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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes1. Number & AlgebraPartial Fractions, Indices & Standard FormStandard Form

Standard Form (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Mark Curtis

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Standard Form

What is standard form?

  • Standard form means writing a number in the form

a cross times 10 to the power of n

  • where:

    • 1 less or equal than a less than 10 (a is between 1 and 10)

    • n greater than 0 (n is positive) for large numbers

    • n less than 0 (n is negative) for small numbers

How do I write numbers in standard form?

  • To write large numbers like 3 240 000 in standard form

    • a equals 3.24

    • n equals 6 as you multiply 3.24 by 10 six times to 3 240 000

      • the decimal point jumps 6 places to the right (positive)

    • 3.24 cross times 10 to the power of 6

  • To write small numbers like 0.000567 in standard form

    • a equals 5.67

    • n equals negative 4 as you divide 5.67 by 10 four times to get 0.000567

      • the decimal point jumps 4 places to the left (negative)

    • 5.67 cross times 10 to the power of negative 4 end exponent

How do I multiply in standard form?

  • Multiply separately the "a" parts and the "ten" parts

    • If the new "a" part is not in the range 1 less or equal than a less than 10 then write it in standard form

  • Write the final answer in the form a cross times 10 to the power of n

    • e.g. open parentheses 3 cross times 10 squared close parentheses cross times open parentheses 4 cross times 10 to the power of 5 close parentheses

      • 3 cross times 4 equals 12 but 1 less or equal than a less than 10 so write in standard form: 12 equals 1.2 cross times 10 to the power of 1

      • Tens: 10 squared cross times 10 to the power of 5 equals 10 to the power of 2 plus 5 end exponent equals 10 to the power of 7 (add the powers)

      • This gives 1.2 cross times 10 to the power of 1 cross times 10 to the power of 7

      • 1.2 cross times 10 to the power of 8

How do I divide in standard form?

  • Divide separately the "a" parts and the "ten" parts

    • If the new "a" part is not in the range 1 less or equal than a less than 10 then write it in standard form

  • Write the final answer in the form a cross times 10 to the power of n

    • e.g. open parentheses 2 cross times 10 to the power of negative 3 end exponent close parentheses divided by open parentheses 8 cross times 10 to the power of negative 5 end exponent close parentheses

      • 2 divided by 8 equals 0.25 but 1 less or equal than a less than 10 so write in standard form: 0.25 equals 2.5 cross times 10 to the power of negative 1 end exponent

      • Tens: 10 to the power of negative 3 end exponent divided by 10 to the power of 5 equals 10 to the power of negative 3 minus open parentheses negative 5 close parentheses end exponent equals 10 squared (subtract the powers)

      • This gives 2.5 cross times 10 to the power of negative 1 end exponent cross times 10 squared equals 2.5 cross times 10 to the power of negative 1 plus 2 end exponent

      • 2.5 cross times 10 to the power of 1

How do I add or subtract in standard form?

  • First find the highest power of 10 and adjust the other power of 10 to be the same

    • e.g. for open parentheses 4 cross times 10 to the power of 50 close parentheses plus open parentheses 2 cross times 10 to the power of 48 close parentheses the highest power is 10 to the power of 50

      • 2 cross times 10 to the power of 48 has a power 100 times smaller, so in terms of 10 to the power of 50 it is 0.02 cross times 10 to the power of 50

  • Then add (or subtract) the "like powers of 10"

    • open parentheses 4 cross times 10 to the power of 50 close parentheses plus open parentheses 0.02 cross times 10 to the power of 50 close parentheses equals open parentheses 4 plus 0.02 close parentheses cross times 10 to the power of 50

      • 4.02 cross times 10 to the power of 50

  • With negative powers, still find the highest power of 10 (a larger negative number is smaller)

    • e.g. for open parentheses 8 cross times 10 to the power of negative 20 end exponent close parentheses minus open parentheses 5 cross times 10 to the power of negative 21 end exponent close parentheses the highest power is 10 to the power of negative 20 end exponent

      • 5 cross times 10 to the power of negative 21 end exponent has a power 10 times smaller, so in terms of 10 to the power of negative 20 end exponent it is 0.5 cross times 10 to the power of negative 20 end exponent

    • open parentheses 8 cross times 10 to the power of negative 20 end exponent close parentheses minus open parentheses 0.5 cross times 10 to the power of negative 20 end exponent close parentheses equals open parentheses 8 minus 0.5 close parentheses cross times 10 to the power of negative 20 end exponent

      • 7.5 cross times 10 to the power of negative 20 end exponent

Examiner Tips and Tricks

You can use your GDC to multiply, divide, add and subtract numbers in standard form.

Worked Example

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

 

i) 3780 space cross times space 200

ai-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 parenthesis

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

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    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.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.