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Standard Form: Converting and Comparing
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Foundational Skills
Evaluate positive powers of 10
\[ 10^4 = \square \]
Work out the value of 10 raised to a positive whole number power.
Evaluate 10 to the power of zero
\[ 10^0 = \square \]
Recall that 10 to the power of zero equals 1.
Evaluate negative powers of 10
\[ 10^{-3} = \square \]
Work out the value of 10 raised to a negative power as a decimal.
Calculate a number multiplied by 10^0
\[ 5.2 \times 10^0 = \square \]
Calculate the value of a number multiplied by 10 to the power of zero.
Identify if a number is in standard form
\[ \text{Is } 4.5 \times 10^6 \text{ standard?} \]
Decide whether a given number is correctly written in standard form.
Converting to Standard Form: Large Numbers
2-3 digit (one sig fig) to standard form
\[ 400 \rightarrow \text{std form} \]
Convert a two or three digit number with one significant figure.
4+ digit (one sig fig) to standard form
\[ 50\,000 \rightarrow \text{std form} \]
Convert a large number with trailing zeros and one significant figure.
3-4 digit (two sig figs) to standard form
\[ 3400 \rightarrow \text{std form} \]
Convert a three or four digit number with two significant figures.
5+ digit (two sig figs) to standard form
\[ 78\,000 \rightarrow \text{std form} \]
Convert a large number with two significant figures.
Whole number (three sig figs) to standard form
\[ 4720 \rightarrow \text{std form} \]
Convert a number with three significant figures.
Decimal greater than 10 to standard form
\[ 45.3 \rightarrow \text{std form} \]
Convert a decimal number between 10 and 1000.
Converting to Standard Form: Small Numbers
Decimal 0.1-0.99 to standard form
\[ 0.45 \rightarrow \text{std form} \]
Convert a decimal between 0.1 and 0.99.
Decimal 0.01-0.099 to standard form
\[ 0.045 \rightarrow \text{std form} \]
Convert a decimal with one zero after the decimal point.
Decimal 0.001-0.0099 to standard form
\[ 0.0072 \rightarrow \text{std form} \]
Convert a decimal with two zeros after the decimal point.
Decimal 0.0001-0.00099 to standard form
\[ 0.00035 \rightarrow \text{std form} \]
Convert a decimal with three zeros after the decimal point.
Very small decimal to standard form
\[ 0.000\,006 \rightarrow \text{std form} \]
Convert a decimal with four or more zeros after the decimal point.
Converting from Standard Form
Positive power (integer coef) to ordinary
\[ 4 \times 10^3 \rightarrow \text{ordinary} \]
Convert with a whole number coefficient and positive power.
Positive power (decimal coef) to ordinary
\[ 3.5 \times 10^4 \rightarrow \text{ordinary} \]
Convert with a decimal coefficient and positive power.
Negative power to ordinary number
\[ 5.2 \times 10^{-3} \rightarrow \text{ordinary} \]
Convert a standard form number with a negative power to a small decimal.
Power of zero to ordinary number
\[ 7.25 \times 10^0 \rightarrow \text{ordinary} \]
Convert a standard form number with power zero.
Adjusting to Standard Form
Adjust when coefficient is 10 or more
\[ 45 \times 10^3 \rightarrow \text{correct} \]
Convert to proper standard form when the coefficient is 10 or greater.
Adjust when coef < 1 (positive power)
\[ 0.45 \times 10^5 \rightarrow \text{correct} \]
Convert when the coefficient is less than 1 and the power is positive.
Adjust when coef < 1 (negative power)
\[ 0.8 \times 10^{-3} \rightarrow \text{correct} \]
Convert when the coefficient is less than 1 and the power is negative.
Comparing and Ordering
Compare: same positive power
\[ 3.2 \times 10^5 \text{ vs } 7.8 \times 10^5 \]
Compare two standard form numbers with the same positive power.
Compare: same negative power
\[ 4.5 \times 10^{-3} \text{ vs } 2.1 \times 10^{-3} \]
Compare two standard form numbers with the same negative power.
Compare: different positive powers
\[ 8.5 \times 10^3 \text{ vs } 2.1 \times 10^4 \]
Compare two standard form numbers with different positive powers.
Compare: different negative powers
\[ 6.2 \times 10^{-2} \text{ vs } 9.1 \times 10^{-4} \]
Compare two standard form numbers with different negative powers.
Compare: standard form vs ordinary
\[ 4.5 \times 10^3 \text{ vs } 5200 \]
Compare a number in standard form with an ordinary number.
Order 3 numbers: positive powers
\[ \text{smallest} \rightarrow \text{largest} \]
Arrange three standard form numbers with positive powers.
Order 3 numbers: negative powers
\[ \text{smallest} \rightarrow \text{largest} \]
Arrange three standard form numbers with negative powers.
Order 4 numbers: mixed powers
\[ \text{+ve, -ve, } 10^0 \text{ powers} \]
Arrange four standard form numbers with positive, negative, and zero powers.
Order mixture: standard & ordinary
\[ \text{std form + ordinary} \]
Arrange a mixture of standard form and ordinary numbers.
Special Cases
Estimate closest power of 10
\[ 7500: 10^3 \text{ or } 10^4? \]
Identify which of two consecutive powers of 10 a number is closest to.
Contextual large number to standard form
\[ \text{Population: } 2\,350\,000 \]
Convert a large number from a real-world context.
Contextual small number to standard form
\[ \text{Bacteria: } 0.000\,004 \text{ m} \]
Convert a small number from a real-world context.
Contextual standard form to ordinary
\[ 6.5 \times 10^{-5} \text{ kg} \rightarrow \text{?} \]
Convert a standard form number from a real-world context to an ordinary number.
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