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

Standard form

What is standard form?

• Standard form means writing a number in the form

$a\times {10}^n$

• where:

  • $1\le a<10$ ($a$ is between 1 and 10)

  • $n>0$ ($n$ is positive) for large numbers

  • $n<0$ ($n$ is negative) for small numbers

How do I write numbers in standard form?

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

  • $a=3.24$

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

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

  • $3.24\times {10}^6$

• To write small numbers like 0.000567 in standard form

  • $a=5.67$

  • $n=-4$ as you divide 5.67 by 10 four times to get 0.000567

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

  • $5.67\times {10}^{-4}$

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 $1\le a<10$ then write it in standard form

• Write the final answer in the form $a\times {10}^n$

  • e.g. $\left(3\times {10}^2\right)\times \left(4\times {10}^5\right)$

    • $3\times 4=12$ but $1\le a<10$ so write in standard form: $12=1.2\times {10}^1$

    • Tens: ${10}^2\times {10}^5={10}^{2+5}={10}^7$ (add the powers)

    • This gives $1.2\times {10}^1\times {10}^7$

    • $1.2\times {10}^8$

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 $1\le a<10$ then write it in standard form

• Write the final answer in the form $a\times {10}^n$

  • e.g. $\left(2\times {10}^{-3}\right)÷\left(8\times {10}^{-5}\right)$

    • $2÷8=0.25$ but $1\le a<10$ so write in standard form: $0.25=2.5\times {10}^{-1}$

    • Tens: ${10}^{-3}÷{10}^{-5}={10}^{-3-\left(-5\right)}={10}^2$ (subtract the powers)

    • This gives $2.5\times {10}^{-1}\times {10}^2=2.5\times {10}^{-1+2}$

    • $2.5\times {10}^1$

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 $\left(4\times {10}^{50}\right)+\left(2\times {10}^{48}\right)$ the highest power is ${10}^{50}$

    • $2\times {10}^{48}$ has a power 100 times smaller, so in terms of ${10}^{50}$ it is $0.02\times {10}^{50}$

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

  • $\left(4\times {10}^{50}\right)+\left(0.02\times {10}^{50}\right)=\left(4+0.02\right)\times {10}^{50}$

    • $4.02\times {10}^{50}$

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

  • e.g. for $\left(8\times {10}^{-20}\right)-\left(5\times {10}^{-21}\right)$ the highest power is ${10}^{-20}$

    • $5\times {10}^{-21}$ has a power 10 times smaller, so in terms of ${10}^{-20}$ it is $0.5\times {10}^{-20}$

  • $\left(8\times {10}^{-20}\right)-\left(0.5\times {10}^{-20}\right)=\left(8-0.5\right)\times {10}^{-20}$

    • $7.5\times {10}^{-20}$