PROGRAMME F1 ARITHMETIC.

PROGRAMME F1 ARITHMETIC

Programme F1: Arithmetic
Types of number Factors and prime numbers Fractions Decimal numbers Powers Number systems Change of base from denary to a new base

Types of number The natural numbers Numerals and place value Points on a line and order The integers Brackets Addition and subtraction Multiplication and division Brackets and precedence rules Basic laws of arithmetic Estimating and rounding

Types of number The natural numbers The first numbers we meet are the whole numbers, also called the natural numbers and these are written down using numerals.

Types of number Numerals and place value The whole numbers or natural numbers are written down using the ten numeral 0, 1, 2, …, 9 where the position of a numeral dictates the value that it represents.

Types of number Points on a line and order The natural numbers can be represented by equally spaced points on a straight line. Numbers to the left of a given number are less than (<) the given number and points to the right are greater than (>) the given number.

Types of number The integers If the straight line displaying the natural numbers is extended to the left we can plot equally spaced points to the left of zero. These points represent negative numbers which are written as the natural number preceded by a minus sign.

Types of number Brackets Brackets should be used around negative numbers to separate the minus sign attached to the number from the arithmetic operation symbol. Never write two arithmetic operation signs together without using brackets. For example:

Types of number Addition and subtraction Adding two numbers gives their sum and subtracting gives their difference. Adding a negative number is the same as subtracting its positive counterpart and subtracting a negative number is the same as adding its positive counterpart. For example:

Types of number Multiplication and division Multiplying two numbers gives their product and dividing two numbers gives their quotient. Multiplying or dividing two positive or two negative numbers gives a positive number. Multiplying or dividing a positive and a negative number gives a negative number.

Types of number Brackets and precedence rules Brackets and the precedence rules are used to remove ambiguity in a calculation. Working from the left evaluate divisions and multiplications as they are encountered. This leaves a calculation just involving addition and subtraction. Working from the left evaluate additions and subtractions as they are encountered. Brackets are evaluated first.

Types of number Basic laws of arithmetic Addition and multiplication are commutative operations Subtraction and division are not commutative operations Addition and multiplication are associative operations Subtraction and division are not associative operations Multiplication is distributed over addition and subtraction from both the left and the right Division is distributed over addition and subtraction from the right but not the left

Types of number Estimating Every calculation should at least be checked for reasonableness and this can be done by estimating the result using rounding.

Types of number Rounding An integer can be rounded to the nearest 10 as follows: If the number is less than halfway to the next multiple of 10 then the number is rounded down to the previous multiple of 10. If the number is more than halfway to the next multiple of 10 then the number is rounded up to the next multiple of 10. If the number is exactly halfway to the next multiple of 10 then the number is rounded up or down to the next even multiple of 10. The principle applies when rounding to the nearest 100, 1000, etc.

Factors and prime numbers Factors Prime numbers Prime factorization Highest common factor (HCF) Lowest common multiple (LCM)

Factors and prime numbers Factors Any two natural numbers are called the factors of their product.

Factors and prime numbers Prime numbers If a natural number has only two factors which are itself and unity, the number is called a prime number. The number 1 is not a prime number.

Factors and prime numbers Prime factorization Every natural number can be written uniquely as a product involving only prime factors. This is called the prime factorization of the number. This can be obtained by dividing by progressively increasing prime numbers.

Factors and prime numbers Highest common factor (HCF) The highest common factor (HCF) of two natural numbers is the highest factor that they have in common.

Factors and prime numbers Lowest common multiple (LCM) The smallest natural number that each one of a pair of natural numbers divides into a whole number of times is called their lowest common multiple (LCM).

Fractions Division of integers Multiplying fractions Of Equivalent fractions Dividing fractions Adding and subtracting fractions Fractions on a calculator Ratios Percentages

Fractions Division of integers A fraction is a number which is represented by one integer – the numerator – divided by another integer – the denominator (or the divisor). Because fractions are written as a ratio they are called rational numbers. Fractions are either: proper the numerator is smaller than the denominator improper the numerator is larger than the denominator or mixed written as an integer and a fraction

Fractions Multiplying fractions Two fractions are multiplied by multiplying their respective numerators and denominators independently. For example:

Fractions Of The word ‘of’ when interposed between two fractions means multiply.

Fractions Equivalent fractions Multiplying the numerator and the denominator of a fraction by the same number is equivalent to multiplying the fraction by unity. So, for example Since and represent the same number they are called equivalent fractions.

Fractions Dividing fractions Two fractions are divided by switching the numerator and the denominator of the divisor and multiplying. For example:

Fractions Adding and subtracting fractions Two fractions can only be added or subtracted immediately if they both possess the same denominator. If they do not possess the same denominator they must be written using equivalent fractions that do possess the same denominator. For example:

Fractions Fractions on a calculator The button on a calculator enables fractions to be entered and manipulated with the results given in fractional form.

Fractions Ratios If a whole number is separated into a number of fractional parts where each fraction has the same denominator, the numerators of the fractions form a ratio. For example: If a quantity of brine in a tank contains salt and water then the salt and water are said to be in the ratio

Fractions Percentages A percentage is a fraction whose denominator is 100.

Decimal numbers Division of integers Rounding Significant figures Decimal places Trailing zeros Fractions as decimals Decimals as fractions Unending decimals Unending decimals as fractions Rational, irrational and real numbers

Decimal numbers Division of integers If one integer is divided by another integer that is not one of the first integer’s factors, then the result will not be another integer but will lie between two integers. For example, using a calculator: Here the decimal point separates the units from the tenths, hundreds and thousandths. Numbers written in this way are called decimal numbers.

Decimal numbers Rounding To make the manipulation of decimal numbers more manageable in calculations it is common practice to round to a given number of decimal places or to a given number of significant figures.

Decimal numbers Significant figures Significant figures are counted from the first non-zero numeral encountered starting from the left of the number. If the first of a group of numerals to be deleted is a 5 or more the last significant numeral is increased by 1, otherwise the last significant numeral is left unaltered. If the only numeral to be deleted is a 5 then the last significant numeral is rounded to the nearest even numeral.

Decimal numbers Decimal places Decimal places are counted to the right of the decimal point. Rounding follows the same rules as those that are applied to rounding to significant figures.

Decimal numbers Trailing zeros Sometimes zeros must be inserted within a number to satisfy a condition for a specified number of either significant figures or decimal places. For example, 13.1 to three decimal places is Such zeros are called trailing zeros.

Decimal numbers Fractions as decimals A fraction can be represented as a decimal number by executing the division.

Decimal numbers Decimals as fractions A decimal number can be represented as a fraction. For example:

Decimal numbers Unending decimals Converting a fraction into its decimal form by performing the division always results in an infinite string of numerals after the decimal point. This string may consists of an infinite sequence of zeros or it may contain an infinitely repeated pattern of numerals. An infinite sequence of zeros is deleted and an infinitely repeated pattern is written as the pattern with a dot placed over the first and last numeral of the pattern. For example:

Decimal numbers Unending decimals as fractions Any decimal that displays an unending repeated pattern can be converted to its fractional form.

Decimal numbers Rational, irrational and real numbers A number that can be expressed as a fraction is called a rational number. An irrational number is one that cannot be expressed as a fraction and has a decimal form consisting of an infinite string of numerals after the decimal point that does not display a pattern. The complete collection of rational and irrational numbers form the real numbers.

Powers Raising a number to a power The laws of powers Powers on a calculator Fractional powers and roots Surds Multiplication and division by integer powers of 10 Precedence rules

Powers Raising a number to a power The arithmetic operation of raising a number to a power is devised from repetitive multiplication. The power is alternatively called the index and the number raised to the power is called the base.

Powers The laws of powers Power unity Any number raised to the power 1 equals itself. Multiplication of numbers and the addition of powers If two numbers are each written as a given base raised to a power then the product of the two numbers is equal to the same base raised to the sum of the powers.

Powers The laws of powers Division of numbers and the subtraction of powers If two numbers are each written as a given base raised to a power then the quotient of the two numbers is equal to the same base raised to the difference of the powers. Power zero Any number raised to the power zero equals unity.

Powers The laws of powers Negative powers A number raised to a negative power denotes the reciprocal. Multiplication of powers If a number is written as a given base raised to some power then the number raised to a further power is written as the given base raised to the product of the powers.

Powers Powers on a calculator Powers on a calculator can be evaluated using the xy key.

Powers Fractional powers and roots Fractional powers denote roots.

Powers Surds The surd is an alternative notation for the positive square root. The notation can be extended to cater for other roots. For example, the seventh root of 4 is written:

Powers Multiplication and division by integer powers of 10 If a decimal number is multiplied by 10 raised to an integer power, the decimal point moves the integer number of places to the right if the integer is positive and to the left if the integer is negative.

Powers Precedence rules With the introduction of the arithmetic operation of raising to a power we need to amend our precedence rules – evaluating powers is performed before dividing and multiplying.

Powers Standard form Working in standard form Using a calculator Preferred standard form Checking calculations and accuracy

Powers Standard form Any decimal number can be written as a decimal number greater than or equal to 1 and less than 10 (called the mantissa) multiplied by the number 10 raised to an appropriate power (the power being called the exponent).

Powers Working in standard form Numbers written in standard form can be multiplied or divided by multiplying or dividing the respective mantissas and adding or subtracting the respective exponents.

Powers Using a calculator Numbers given in standard form can be manipulated on a calculator by making use of the EXP key. For example, to enter the number: enter 1.234, press the EXP key and then enter the power 3.

Powers Preferred standard form Numbers given in preferred standard form are written as in standard form but with the exponent restricted to being a multiple of 3.

Powers Checking calculations and accuracy When performing calculations involving decimal numbers it is always a good idea to check that your result is reasonable and that an arithmetic blunder or an error in using the calculator has not been made. This can be done using standard form. Any calculation involving measured values will not be accurate to more significant figures than the least number of significant figures in any measurement.

Number systems Denary Binary Octal Duodecimal Hexadecimal

Number systems Denary The denary system is our basic system in which quantities large or small can be represented by use of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 together with appropriate place values according to their positions. The place values are powers of 10 – the denary system has a base of 10.

Number systems Binary The binary system uses just the two symbols 0 and 1 together with appropriate place values according to their positions. The place values are powers of 2 – the binary system has a base of 2.

Number systems Octal The octal system uses the symbols 0, 1, 2, 3, 4, 5, 6, 7 together with appropriate place values according to their positions. The place values are powers of 8 – the octal system has a base of 8.

Number systems Duodecimal The duodecimal system uses the symbols together with appropriate place values according to their positions. The place values are powers of 12 – the duodecimal system has a base of 12.

Number systems Hexadecimal The hexadecimal system uses the symbols: together with appropriate place values according to their positions. The place values are powers of 16 – the hexadecimal system has a base of 16.

Change of base from denary to a new base
Denary to binary This can be done by repeated division by 2, noting the remainder in each division. This continues until a final 0 quotient is obtained. The resultant binary form is then given by the list of remainders in reverse order to that in which they were obtained.

Denary to octal This can be done by repeated division by 8, noting the remainder in each division. This continues until a final 0 quotient is obtained. The resultant octal form is then given by the list of remainders in reverse order to that in which they were obtained.

Denary to duodecimal This can be done by repeated division by 12, noting the remainder in each division. This continues until a final 0 quotient is obtained. The resultant duodecimal form is then given by the list of remainders in reverse order to that in which they were obtained.

Denary decimal to octal The integer part of the decimal number can be written in octal form in the usual manner. The decimal part is then multiplied by 8 and the whole-number numeral that results is noted. This is repeated but only multiplying the decimal part and again, noting the whole number numeral that results. This continues until the required accuracy and the octal form of the decimal part then consists of the whole number numerals written in the order in which they were obtained.

Denary decimal to duodecimal The integer part of the decimal number can be written in duodecimal form in the usual manner. The decimal part is then multiplied by 12 and the whole-number numeral that results is noted. This is repeated but only multiplying the decimal part and again, noting the whole number numeral that results. This continues until the required accuracy and the octal form of the decimal part then consists of the whole number numerals written in the order in which they were obtained.

Use of octals as an intermediary step This gives an easy way of converting denary numbers into binary and hexadecimal forms. First we change the denary number into octal form in the usual way. Binary The numerals in the octal form are each written in binary form to give the required conversion.

Use of octals as an intermediary step This gives an easy way of converting denary numbers into binary and hexadecimal forms. First we change the denary number into octal form in the usual way. Hexadecimal The binary digits are arranged in groups of four starting from the decimal point. Each group of four binary digits is then converted into the appropriate hexadecimal numeral to give the required conversion into hexadecimal form.

Reverse method Starting with an hexadecimal number each numeral can be written in binary form. The binary form is then separated into groups of three binary digits starting from the decimal point. The octal numerals are then written down for each group of three binary digits to give the octal form. The octal number is then converted into denary form.

Learning outcomes Carry out the basic rules of arithmetic with integers Write a natural number as a product of prime numbers Find the highest common factor and lowest common multiple of two natural numbers Check the result of a calculation making use of rounding Manipulate fractions, ratios and percentages Manipulate decimal numbers Manipulate powers Use standard or preferred standard form and complete a calculation to the required level of accuracy Understand the construction of various number systems and convert from one number system to another

PROGRAMME F1 ARITHMETIC.

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