Standard Form

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Standard Form

Standard form is a shorthand way of expressing VERY LARGE or VERY SMALL numbers. There are 6 key skills that you need to learn.

Standard from to ordinary numbers and ordinary numbers to standard form are two skills you will be taught.

Make sure you are happy with the following topics before continuing:

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What is Standard Form?

We always write a number in standard form exactly like this:

A×10ntextcolor{red}{A}timestextcolor{blue}{10}^textcolor{limegreen}{n}

The Three Key Rules

1)   Atextcolor{red}{A} must be any between 1mathbf{1} and 10mathbf{10}, in other words, 1≤A<101leq A<10.

2)   Standard form is always 10textcolor{blue}{10} to the power of something (ntextcolor{limegreen}{n})

3)   ntextcolor{limegreen}{n} must be a whole number, this is the number of places the decimal point moves.

e.g. 4.2×105textcolor{red}{4.2}timestextcolor{blue}{10}^textcolor{limegreen}{5}

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Skill 1: Standard Form into Large Numbers

Example: Express 4.2×1054.2timestextcolor{limegreen}{10^5} as a number not in standard form.

Standard Form to Large Numbers
Standard Form to Large Numbers

Firstly, recall that 105=10×10×10×10×10textcolor{limegreen}{10^5}=textcolor{limegreen}{10}timestextcolor{limegreen}{10}timestextcolor{limegreen}{10}timestextcolor{limegreen}{10}timestextcolor{limegreen}{10}.

Then, we get  4.2×105=4.2×10×10×10×10×104.2times10^5=4.2times10times10times10times10times10

Multiplying by 1010 means moving the decimal place to the right, here we must do it 55 times:

Step 1: Write 4.24textcolor{red}{.}2 out and move the decimal place 5textcolor{red}{5} jumps to the right.

Step 2: Add 0textcolor{limegreen}{mathbf{0}}‘s to fill in the spaces created as the decimal point has moved.

Step 3: Remove the original decimal point.

This gives the answer to be:

4.2×105=4200004.2times10^{5}=420000

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Skill 2: Standard Form into Small Numbers

Example: Write 2.8×10−42.8timestextcolor{limegreen}{10^{-4}} in decimal notation.

Standard Form to Small Numbers
Standard Form to Small Numbers

This is different because the power is negative, but it’s actually no harder. We know,

10−4=1104textcolor{limegreen}{10^{-4}}=textcolor{limegreen}{dfrac{1}{10^4}}

This means we are dividing by 10textcolor{limegreen}{text{dividing by }10} four timestextcolor{limegreen}{text{four times}} which means we move the decimal point 44 spaces to the left.

Step 1: Write 2.82textcolor{red}{.}8 out and move the decimal place 4textcolor{red}{4} jumps to the left.

Step 2: Add 0textcolor{limegreen}{mathbf{0}}‘s to fill in the space created as the decimal point has moved.

Step 3: Remove the original decimal point.

Therefore, we have concluded that

2.8×10−4=0.000282.8times10^{-4}= 0.00028

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Skill 3: Writing Large Numbers in Standard Form

Example: Write 56,700,00056,700,000 in standard form.

Step 1: Move the decimal point to the left until the number becomes 5.675.67 (1≤A<10)(1leq A<10)

Large Numbers to Standard Form

Step 2: Count the number of times the decimal point has moved to the left, this will become our power (ntextcolor{limegreen}{n}), in this case 7textcolor{limegreen}{7}.

Step 3: We have moved to the left meaning it will be +7textcolor{limegreen}{+7} not −7-7

So,

56,700,000=5.67×10756,700,000 = 5.67times10^{textcolor{limegreen}{7}}

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Skill 4: Writing Small Numbers in Standard Form

Example: Write 0.00000990.0000099 in standard form.

Step 1: Move the decimal point to the right until the number becomes 9.99.9 (1≤A<10)(1leq A<10)

Small Numbers to Standard Form

Step 2: Count the number of times the decimal point has moved to the right, this will become our power (ntextcolor{limegreen}{n}), in this case 6textcolor{limegreen}{6}.

Step 3: We have moved to the right meaning it will be −6textcolor{limegreen}{-6} not +6+6

So,

0.0000099=9.9×10−60.0000099 = 9.9times10^{textcolor{limegreen}{-6}}

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Skill 5: Multiplying Standard Form

Example: Find the standard form value of (3×108)×(7×104)(3times10^8)times(7times10^4), without using a calculator.

Step 1: Change the order around of the things being multiplied.

(3×108)×(7×104)=3×7×108×104(textcolor{red}{3}timestextcolor{limegreen}{10^8})times(textcolor{red}{7}timestextcolor{limegreen}{10^4})=textcolor{red}{3}timestextcolor{red}{7}timestextcolor{limegreen}{10^8}timestextcolor{limegreen}{10^4}

Step 2: Multiply the numbers and the powers out separately.

(3×7)×(108×104)=21×1012(textcolor{red}{3}timestextcolor{red}{7})times(textcolor{limegreen}{10^8}timestextcolor{limegreen}{10^4})=textcolor{red}{21}timestextcolor{limegreen}{10^{12}}

Step 3: Convert the number at the front to standard form if necessary (1≤A<101 leq A < 10).

This answer is not in standard form (2121 is not between 11 and 1010), and we need it to be. Fortunately, if we recognise that 21=2.1×10textcolor{red}{21}=textcolor{red}{2.1}textcolor{limegreen}{times10}, then we get that

21×1012=2.1×10×1012=2.1×1013textcolor{red}{21}textcolor{limegreen}{times10^{12}}=textcolor{red}{2.1}textcolor{limegreen}{times10}textcolor{limegreen}{times10^{12}}=textcolor{red}{2.1}textcolor{limegreen}{times10^{13}}

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Skill 6: Dividing Standard Form

Example: Find the standard form value of (8×10−5)÷(2×106)(textcolor{red}{8}timestextcolor{limegreen}{10^{-5}})div(textcolor{red}{2}timestextcolor{limegreen}{10^6}), without using a calculator.

Step 1: Break up the problem and change the order of how we divide things.

8×10−52×106=82×10−5106=4×10−11dfrac{textcolor{red}{8}timestextcolor{limegreen}{10^{-5}}}{textcolor{red}{2}timestextcolor{limegreen}{10^6}}=dfrac{textcolor{red}{8}}{textcolor{red}{2}}timesdfrac{textcolor{limegreen}{10^{-5}}}{textcolor{limegreen}{10^6}}=textcolor{red}{4}timestextcolor{limegreen}{10^{-11}}

Step 2: Convert the number at the front to standard form if necessary (1≤A<101 leq A < 10).

4textcolor{red}{4} is between 11 and 1010, so this answer is in standard form, and so we are done.

(Remember: 10−5−106=10−5−6=10−1110^{-5} – 10^6 = 10^{-5-6} = 10^{-11})

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Standard Form Example Questions

Question 1: What is 1.15×10−61.15times10^{-6} in decimal notation?

[1 mark]

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The power is negative, so this is going to be a very small number. As the power of ten is −6-6, we want to divide the number 1.151.15 by 1010 six times, and so we will move the decimal point six places to the left.

 

1.15×10−6=0.000001151.15times10^{-6}=0.00000115.

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Question 2: What is 5,980,0005,980,000 in standard form?

[1 mark]

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In this case, the power of 1010 is going to be positive.

So, if we move the decimal point in 5,980,0005,980,000 to the left six places it becomes 5.985.98. Therefore, we get that,

 

5,980,000=5.98×1065,980,000=5.98times10^{6}

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Question 3: What is 0.00680.0068 in standard form?

[1 mark]

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By considering the position where the first non-zero digit is compared to the units column we find,

 

0.0068=6.8×10−30.0068=6.8times10^{-3}

 

as the 66 is 33 places away from the units column.

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Question 4: Calculate 5,600,000÷8005,600,000div800

Give your answer in standard form.

[3 marks]

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First, write each of the numbers in standard form i.e.  5.6×1065.6times10^6 and 8×1028times10^2

 

(5.6×106)÷(8×102)=(5.6÷8)×(106÷102)(5.6times10^6)div(8times10^2)=(5.6div8)times(10^6div10^2)

 

Using the formula 10a÷10b=10a−b10^adiv10^b=10^{a-b} we can rewrite the eqaution as,

 

 (5.6÷8)×106−2=0.7×104(5.6div8)times10^{6-2}=0.7times10^4

 

Standard form requires the number be between 11 and 1010, so adjusting by a factor of 1010, we have,

 

0.7×104=7×10−1×104=7×1030.7times10^4=7×10^{-1} times10^4=7times10^3

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Question 5: Calculate (2.5×104)×(6×10−9)(2.5times10^{4})times(6times10^{-9}).

Give your answer in standard form.

[3 marks]

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We will split up this multiplication, multiplying the initial numbers together and the powers of 1010 together separately. Firstly,

 

2.5×6=152.5times6=15.

 

Secondly, using the multiplication law of indices,

 

104×10−9=104+(−9)=10−510^4times10^{-9}=10^{4+(-9)}=10^{-5}

So, we get

 

(2.5×104)×(6×10−9)=2.5×6×104×10−9=15×10−5(2.5times10^{4})times(6times10^{-9}) =2.5times6times10^4times10^{-9}\ =15times10^{-5}

 

Standard form requires the number be between 11 and 1010, so adjusting by a factor of 1010, we have, 

 

15×10−5=1.5×10×10−5=1.5×10−415times10^{-5}=1.5times10times10^{-5}=1.5times10^{-4}

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Specification Points Covered

Number – 9. calculate with and interpret standard form A×10nA times 10^n, where 1≤A<101 leq A < 10 and nn is an integer

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