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E1.8 Standard form

Syllabus
0580–2028–2029
Topic
E1.8
Level
Extended

Recognise a valid number in standard form

Standard form writes a non-zero number as a coefficient multiplied by an integer power of 10. The coefficient carries the significant digits and the power records the scale.

A×10n,1≤A<10,n∈ZA\times10^n,\qquad 1\le A<10,\qquad n\in\mathbb Z

Part Requirement Meaning
AA at least 1 and less than 10 exactly one non-zero digit before the decimal point
10n10^n nn is an integer positive nn gives a large scale; negative nn gives a scale below 1

The expression 85.1×10485.1\times10^4 has a clear value but is not in standard form because 85.1≥1085.1\ge10. Move the decimal one place left and increase the exponent by 1: 85.1×104=8.51×10585.1\times10^4=8.51\times10^5.

A power of 10 alone does not make a representation standard form: 0.3×10−20.3\times10^{-2} is invalid because its coefficient is below 1. Standard form preserves the exact value; it is not automatically a rounded approximation.

Convert between ordinary numbers and standard form

To convert into standard form, move the decimal point until the coefficient lies from 1 inclusive to 10 exclusive, then choose the power of 10 that restores the original place value.

Ordinary number Coefficient Movement used to make it Standard form
510100000510100000 5.1015.101 8 places left 5.101×1085.101\times10^8
0.06050.0605 6.056.05 2 places right 6.05×10−26.05\times10^{-2}
0.00000003470.0000000347 3.473.47 8 places right 3.47×10−83.47\times10^{-8}

A positive exponent restores a large number by moving the decimal point right. A negative exponent restores a small number by moving it left. The exponent reverses the movement used to make the coefficient.

Standard form Apply the scale Ordinary number
4.73×1064.73\times10^6 move 6 places right 47300004730000
2.06×10−22.06\times10^{-2} move 2 places left 0.02060.0206
3.47×10−83.47\times10^{-8} move 8 places left 0.00000003470.0000000347

Do not choose the exponent by counting visible zeros alone; count place-value moves from the original decimal point. Check both value and format by converting back and confirming 1≤A<101\le A<10.

Calculate with values in standard form

In standard-form calculations, operate on coefficients and powers separately, then normalise the result so the coefficient returns to 1≤A<101\le A<10.

Operation Method Example before final normalisation
multiply multiply coefficients; add exponents (4.1×10−3)(8.9×107)=36.49×104(4.1\times10^{-3})(8.9\times10^7)=36.49\times10^4
divide divide coefficients; subtract exponents (6.39×104)÷(2.45×106)=2.608…×10−2(6.39\times10^4)\div(2.45\times10^6)=2.608\ldots\times10^{-2}
add or subtract first rewrite terms with the same power 3×105+4×104=3×105+0.4×1053\times10^5+4\times10^4=3\times10^5+0.4\times10^5

36.49×104=3.649×10536.49\times10^4=3.649\times10^5

3×105+0.4×105=3.4×1053\times10^5+0.4\times10^5=3.4\times10^5

For a power, apply it to both parts: (3×10−3)3=33×10−9=27×10−9=2.7×10−8(3\times10^{-3})^3=3^3\times10^{-9}=27\times10^{-9}=2.7\times10^{-8}.

Units must be consistent before calculating. If one atom has mass 7.95×10−237.95\times10^{-23} g, then 1 kg is 1000 g and the atom count is 1000÷(7.95×10−23)=1.257…×10251000\div(7.95\times10^{-23})=1.257\ldots\times10^{25}.

Keep full calculator precision, normalise first, and round only the final coefficient when accuracy is requested. Never add coefficients until powers match, and never add exponents when adding numbers.