Numbers

[Test Yourself to Improve your understanding] [Useful Links]

What do we mean by Numbers and why are they important

Some basic arithmetic

Conversions

Estimating

Some basic factual information:

Some very large and very small numbers:

Useful or interesting Links

Test yourself to improve your understanding:

Numbers -> Standard Form
Numbers -> Some Real World Numbers
Numbers -> Estimating
Numbers -> Percentages
Numbers -> Miscellaneous

What do we mean by Numbers and why are they important

Numbers are all around us, and if you don’t understand numbers you don’t really understand the world. In particular anything to do with finance uses numbers and if you can’t do basic arithmetic or have a feel for numbers you will be liable to make mistakes or be taken advantage of.

Knowing a bit more than basic arithmetic can be useful when it comes to making decisions. Many decisions come down to numbers and understanding and being able to manipulate and understand numbers will help you make better decisions.

Having an understanding of numbers also adds an extra dimension to your life. You can better appreciate how the world works and just the wonder of it all. It can also be a source of entertainment.

Some basic arithmetic

Being able to do some basic arithmetic and estimating is all the maths most people need for their daily lives.

There is very little you need to know, and people’s difficulties with numbers is nothing to do with it being difficult or them not being very good with numbers. It is the psychological blocks they have which prevent them from trying in the first place.

If you don’t know basic arithmetic, including multiplication and division, then you need to learn it. Look to take some adult classes, get someone to help you, or use readily available on-line resources, such as:

http://www.bbc.co.uk/skillswise/maths

However it is easy and useful to go beyond basic arithmetic, and it is not difficult. It is mostly simply a case of taking your time, not panicking about it, and just following simple steps. What might be seemingly difficult questions are actually very straight forward.

Some examples:

E1. If you could walk none stop, at say 2 yards a second, how long would it take you to walk a thousand miles.

Answer:

A case of calculating how many yards in a thousand miles, which gives you the number of seconds it would take, and then just converting seconds into days and hours.

Thus:

1000 miles = 1000 x 1760 yards, noting that there are 1760 yards in a mile, ie. 1,760,000 yards.

Noting we are walking at 2 yards per second it will thus take 880,000 seconds.

Which is 14,666 minutes (with 40 seconds left over)

Which is 243 hours (with 26 minutes, and the 40 seconds left over)

Which is 10 days (and nights), 3 hours, 26 minutes and 40 seconds.

E2. The great wall of china is of the order 10,000 kilometers long. Assuming it is made of bricks of about 50 cm x 25 cm x 25 cm and it is 3 metres high and 4 metres deep and one man lays one brick every 1 minute on average and there are 1,000 men working 12 hours a day every day none stop, how long would it take to have been built?

A case of calculating how many bricks in the great wall, and then noting that 1,000 bricks are being laid per minute calculating how many minutes it would take to build. These minutes can then be turned into hours and days and years, noting work progressing for 12 hours per day none stop.

Number of bricks = 10,000 (kilometres)

x 1,000 (to convert to metres)

x 3 x 4 (number of square metres in each 1 metre length of wall)

x 2 x 4 x4 (number of bricks in 1 square metre of wall).

= 3,840,000,000 bricks.

1,000 men are laying 1,000 bricks per minutes.

Thus length of time is 3,840,000 minutes

= 64,000 hours.

Noting the men are working 12 hours a day, this

= 5,333 days (with 4 hours left over)

This then equates to a little over 14 ½ years.

E3. Folding paper to reach the sun. Assuming you could fold a piece of paper 0.1mm thick, how many times, about, would you need to fold it to reach the sun, call it 150 million kilometres away.

Ie. how many times do you have to double from a start of 0.1 mm to get to 150 million kilometres.

This may seem a little daunting, but a useful short cut is to know that doubling 10 times, ie. 2x2x2x2x2x2x2x2x2x2 gives 1,024. Since we are happy to approximate we can call this 1,000.

Now it gets simpler:

10 lots of doubling 0.1mm thus gives us 100mm, ie. 0.1 metre.

Another 10 lots of doubling (ie. 20 lots in all) gives us 100 m.

Another 10 lots of doubling (ie. 30 lots in all) gives us 100 kilometres.

Another 10 lots of doubling (ie. 40 lots in all) gives us 100,000 kms.

Another 10 lots of doubling (ie. 50 lots in all) gives us 100 million kilometres.

Almost there. One more lot of doubling gives us 200 million kilometres, and we have thus reached the sun and beyond.

Thus 51 lots of folding our piece of paper gets us to the sun.

As it happens 1,024 is close enough to 1,000 that this would have been the answer even if we had calculated accurately.

Conversions

Basic arithmetic allows us to do conversions. We often have to convert numbers from one form to another. Thus for example miles to kilometres, or pounds (weight) to kilograms, or centigrade to Fahrenheit. Of course you can always look up conversions on the internet, or use a calculator, but these are not hard to do, and if you can do them yourself there is a certain satisfaction and you can do them even when you don’t have the internet or a calculator to hand.

Conversions are simply about multiplying one number by another. The trick is to know which way round to do the multiplying.

Thus consider miles and kilometres.

1 mile is (approximately) 1.6 kilometres.

Thus a mile is longer than a kilometre.

Thus for every mile there are 1.6 kilometres.

Which as an equation means (number of kilometres) = (number of miles) x 1.6

Thus to calculate number of kilometres in 15 miles we calculate 15 x 1.6 = 24.

Ie. 15 miles = 15 x 1.6 = 24 kilometres.

To convert kilometres to miles then (number of miles) = (number of kilometres) / 1.6

Thus 80 kilometres = 80 / 1.6 miles, ie. 50 miles.

Miles per hour conversion to kilometres per hour or visa versa, is exactly the same conversion.

Thus 80 kilometres per hour (80 km/hr) is 50 miles per hour (50 m/hr).

Other seemingly more complex conversions are just as straight forward, you just need to do them one step at a time and check you’ve got the conversions the right way round.

Thus, for example, consider converting 70 miles per hour into metres per second.

70 miles = 70 x 1.6 = 112 km.

Thus 70 miles/hr = 112 km/hr, which itself is then 112,000 metres/hr.

If we travel 112,000 metres in one hour then we travel 112,000/60 metres in 1 minute, ie. 1866.66 metres / minute.

And to convert to metres / second we divide by 60 again, ie. 1866.66 / 60 = 31.111 metres / second.

Ie. 70 miles / hour is just a little over 30 metres / second.

Some other common conversions include:

Pounds to kilograms: 2.2 pounds (lbs) = 1 kilogram (kg) [approximately]

Thus 1lb = 1/(2.2) kg = 0.454.5kg [approximately, actually 0.4536 = 453.6 grams to 4 significant figures]

Thus 1 ounce (oz) = 1/16 lbs = 453.6/16 = 28.35 grams

In UK

1 ton = 160 stone = 160 x 14 lbs = 2,240 lbs = 2,240 x 0.4536 kg = 1,016 kgs

The UK ton is sometimes called a long ton, to differentiate if from a US ton which is 2,000 lbs and which is sometimes called a short ton. Ie. a UK ton is heavier than a US ton. (A US ton is 908 kgs)

Temperatures

0 degrees centigrade = 32 degrees Fahrenheit

A 1 degree change in centigrade is equivalent to a 1.8 degree change in Fahrenheit.

Thus 100 degrees centigrade = 32 + 1.8 x 100 = 212 degrees Fahrenheit

(degrees Fahrenheit) = 32 + (degrees centigrade) * 1.8

Estimating

Being able to estimate is both a useful daily living skill, and also a skill that helps us better appreciate and understand the world we live in. Estimating is where we are looking for numbers or answers that are close enough without any concern about them being exactly right.

There are two types of estimating we concern ourselves with here: general day to day estimating such as estimating a shopping bill, and being able to estimate something we don’t know by relating it to something we do know.

Day to day estimating is commonly encountered where we have some numbers and an answer and we want to be know whether the answer is about right.

For example:

You buy some goods at £5.27, £10.20, £4.45. You are charged £24.92. Quickly, do you think it’s about right. Without adding the numbers precisely you should know it’s not right. The three numbers can very quickly be rounded to £5, £10, and £4, plus an extra £1 because you notice all your rounding is slightly less than the number given, giving an answer of about £20. An answer of about £25 clearly isn’t right.
You multiple 1027 by 42 on a calculator and get the answer 5,334. Is that likely to be right. No, because 1000 x 40 is already 40,000 and since both numbers are bigger therefore the answer must be bigger than 40,000. In this case you’ve probably typed in 127 rather than 1027.

Some tips relating to day to day estimating include:

Round numbers to approximate integers or multiples of some convenient numbers such as 5 or 10, and then add them or multiply them together.
If you have a mix of big and small numbers you can ignore the small numbers and focus on what the big numbers approximate to.
If adding numbers together with the numbers being reasonably close together, look at the numbers and identify what looks like an average which you can then multiply by the number of numbers. If the list of numbers includes some much smaller than others you can simply ignore them. If you have one or two much larger, take these separately, average the rest, and then add back in the larger numbers or approximations to them.
You can estimate number of objects in a given area by estimating the number in a representative smaller area and multiplying up by approximately how many of the smaller areas make up the larger area.

Get a feel for numbers by practicing day to day estimating:

A second type of estimating is about things we don’t know. In this case we need to look to relate what we are estimating to what we do know, as best we can.

For example, what is the population of Australia? How big (square miles) is the UK? I’m assuming you don’t already know.

With regards the population of Australia. What do you know? You may know, or at least be pretty sure, the population is smaller than that of the United Kingdom, which you know to be something like 65 million. Is it a lot smaller? Well you know it’s a very big country, and whilst you know much of it is very sparsely populated, neither have you heard that it has a very small population. Thus an estimate of about half the population of the United Kingdom might not seem unreasonable. Ie. a population of about 32 million. [Actually the population is about 24 million.]

With regards how big in square miles is the UK? Well you might think about the distance from the south coast to the top of Scotland. Imagine driving up from London. You might estimate it taking about 6 hours to get up to the Scottish border. Say about 60 miles/hr. About 360 miles. Double it because you know Scotland is quite long. But you didn’t think it was quite that long so consider it as also accounting for the fact that London isn’t on the South coast. So 700 miles say. Clearly the width of Great Britain varies, but you might say across from London to Bristol is about a two hour drive without traffic, say 120 miles. Generally much of the country is more than this, though less so once you get further up North. Call it 150 miles. Approximate size of Great Britain 105,000 square miles. The UK also includes Northern Ireland. Add another 1/10^th. Overall estimate about 115,000 square miles. [Actually the area is about 95,000 square miles.]

This is not about whether or not these estimates are right, it is about just making best use of whatever information you do know to make an estimate. Of course you could just look it up on the internet. However if you are not on the internet and for whatever reason you want to estimate, then the key is to relate it to what you do know, and don’t worry about precision.

Some basic factual information:

In order to be able to make good general estimates about things we don’t know, and also to have a good feel for the way the world really is, it is useful knowing, by heart, some basic information. You don’t need to know it precisely, but should know it roughly. The following is some such information that you should look to learn by heart and thus be able to use when estimating other information.

Circumference of the earth: about 40,000 km (or 25,000 miles)

Mass of the earth: 6*10²⁴ kg (If you are not familiar with the format of 10²⁴ then see below under Standard Form.

Volume of the earth: 10¹² km³

Distance from the Earth to the Moon: average of about 384,000 km or 240,000 miles, though this varies over a range of about 50,000 kms or 30,000 miles.

Average distance from the Earth to the Sun: 149,000,000 km, or 93,000,000 miles. This varies from about 146 million kilometres (91 million miles) to about 152 million kilometres (94.5 million miles).

1 Astronomical Unit (au) is the average distance from the sun to the earth, approximately 150 million km.

Distance from the Sun to Pluto is 30 – 50 au, average about 40 au.

1 light year is approx. 9.46*10¹² km, or 63,200 au.

Distance to the nearest star, Proxima Centauri, is about 270,000 au, or 4.2 light years.

Distance to the centre of our galaxy, the milky way, is about 25,000 light years.

Distance to the nearest major galaxy, Andromeda, is about 2.5 million light years. Note that our galaxy and the Andromeda galaxy are on a collision course, though this collision is some 4 billion years in the future.

Approximate number of grains in a 1lb packet of sugar = about 8 million.

A litre of water weighs 1 kg. Note that is also a cubic volume of 100cm x 100 cm x 100 cm (ie. 1 cubic metre contains 1,000 litres).

A large car weighs approximately 2,000 kg.

A snail travels at about 50 metres per hour.

A German shephard dog can run at up to about 50 km/hr.

The fastest man can run at about 40 km/hr, though most of us would be unlikely to get much above about 15 km/hr.

Basic Percentages

Percentages are simple. If you feel uncomfortable about them you just need to get a bit of practice with them.

To calculate a given percentage of a number you multiply the number by the percentage value and divide by 100. This is often a simple fraction. Thus 50% is a half. 10% is a tenth. 25% is a quarter.

You can have percentages greater than 100. Thus 200% is multiplying by 2.

Take a number. 250. 1% of 250 is 2.5. 10% of 250 is 25. 40% of 250 is 100. 100% of 250 is 250. 150% of 250 is 375.

Net Percentages

Here are some slightly more complex percentages but if you just take them a step at a time they are straight forward.

Something is reduced by 10%, and then reduced by another 10%. What is the total reduction?

A reduction by 10% means 10% taken away, which is 1/10^th taken away from the original number and thus 1-1/10^th = 9/10^th of the original number.

Thus 10% taken off from something that was £200 becomes £200 x 9/10 = £180.

Alternatively 10% of £200 is £20, and thus 10% off £200 is £200 – £20 = £180.

Either approach gets you to the same result.

The statement of a further 10% reduction is ambiguous. Is this a further £20 off, ie. £160, or is it 10% of £180, which would then be £162.

Ambiguity about whether or not a second percentage uses the original value as its base measure or the result of the first percentage can explain seemingly odd results.

For example if an amount is reduced by 50% and then increased by 50%, the resulting value could either be back where you started, or it could be just 75% of where you started.

Compound Interest

What is termed compound interest is simply multiplying percentages over a period of time. It works to your advantage if it relates to your savings, but to your disadvantage if it relates to your debts.

For example if you have savings which earn you an increase of 5% a year, then assuming you start with £1000, this would grow according to the following:

After 1 year £1000 x 1.05 = £1050

After 2 years £1050 x 1.05 = £1102.5

After 10 years this becomes approximately £1,629

Higher percentages compound at increasingly faster rates.

A yearly increase of 10%, would, after 10 years, become approximately £2,594.

A yearly increase of 100%, would, after 10 years, become over £1 million.

Which is why debts can rapidly accumulate, since the rates you are charged for loans are higher than the rates you get for savings. And of course if percentages are applied monthly they will lead to compounded increases faster than if applied yearly.

Recognition of misleading interpretation of monthly rates means that financial institutions are required to quote the yearly equivalent rate, known as Annual Percentage Rate, ie. APR. Thus a monthly rate of 2% is an APR of approximately 26.8%. Sometimes a ‘yearly’ rate is quoted, maybe 24%, but the statement made that it is compounded monthly. This in fact means a 2% is applied monthly so the real yearly rate is 26.8%, not the quoted 24%.

Understanding compound interest can help you make wise financial decisions. Loans for example will have different interest rates quoted. Your savings might be earning at different interest rates. Often the rates at which you borrow will depend upon the period over which you want the loan. You can thus work out how much in total a given loan may cost you and make your choice accordingly.

Thus, for example, say you loan £10,000 at a monthly interest rate of 1%. If you were to wait 5 years before paying it back you would have to pay £18,167 (ie. 1.01 multiplied together 60 times, x the original loan).

Of course loans are generally paid back so much a month. If you were to pay back the £10,000, with the monthly 1% interest rate, at £225 a month, then you would pay the full loan back in 5 years. By which time you will have paid back about £13,280. [The spreadsheet calculation is to start with 10,000. End of month 1 this has increased to £10,100 as a result of the 1% interest (multiply by 1.01), but you then pay back £225 (ie. take away 225). Thus at the end of the 1^st month you still owe £9,875, despite having paid back £225. The calculation for each subsequent month is then taking the previous month value, multiplying by 1.01, and then taking off 225. Do this in a spreadsheet and at month 59 you will owe just over £16, which is then less than your remaining monthly payment.

A characteristic of such a loan is that during the early pay back periods you are paying back per month significantly less of the actual loan itself than you will be later during the pay back period, due to the greater impact of the interest rate on the larger amount that you are still owing.

Take a loan of £100,000, which might be more usual for a house purchase. If you were to take such a loan at a monthly rate of 1% then you would need to pay a monthly rate of £1,000 just to pay the interest, without reducing your loan at all. Mortgage rates however are generally far lower than other loan rates (the house itself is taken as security on the loan), so a more typical monthly rate might be 0.3%. In this case a monthly repayment of £500 will see the loan repaid in just a little over 25 years (25 years and 6 months). By this time you will have paid back of the order £153,000.

This is made more complicated by the fact that inflation changes the value of money over time, and, in support of making decisions about financial options which involve monies over many years, this should be included in the calculation.

For example, considering the above calculation for the loan of £100,000 at a monthly interest rate of 0.3%. Let us say we have a monthly inflation rate of 0.1%. Whilst this does not affect our calculations of what we owe or how long we owe it for, we can consider that the £500 we pay a month later is actually only equivalent to £500 * (1-0.001), ie. £499.50 relative to the month in which we actually got the £100,000 loan. And each month the £500 we continue to pay is progressively worth less (by an additional (1-0.001) each month). With our spreadsheet calculation we can thus calculate the ‘present value’ worth of each £500 we pay. By the time we finish paying our loan after just over 25 years, the worth of our £500 has dropped to about £369. We can thus sum up the ‘present values’ of each of the £500s and the equivalent repayment is actually about £131,500 rather than the £153,000 we calculated earlier.

Note that the term Net Present Value (NPV) is used for the current day value of some future (or past) monetary value.

Spreadsheets makes all these calculations relatively simple, assuming you set them up correctly.

Standard Form

A convenient way of writing down very large or very small numbers is in what is termed Standard Form, or Standard Index Form.

A number in standards (index) form is written in the form A x 10ⁿ, where A is a number bigger or equal to 1 but smaller than 10, and n is an integer. Note that sometimes we might write 10ⁿ as 10(n) since the format of 10ⁿ does not always appear correctly, depending upon the particular browser or app that you are using.

The n is the number of times 10 is multiplied by itself.

Thus:

1 hundred = 100 = 1 x 10(2), or 1 x 10², or just 10(2) or 10².

1 million = 1,000,000 = 10(6), or 1 x 10(6) or 10⁶.

1 billion = 1,000,000,000 = 10(9) or 10⁹ [Note that a billion in the UK used to be 10(12) or 10¹², but it is now far more common to use what was originally the US usage of 10(9). Nevertheless on occasion you may come across the old usage.

1 trillion = 1,000,000,000,000 = 10¹² [Old UK usage would have had this as 10¹⁸.]

Any number can be written in the standard form.

For example 2,000 can be written as 2 x 10³.

We can write 2,300 as 2.3 x 10³

2,350 as 2.35 x 10³

999 as 9.99 x 10²

This principle also extends to numbers less than 1, where we use a negative power to represent 1/10ths, thus 0.1 = 1 x 10^-1, and 1/1000^th = 1 x 10^-3.

Thus 0.0023 can be written as 2.3 x 10^-3

Note that the integer number associated with the multiples of 10 (ie. ‘n’) relates to the number of decimal places. If you take the number before the power of 10 (ie. ‘A’), and move the decimal point by the number places associated with the multiple of 10. If the ‘n’ is positive you move the decimal point to the right, if it is negative you move it to the left. You can thus turn any number into its standard form simply by counting the number places you need to move the decimal point to get it to a point where your number is greater or equal to 1 and smaller than 10.

Thus:

45,000: the number ‘A’ is 4.5, and the decimal point must be moved 4 places to get to the number 45,000. Thus it is written in the form 4.5 x 10⁴.

Whilst the standard form might appear cumbersome, it is very useful for representing very large and small numbers. Note that we tend to use it when we only need approximate the number. Thus a number such as 29,167,921 would not be put in standard form. If however we were not interested in the detail, we might show the number as 2.9 x 10⁷. We shall see below that when talking about very large or very small numbers this is how we tend to represent them. It enables us to readily represent and compare large or small numbers and think about ‘orders of magnitude’.

Some very large and very small numbers:

You should have a feel for some very large numbers and small numbers.

Number of galaxies in the universe: 10¹¹, ie. 100 billion

Number of stars in the universe: 7 x 10²², ie. 70,000 million, million, million

Number of atoms in a human being: 7 x 10²⁷

Note that 2/3 of these atoms are hydrogen, ¼ oxygen, and 1/10 carbon, which together thus make up about 99% of the atoms we are made up of.

Age of the universe: 1.38 x 10 ¹⁰, ie. 13.8 billion years.

Number of seconds since the universe began = 4.35 x 10¹⁷

Number of seconds in a human life of 80 years = 2.5 x 10⁹ or 2.5 billion

time taken by light to travel one meter: roughly 3 × 10⁻⁹ seconds

radius of a hydrogen atom: 2.5 × 10⁻¹¹ metres

Useful or interesting Links

http://www.skillsyouneed.com/numeracy-skills.html

Estimating: http://www.mathsisfun.com/numbers/estimation.html

Quiz/Tests

Numbers -> Standard Form

Numbers -> Some Real World Numbers

Numbers -> Estimating

Numbers -> Percentages

Numbers -> Miscellaneous

Reminder on taking tests: It’s not about trying to prove you already know it, it’s about learning.

Numbers -> Standard Form

[Alternative version of Quiz with Scoring]: Requires ActiveX enabled

Note that we use the form 10(X) to refer to 10 raised to the power of X.

Question SF.1

Which is the correct standard form for: 23,400,000

(a) 23.4 x 10(6)
(b) 23.4 x 10(7)
(c) 2.34 x 10(7)
(d) 2.34 x 10(8)

Delete this and type in here.

	1	2	3	4
USA	500,000,000	350,000,000	800,000,000	150,000,000
UK	55,000,000	60,000,000	75,000,000	65,000,000
England	55,000,000	50,000,000	65,000,000	45,000,000
India	500,000,000	1,000,000,000	1,800,000,000	1,300,000,000
China	1,000,000,000	1,400,000,000	1,700,000,000	2,200,000,000
Russia	150,000,000	100,000,000	300,000,000	250,000,000
Japan	95,000,000	130,000,000	500,000,000	100,000,000
Germany	60,000,000	120,000,000	100,000,000	80,000,000
Brazil	210,000,000	300,000,000	450,000,000	620,000,000
South Africa	55,000,000	50,000,000	65,000,000	45,000,000
The World	6,500,000,000	7,300,000,000	8,100,000,000	9,000,000,000
Iceland	150,000	350,000	500,000	3,000,000
Saudi Arabia	12,000,000	22,000,000	32,000,000	42,000,000
Australia	18,000,000	23,000,000	35,000,000	40,000,000
Singapore	12,400,000	3,500,000	1,800,000	5,500,000