NUMBER SYSTEMS

Objectives Identify some different number systems Round-up numbers and correct them to significant figures Carry out calculations involving the processes of addition,

subtraction, multiplication and division Manipulate negative numbers Fractions down to their simplest terms Change improper fractions to mixed numbers or integers,

and vice versa Add, subtract, multiply and divide fractions and mixed

numbers Carry out calculations using decimals Calculate ratios and percentages, carry out calculations using

percentages and divide a given quantity according to a ratio Explain the terms index, power, root, reciprocal and factorial.

Denary Number System Because ten figures (i.e. symbols 0 to 9) are used, we call

this number system the base ten system or denary system.

Numbers which are more than zero are positive. We indicate a negative number by a minus sign (-). A common example of the use of negative numbers is in the measurement of temperature – for example, 20°C (i.e. 20°C below zero).

We could indicate a positive number by a plus sign (+), but in practice this is not necessary and we adopt the convention that, say, 73 means 73.

Roman Number System

We have seen that position is very important in the number system that we normally use. This is equally true of the Roman numerical system, but this system uses letters instead of figures.

Roman numerals are in occasional use in, for example, the dates on film productions, tabulation, house numbers, etc.

Binary System

As we have said, our numbering system is a base ten system. By contrast, computers use base two, or the binary system. This uses just two symbols (figures), 0 and 1.

This system is necessary because computers can only easily recognise two figures – zero or one, corresponding to "components" of the computer being either switched off or on. When a number is transferred to the computer, it is translated into the binary system.

NUMBER SYSTEMS

Denary Number System Roman Number System Binary System

NUMBERS – APPROXIMATION AND INTEGERS

Approximation (a) Rounding off (b) Significant figures Integers

ARITHMETIC

Addition Subtraction Multiplication Division

The Rule of Priority

Division and multiplication are carried out before addition and subtraction.

Brackets

Brackets are used extensively in arithmetic We use brackets to separate particular elements in the

calculation so that we know exactly which operations to perform on which numbers.

We can put brackets within brackets to further separate elements within the calculation and clarify how it is to be performed.

The rule of priority explained above now needs to be slightly modified. It must be strictly followed except when brackets are included. In that case, the contents of the brackets are evaluated first. If there are brackets within brackets, then the innermost brackets are evaluated first.

DEALING WITH NEGATIVE NUMBERS

Addition and Subtraction

FRACTIONS

When two or more whole numbers are multiplied together, the product is always another whole number. In contrast, the division of one whole number by another does not always result in a whole number

The remainder is not a whole number, but a part of a whole number – so, in each case, the result of the division is a whole number and a fraction of a whole number. The fraction is expressed as the remainder divided by the original divisor

(a) The numerator is larger than the denominator :In this case, the fraction is known as an improper fraction. The result of dividing out an improper fraction is a whole number plus a part of one whole

(b) The denominator is larger than the numerator :In this case, the fraction is known as a proper fraction. The result of dividing out a proper fraction is only a part of one whole and therefore a proper fraction has a value of less than one.

(c) The numerator is the same as the denominator :In this case, the expression is not a true fraction. The result of dividing out is one whole part and no remainder.

Basic Rules for Fractions

Cancelling down = Changing the denominator to required

number

Changing an improper fraction into a whole or mixed number

Changing a mixed number into an improper fraction

Adding and Subtracting with Fractions

(a) Proper fractions

Mixed numbers

Multiplying and Dividing Fractions (a) Multiplying a fraction by a whole

number

(b) Dividing a fraction by a whole number

(c) Multiplying one fraction by another

(d) Dividing by a fraction

(e) Dealing with mixed numbers

(f) Cancelling during multiplication

DECIMALS

Decimals are an alternative way of expressing a particular part of a whole. The term "decimal" means "in relation to ten", and decimals are effectively fractions expressed in tenths, hundredths, thousandths, etc.

Decimals do not have a numerator or a denominator. Rather, the part of the whole is shown by a number following a decimal point:

Arithmetic with Decimals

(a) Adding and subtracting : (b) Multiplying decimals (c) Division with decimals (d) Converting fractions to decimals, and vice

versa

PERCENTAGES

A percentage is a means of expressing a fraction in parts of a hundred – the words "per cent“ simply mean "per hundred", so when we say "percentage", we mean "out of a hundred".

Percentages are used extensively in many aspects of business since they are generally more convenient and straightforward to use than fractions and decimals

Arithmetic with Percentages

Take an example: if Peter spends 40% of his income on rent and 25% on household expenses, what percentage of his income remains?

Percentages and Fractions

Percentages and Decimals Calculating Percentages

RATIOS

A ratio is a way of expressing the relationship between two quantities. It is essential that the two quantities are expressed in the same units of measurement – pence, number of people, etc. – or the comparison will not be valid.

Note that the ratio itself is not in any particular unit – it just shows the relationship between quantities of the same unit

We use a special symbol (the colon symbol : ) in expressing ratios, so the correct form of showing the above ratio would be: 200: 87 (pronounced "200 to 87")

They should always be reduced to their lowest possible terms

Ratios should always be expressed in whole numbers. There should not be any fractions or decimals in them.

Consider the case of three partners – X, Y and Z – who share the profits of their partnership in the ratio of 3 : 1 : 5. If the profit for a year is 18,000, how much does each partner receive?

Indices

Indices are found when we multiply a number by itself one or more times. The number of times that the multiplication is repeated is indicated by a superscript number to the right of the number being multiplied

Expressing Numbers in Standard Form

The exploration of indices leads us on to this very useful way of expressing numbers. Working with very large (or very small) numbers can be difficult, and it is helpful if we can find a different way of expressing them which will make them easier to deal with. Standard form provides such a way.

93,825,000,000,000 843,605,000,000,000

Factorials

A factorial is the product of all the whole numbers from a given number down to one.

For example, 4 factorial is written as 4! (and is sometimes read as "four bang") and is equal to 4 x3x2x1

You may encounter factorials in certain types of statistical operations.

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