Ordering, Standard Form | O Level Mathematics 4024 & IGCSE Mathematics 0580 | Detailed Free Notes To Score An A Star (A*)

General Principles

• Ordering means listing the given data either from highest to lowest or from lowest to highest.
• Ascending Order
  • From Lowest to Highest
  • It means you will write the lowest one first, then move on to the highest value
• Descending Order
  • From Highest To Lowest
  • This means you will move from the highest value to the lowest.
  • Write the highest value first.
  • Then go down to the lowest one.
• Separate values by commas

The Main Trouble Issue

• Ordering questions are never straight forward.
• Instead,
  • They will give you values in different forms
  • Some of the values will be a square root.
  • There will be decimals
  • Fractions and even symbols
• The trick to solve the question
  • The easiest trick to solve such questions is as follows
  • Say, you are given these numbers
  • Step 1
    • Convert all these values to decimals
      • Stay to 3 decimal places at maximum
      • Do not change a value already given in more than 3 decimal places to three decimal places
    • So
      • 10/3 = 3.33
      • 3.1582
      • 3
      • 100/33 = 3.030
      • = 3.162
  • Step 2
    • Now order them
    • If he asked for descending order
      • The largest value first i.e. 10/3, , 3.1582, 100/33, 3
    • If he asked for ascending order
      • The smallest value first i.e. 3, 100/33, 3.1582, , 10/3
    • Never try to do this question directly otherwise you will make mistakes
    • REMEMBER
      • Common mistake
      • Students usually write the final answer in the DECIMAL VALUES they have calculated instead of the original values.
      • HUGE BLUNDER.
      • Instead, use those decimal values to check which one is larger and which one is smaller
        • BUT WRITE THE ANSWER in the original values given in the question

What is the Standard Form

• Basically
  • In Mathematics of O Level, we have to go through two types of forms: the indices form and the standard form.
• In the standard form
  • You are basically expressing the number given in terms of 10.
  • In other words, you are leaving just one number (WHICH MUST NOT BE ZERO) on the LEFT side of the decimal point and taking everything else to the right side of the decimal point.
  • Based on this method, you will decide the exponential power of 10 with which the entire number has to be multiplied.
• Remember
  • With each place you move towards the left, you reduce the power of 10 by 1
  • With each place your move towards the right, you will add the power of 10 by 1.
• Let us do an example
  • Say we have a number 100.
  • We need to convert it to standard form.
  • Step 1
    • Basically, we have a whole number here.
    • It can be written as 100.0 (as there is nothing on the right side of decimal)
    • So now, we have to decide do we need to move towards the left or right to make sure there is just one number remaining on the left side of the decimal
    • Here, we need to move towards the right.
  • Step 2
    • Remember, I told you above that when we move to the right, we add 1 to the power of 10 based on the number of steps we move.
    • So first, we will write it as
  • Step 3
    • Move each step to the right and add 1 to the power of 10
    • So we have
    • 
    • Then, we move another step and we have
    • 
    • And that is completed – we are left with one number (non zero number) on the left of decimal place and the standard form has been created.

The Other Example

• For example, we are given a number 0.00000251850
• Now we follow the same steps again:
• Step 1
  • Basically, we have a decimal number here.
  • There is already material to the right of the decimal and there is a zero on the left. We need a number to the left to make it in standard form.
  • So now, we have to decide do we need to move towards the left or right to make sure there is just one number remaining on the left side of the decimal
  • Here, we need to move towards the left.
• Step 2
  • Remember, I told you above that when we move to the left, we subtract 1 to the power of 10 based on the number of steps we move.
  • So first, we will write it as 
• Step 3
  • Move each step to the right and subtract 1 to the power of 10
  • So we have
  • 
  • Then, we move another step and we have
  • 
  • Then we move another step and we have
  • 
  • Then we move another step and we have
  • 
  • Then we move another step and we have
  • 
  • Then we move another step and we have
  • 
  • And that is completed – we are left with one number on the left of decimal place that is not a zero and the standard form has been created.