The Simplifier

Example Three 3. Solve: {1 – [1 – 1 x (1 + 1) – 1 ] of 1} ÷ 1 + 1 This has nested brackets (i.e. bracket inside another bracket and we must start from the innermost bracket). Starting from innermost bracket: {1 – [1 – 1 x (1 + 1)   – 1] of 1} ÷ 1 + 1 = {1 – [1 – 1 x 1 – 1] of 1} ÷ 1 + 1 Apply the Precedence Rule also in the Innermost Bracket:  {1 – [1 – 1 x 1 – 1] of 1} ÷ 1 + 1 = {1 – [1 – 1 – 1] of 1} ÷ 1 + 1 Applying Precedent rule from left to right inside the bracket we have: {1 – [1 – 1 – 1] of 1} ÷ 1 + 1 = {1 – [0 – 1] of 1} ÷ 1 + 1 Furthermore we have: {1 – [0 – 1] of 1} ÷ 1 + 1 = {1 – [–1] of 1} ÷ 1 + 1 Next is to consider the of in the Innermost Bracket (if it exists): {1 – (–1) of 1 } ÷ 1 + 1 = {1 – (–1) } ÷ 1 + 1 Notice that we do not write two mathematical operators together without a bracket to separate them: e.g. instead of 1  –  –  1 we write 1 – ( - 1) . Next Step finalize the operations in bracket...

Example Two 2. Evaluate: 1 of 1 – (1 – 1) + 1 ÷ 1 x 1 Start with the bracket: 1 of 1 – (1 – 1) + 1 ÷ 1 x 1 = 1 of 1 – 0 + 1 ÷ 1 x 1 Next is to handle the of ( if it exists ): 1 of 1 – 0 + 1 ÷ 1 x 1 = 1 – 0 + 1 ÷ 1 x 1 Next is to deal with Division ( if it exists ): 1 – 0 + 1 ÷ 1 x 1 = 1 – 0 + 1  x 1 Next is to handle Multiplication ( if it exists ) 1 – 0 + 1 x 1 = 1 – 0 + 1 Next is to consider Addition & Subtraction ( if it exists ) 1– 0 + 1 = 1 + 1 = 2 Keynote It is not a must that all the operators must exist in one problem but the sequence is always followed while the operator not in the operators are skipped to the next available one!

Example One 1. Simplify: 1 + 1 – 1 x 1 ÷ (1 + 1) of 1 Step 1: Recall the precedence Rule as: • Starting from left to right and • Following the BODMAS sequence  We start the part with bracket : 1 + 1 – 1 x 1 ÷ (1 + 1) of 1 = 1 + 1 – 1 x 1 ÷ 1 of 1 Next we move to the part with the of : 1 + 1 – 1 x 1 ÷ 1 of 1 = 1 + 1 – 1 x 1 ÷ 1 (Note: 1 of 1 = 1 * 1 = 1) Next we move to where we have a division : 1 + 1 – 1 x 1 ÷ 1 = 1 + 1 – 1 x 1 Next is to handle the Multiplication : 1 + 1 – 1 x 1 = 1 + 1 – 1 (Note the negative sign in front of 1 i.e. – 1 x 1) Next to be considered is Addition and Finally Subtraction : 1 + 1 – 1 = 2 – 1 = 1 Keynote This sequence is maintained even if more than one type of the same operator exists at different locations - finish with all additions  before subtractions. Applicable to all others operators!

SIMPLE EXAMPLES 1. Simplify: 1 + 1 – 1 x 1 ÷ (1 + 1) of 1 2. Evaluate:1 of 1 – (1 – 1) + 1 ÷ 1 x 1 3. Solve: {1 – [1 – 1 x (1 + 1) – 1] of 1} ÷ 1 + 1 SOLUTION Rule1: Don’t be scared Rule2: Follow the principles of operation (precedence rule). Note : do not be confused by the different terms: simplicity, evaluate or solve. For now we can assume them to all mean the same thing. Simply see them as saying “ manipulate ” the digits using the right principles until the simplest form is reached!

THE PRECEDENCE RULE The first rule of mathematics is the rule that guides us on how we respect the operators. This rule tells us which operator to consider first before the other in cases where more than just one of the operators exist together.  Note that we most often handle mathematical processes from left to right except in few special occasions and we do this by obeying a rule known as precedence rule. More generally we know this rule as BODMAS . BODMAS is an acronym for: 1. B  for Bracket ⇒ (parenthesis), [square bracket], {curly braces} 2. O for Of (*) ⇒ this a statement that tells us to also multiple 3. D for Division (÷) 4. M for Multiplication (*) 5. A for Addition (+) 6. S for Subtraction ( - ) Note also that we could be faced with roots, powers and other operators and such special operators are considered first before these ones stated above, however for now let's deal with the basics. Keynote Every mathematical problem f...

See How Much You Know Already! Do you know that you already know so much about mathematics? For instance you know that: (i) 1 + 1 = 2 (ii) 1 – 1 = 0 (iii) 1 x 1 = 1 (iv) 1 ÷ 1 = 1 (v) – 1 – 1 = -2 (a bit funny right?) (vi) – 1 + 1 = 0 (another funny one) Let’s try kids rule:  If you are owing your friend ₦1 (i.e. - ₦1) and then you borrow another ₦1 (i.e. - ₦1) from him again, how much are you owing him in total?  Answer is that you now owe him ₦2(⇒ - ₦2) What if you borrowed ₦1 from your friend ( - ₦1) yesterday and then you got ₦1 and  repaid him his  ₦1 today. How much do you have left in your pocket? And how much are you no owing? Answer: you have ₦0 left in your pocket and you now owe ₦0. If you could understand all these, then you know mathematics because mathematics begins and ends with one plus one. Keynote You already know so much about mathematics, so don't be scared!

Rule 2: Know the Mathematical Operators and their Operations Mathematical operators are simply tools that determine the behaviour of a mathematical process. The operators tell us to join, separate, etc. What we want or are given determine the tools to be used. The basic mathematical tools are: + Plus [Addition (Summation)] - Minus [(Subtraction (Difference)] x Times [(Multiplication (Product)] ÷ Divide [Division (Quotient)] There are lots of other operators, such as roots, power, trigonometric functions, etc, but knowing the basic ones is a great step forward. Keynote Start by knowing the basic operators and how they behave.