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Arithmetic

Arithmetic is adding, multiplying, subtracting, and dividing numbers, without using any special symbols to represent numbers. There are a number of rules to help keep our equations organised, and legible for other people. Remember, mathematics is a language with rules and conventions which everybody must follow, so that anybody, anywhere in the world can read it and understand easily.

Fractions

The top part of a fraction is called the numerator. The bottom part is the denominator.

To add or subtract fractions, the denominator must be made the same: e.g. $1/2 + 1/3 = 3/6 + 2/6 = 5/6$.

To multiply fractions, the denominators can be multiplied and the numerators can be multiplied: e.g. $4/5 ⋅ 3/7 = {4 ⋅ 3}/{5 ⋅ 7} = {12}/{35}$.

To divide two fractions, the fraction which is the denominator is brought to the numerator position and inverted, before multiplying: e.g. ${(4/{51})}/{(3/{17})} = {4/51} ⋅ {17/3} = {4/3} ⋅ {1/3} = {4}/{9}$.

The lowest common multiplier (or lowest common denominator, LCD, in the case of fractions) is the lowest number which will be divided evenly (as a whole number) by each of a set of numbers. Multiplying the set of numbers together will provide a common multiplier, but first check to see if all the numbers can be divided by a common number before multiplying them:

e.g. 12, 8, 4: first notice that 4 divides evenly into 8, so any multiple of 8 will be divisible by 4.

12 divided by 8 is $3/2$ so a LCM will be 2 times the highest number and 3 times the lower number: so 3 x 8 = 24, and 2 x 12 = 24. This is the lowest common multiplier.

Lowest common denominator: $1/{12} + 1/8 + 1/4 = {2 + 3 + 6}/{24} = {11}/{24}$

Percentages

Percentage is a commonly used way of expressing one quantity as a proportion of a larger quantity. 1 cent is 1 per cent of a euro.

To convert a fraction to a percentage, multiply by 100:

$4/5$ as a percentage: $4/5 * 100 = {400}/5 = 80%$

Map Scales

1 : 50 000 is a typical map scale. A centimetre on this map represents 50 000 cm in reality. This is 500 m, or 0.5 km.

A 1 : 72 model of an aeroplane is reduced by a factor of 72 in each dimension.

Speed, distance and time

Calculating time is a little more complicated because the units of time are the only S.I. units which do not follow the metric system. There are 60 seconds in a minute, etc. A metric system work have 100 seconds in a minute. This was once proposed, but it never took off.

To convert from $m/s$ to ${km}/h$ use the factor 3.6. This is because there are 3600 seconds in an hour (60 s x 60 min = 3600 s), and 1000m in a kilometer: $1.0 m/s = 1.0 ⋅ {1 km}/{1000m} ⋅ {1s}/{1/{3600}h} = 3.6 {km}/h$.

For example, $14 m/s = 14 ⋅ 3.6 = 50.4 {km}/h$.

Estimation and Mental Arithmetic

To do algebra, it is necessary to be able to make estimates and work without a calculator. This is because a student needs to be able to manipulate equations which contain unknowns (letters instead of numbers), and a calculator is useless with unknowns!

Mental Arithmetic

Rounding up

If you need to calculate with a number like, for example, 19, it is a good first step to round it up to 20 first. So if you need to multiply 19 by 7, first do 7 x 20, then subtract one 7, and you have 140 - 7 = 133. this is much faster and easier than using long division.

Ignore decimal points and zeroes

When dealing with numbers which contain zeros and decimal points, first just calculate with the integers, then add up the zeros and decimal places afterwards and add them in: for example, 1100 ÷ 55 is easier to calculate by making it 110 ÷ 55 = 2, then add the zero to make 20.

In very complex problems, a useful trick is to estimate the answer, so that any mistakes can be identified before they are carried forward to mess up the rest of the work. the last thing a mathematician wants to discover when reviewing his 20-page solution is that there was a simple arithmetic mistake near the top of page 1!

Practice makes perfect in calculating numbers in your head. Try working out maths problems in your head, and only use the calculator to check your answer. Always reaching for the calculator for any problem is a bad habit which will make difficult for you as the subject gets more challenging, and you realise too late how little maths calculators can do on their own.

Sequences

A sequence is a collection of numbers in a defined order. The terms of a sequence follow a certain rule.

2, 5, 8, 11, .... rule: start at 2 and add 3 to each consecutive number. This is an arithmetic sequence.

0, 3, 6, 12, 24, .... rule: start at 3 and double each term to produce the next term. This is a geometric sequence.

Roots or Surds

Surds (or radicals) are expressions which contain square roots. The rules of surds are:

$(√a)^2 = a$ ; $√{a⋅b} = √a ⋅ √b$ ; $√{a/b} = {√a}/{√b}$

It is not possible in the Real realm of numbers to have the square root of a negative number. However, the square root of 4 has two solutions. +2 and -2.

Content © Andrew Bone. All rights reserved. Created : March 11, 2014

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