CourseCourseQ BankQuestion BankDocsDocuments

C1.8 Standard form

Syllabus
0580–2028–2029
Topic
C1.8
Level
Core

Recognise a valid number in standard form

Standard form writes a non-zero number as one coefficient multiplied by an integer power of 10. The coefficient shows the significant digits, while the power of 10 records the number's scale.

Aimes10n,1≤A<10,n∈ZA imes10^n,\qquad 1\le A<10,\qquad n\in\mathbb{Z}

Part Requirement Meaning
AA at least 1 but less than 10 contains exactly one non-zero digit before the decimal point
10n10^n nn is an integer positive nn scales to a large value; negative nn scales to a value between 0 and 1

85.1×10485.1\times10^4 has the correct value but is not in standard form because 85.185.1 is not less than 10. Renormalising gives 8.51×1058.51\times10^5.

For positive standard-form numbers, compare powers of 10 first. Thus 2.04×109>9.78×1082.04\times10^9>9.78\times10^8 because 10910^9 is ten times the scale of 10810^8. If powers match, compare the coefficients.

The power of 10 alone does not make a representation standard form: 0.3×10−20.3\times10^{-2} is not valid 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, reposition the decimal point so the coefficient is at least 1 and less than 10, then use the power of 10 that restores the original place value.

Ordinary number Valid coefficient Decimal-point movement Standard form
153000000153000000 1.531.53 8 places left 1.53×1081.53\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 records the reverse of the movement used to create 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

Check both value and format: the coefficient must satisfy 1≤A<101\le A<10, and converting back must reproduce every zero and significant digit of the ordinary number.

Do not choose the exponent from the number of visible zeros alone; count place-value moves from the original decimal point. Leading zeros in a small decimal are place holders, not significant digits.

Calculate with numbers in standard form

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

Operation Method Example before normalising
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 (6.4×105)÷(2.5×10−7)=2.56×1012(6.4\times10^5)\div(2.5\times10^{-7})=2.56\times10^{12}
add or subtract first rewrite both terms with the same power of 10 3×10199+2×10201=0.03×10201+2×102013\times10^{199}+2\times10^{201}=0.03\times10^{201}+2\times10^{201}

36.49imes104=3.649imes10536.49 imes10^4=3.649 imes10^5

0.03imes10201+2imes10201=2.03imes102010.03 imes10^{201}+2 imes10^{201}=2.03 imes10^{201}

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}.

Keep full calculator precision during the calculation, normalise first, and round only the final coefficient when a degree of accuracy is requested. For example, 4.6×102×6.7×105=3.082×1084.6\times10^2\times6.7\times10^5=3.082\times10^8, which is 3.1×1083.1\times10^8 to 2 significant figures.

Do not add coefficients until the powers match, and do not add exponents when adding numbers. For Core candidates, calculation with standard form is expected only on Paper 3; conversion and recognition remain part of the Topic generally.