Functions, equations and algebra Basic Arithmetic Notation and operations

Functions equations and algebra This week we start the 2 nd part of the course, examining algebra Algebra is the most basic form of formal mathematics It allows the symbolic representation of numbers {1,2,3,…} in terms of variables {a,b,c,…} It allows logical operations to be carried out on these (+,-,×,÷)

Functions equations and algebra Today, we will go over the basic rules of arithmetic No variables will be introduced, we will only use numbers in the calculations The exercises for next week are intended to get you used to manipulating them The slides are meant to provide list of properties you can go back to for reference

Functions equations and algebra Operations and their properties Performing calculations The ordering of operations

Operations and their properties We will use two main groups of operations There are in fact only 2 “operations”, the other two being simply their inverse Addition (+) and subtraction (-) Multiplication (×) and division (÷)

Operations and their properties Addition (+) and subtraction (-) Addition is the basic operation of arithmetic It takes two numbers to give their sum: 8+5=13 Subtraction is the opposite operation 13-8=5 In fact it helps to imagine that subtraction is simply addition of a negative number 13+(-8)=5

Operations and their properties Addition (+) and subtraction (-) Addition is Commutative The ordering of the numbers doesn’t matter 8+5=5+8 Addition is Associative The ordering of the operation doesn’t matter (4+8)+5=4+(8+5)

Operations and their properties Addition (+) and subtraction (-) Subtraction is NOT Commutative The ordering of the numbers matters 8-5≠5-8 Subtraction is NOT Associative The ordering of the operation matters (4-8)-5 ≠ 4-(8-5)

Operations and their properties Multiplication (×) and division (÷) Multiplication is Commutative The ordering of the numbers doesn’t matter 8 × 5=5 × 8 Multiplication is Associative The ordering of the operation doesn’t matter (4 × 8) × 5=4 × (8 × 5)

Operations and their properties Multiplication (×) and division (÷) Multiplication has a third property Multiplication is Distributive 5 × (4 + 8) = 5 × 4+5 × 8

Operations and their properties Multiplication (×) and division (÷) Division is NOT Commutative The ordering of the numbers matters 8 ÷ 4 ≠ 4 ÷ 8 Division is NOT Associative The ordering of the operation matters (4 ÷ 8) ÷ 5 ≠ 4 ÷ (8 ÷ 5)

Functions equations and algebra Operations and their properties Performing calculations The ordering of operations

We have seen that the 2 basic operations of multiplication and addition are commutative and additive Their inverses (Subtraction and Division) are not Furthermore, multiplication is distributive over addition (which is where groupings come in) This influences the order of calculation. So when an equation or calculations contain several different operations, they have to be undertaken in a given order

The ordering of operations A grouping is the isolation of a set of operations using parentheses or fraction bars 1 st : Always carry out the operations in the group If several groups are embedded, start with the innermost and work outwards

The ordering of operations 2 nd : Within the same group, carry out the multiplications / divisions 3 rd : Within the same group, finish with the additions / subtractions

Functions equations and algebra Operations and their properties Performing calculations The ordering of operations

Performing calculations This section will simply present an example you can follow when practising with the exercises Each line of calculation in the following example is a step in the process. The general rule : Use one step per type of operation carried out Take as many steps as you feel you need (i.e. take your time) to avoid making mistakes.

Performing calculations Let’s start with an arithmetic calculation and outline the steps Step 1 : carry out additions in the parenthesis and the fractions

Performing calculations Step 2: Carry out the multiplications in the two fractions Step 3 : Carry out additions in the two fractions

Performing calculations Step 4: Carry out the two divisions Step 5 : Carry out the multiplication Step 6 : Carry out the addition