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IBMaths: AI HLDPRevision Notes1. Number & Algebra1.1 Number Toolkit1.1.1 Standard Form

Standard Form (DP IB Maths: AI HL)

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Standard Form

Standard form (sometimes called scientific notation or standard index form) gives us a way of writing very big and very small numbers using powers of 10.

Why use standard form?

  • Some numbers are too big or too small to write easily or for your calculator to display at all
    • Imagine the number 5050 , the answer would take 84 digits to write out
    • Try typing 5050 into your calculator, you will see it displayed in standard form
  • Writing very big or very small numbers in standard form allows us to:
    • Write them more neatly
    • Compare them more easily
    • Carry out calculations more easily
  • Exam questions could ask for your answer to be written in standard form

 

How is standard form written?

  • In standard form numbers are always written in the form a cross times 10 to the power of k where a and k satisfy the following conditions:
    • 1 space less or equal than a less than 10
      • So there is one non – zero digit before the decimal point
    • k space element of straight integer numbers 
      • So k must be an integer
    • k greater than 0 for large numbers
      • How many times a is multiplied by 10
    • k less than 0 for small numbers
      • How many times a is divided by 10

 

How are calculations carried out with standard form? 

  • Your GDC will display large and small numbers in standard form when it is in normal mode
    • Your GDC may display standard form as aEn
      • For example, 2.1 space cross times space 10 to the power of negative 5 end exponent will be displayed as 2.1 straight E minus 5
      • If so, be careful to rewrite the answer given in the correct form, you will not get marks for copying directly from your GDC
  • Your GDC will be able to carry out calculations in standard form
    • If you put your GDC into scientific mode it will automatically convert numbers into standard form
      • Beware that your GDC may have more than one mode when in scientific mode
      • This relates to the number of significant figures the answer will be displayed in
      • Your GDC may add extra zeros to fill spaces if working with a high number of significant figures, you do not need to write these in your answer
  • To add or subtract numbers written in the form a space cross times space 10 to the power of k without your GDC you will need to write them in full form first
    • Alternatively you can use 'matching powers of 10', because if the powers of 10 are the same, then the 'number parts' at the start can just be added or subtracted normally
      • For example  open parentheses 6.3 cross times 10 to the power of 14 close parentheses space plus space open parentheses 4.9 cross times 10 to the power of 13 close parentheses space equals space open parentheses 6.3 cross times 10 to the power of 14 close parentheses space plus space open parentheses 0.49 cross times 10 to the power of 14 close parentheses space equals space 6.79 cross times 10 to the power of 14
      • Or  open parentheses 7.93 cross times 10 to the power of negative 11 end exponent close parentheses space minus space open parentheses 5.2 cross times 10 to the power of negative 12 end exponent close parentheses space equals space open parentheses 7.93 cross times 10 to the power of negative 11 end exponent close parentheses space minus space open parentheses 0.52 cross times 10 to the power of negative 11 end exponent close parentheses space equals space 7.41 cross times 10 to the power of negative 11 end exponent
  • To multiply or divide numbers written in the form a space cross times space 10 to the power of k without your GDC you can either write them in full form first or use the laws of indices

Examiner Tip

  • Your GDC will give very big or very small answers in standard form and will have a setting which will allow you to carry out calculations in scientific notation
  • Make sure you are familiar with the form that your GDC gives answers in as it may be different to the form you are required to use in the exam

Worked example

Calculate the following, giving your answer in the form a cross times 10 to the power of k, where 1 less or equal than a less than 10 space and k element of straight integer numbers.

 

i)
3780 space cross times space 200ai-sl-1-1-1-we-1-standard-form-part-i

 

 ii)   left parenthesis 7 space cross times space 10 to the power of 5 right parenthesis space minus space left parenthesis 5 space cross times space 10 to the power of 4 right parenthesis

ai-sl-1-1-1-we-1-standard-form-part-ii

 

iii)
left parenthesis 3.6 cross times 10 to the power of negative 3 end exponent right parenthesis left parenthesis 1.1 cross times 10 to the power of negative 5 end exponent right parenthesisai-sl-1-1-1-we-1-standard-form-part-iii

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.