. Radians are used in higher level mathematics and gradians are used in surveying. You will not need either for some time, and you’ll never need gradians, unless you become a surveyor or going into mining.
SLIDE 34
On the display there is usually a D in a small black box. This shows that you are in degree mode.
Here the black box is enlarged for clarity. It is normally smaller than shown above.
SLIDE 35
Find out what happens when you key in
SLIDE 36
You will obtain something like the following
You can learn what these buttons do by experimenting with them.
They control such things as how many decimal places your calculator displays. This may be useful in higher level and working situations.
When you reset the calculator
you return to the default settings, which are all you need for the time being.
SLIDE 37
Understanding the display
? Key into the calculator . Write down what you see on the display.
? Key into the calculator . Write down and interpret what you see on the display.
SLIDE 38
? Key into the calculator . Write down what you see on the display.
The dot indicates the decimal place. This is eight million, three hundred and sixty-nine thousand, nine hundred and ten.
? Key into the calculator . Write down and interpret what you see on the display.
The number is too large to be displayed entirely on screen. It is displayed in standard form.
This is what standard form is for – to display large and small numbers.
SLIDE 39
Observe on the display the right arrow and the up-triangle symbols.
Your calculator is likely to have a replay button
? The right arrow indicates that there are digits to the right of the top line of the display. Click the right triangle of the replay button to show these digits.
? The up-triangle symbol indicates that a previous result is stored. Click the up-triangle of the replay to display it.
SLIDE 40
?
You see the digits you could not see
?
You see the previous calculation
SLIDE 41
Using the memory
? Key in the following and write down what you see.
What does this do?
? Then key in the following and write down what you see
What does this do?
SLIDE 42
Using the memory
?
It stores 5 into the memory
?
It multiplies the stored value of 5 by 7
The memory is called independent memory. By pressing
you can work with the answer to the last calculation. This is called answer memory.
SLIDE 43
Revision
Use long division to find the exact decimal representing the fraction
SLIDE 44
SLIDE 45
We call 0.0625 the reciprocal of
What is the reciprocal of 0.0625?
SLIDE 46
The reciprocal of 0.0625 is 16 and the reciprocal of 16 is 0.0625
They are the inverses of each other
SLIDE 47
The reciprocal is marked by an index with
The minus 1 index is the same as dividing by the number
? Find the reciprocal of 2 expressing your answer as a fraction
? Find as a fraction and as a decimal
? Evaluate
SLIDE 48
?
?
?
SLIDE 49
On a calculator the reciprocal button is shown by one of these two alternatives.
You key in the number first to find its reciprocal
Using a calculator, find the reciprocals of the following numbers, giving your answers to 4 significant figures.
? 9.29 ? 8.247
? 78.6 ? 8252
? 0.1647 ? 0.03857
? 0.00148 ? 27352
SLIDE 50
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? 78.6
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?
SLIDE 51
Square and square root buttons on the calculator.
Sometimes these are placed together on one button, and one has to shift to use one rather than the other.
Use a calculator to find the following, giving your answers to 4 significant figures.
? ?
? ?
SLIDE 52
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?
?
SLIDE 53
Use a calculator to find the following, giving your answers to 4 significant figures.
?
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?
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SLIDE 54
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