Orders of magnitude
Write a positive number as a times a power of 10, with the first factor at least 1 and less than 10. The exponent of 10 is the order of magnitude
\[N = a \times 10^{n},\quad 1 \le a < 10\]
- Use standard form Rewrite the number so that the coefficient is from 1 up to, but not including, 10 \(\displaystyle 2400 = 2.4 \times 10^{3}\)
- Read the exponent The exponent on 10 is the order of magnitude \(\displaystyle 2.4 \times 10^{3} \Rightarrow n = 3\)
- Increase by orders An increase of n orders multiplies a quantity by 10 to the power n \(\displaystyle 10^{2} = 100\)
- Decrease by orders A decrease of n orders divides a quantity by 10 to the power n \(\displaystyle 7 \times 10^{5} \div 10^{2} = 7 \times 10^{3}\)
Worked example
A store receives 8 cartons. Each carton contains 350 fasteners. Find the order of magnitude of the total number of fasteners
\[8 \times 350\]
- \(\displaystyle 8 \times 350 = 2800\)Multiply the number of cartons by the number of fasteners in each carton
- \(\displaystyle 2800 = 2.8 \times 10^{3}\)Move the decimal point so the coefficient is at least 1 and less than 10
- \(\displaystyle \text{order of magnitude} = 3\)Read the exponent of 10 in the standard form
Common mistakes
Counting digits
✕\(\displaystyle 2400 = 2.4 \times 10^{3} \Rightarrow \text{order} = 4\)✓\(\displaystyle 2400 = 2.4 \times 10^{3} \Rightarrow \text{order} = 3\)
The order is the exponent in standard form, not the number of digits in the integerMultiplying the order
✕\(\displaystyle 3\text{ orders} \Rightarrow 10 \times 3 = 30\)✓\(\displaystyle 3\text{ orders} \Rightarrow 10^{3} = 1000\)
An increase of n orders uses 10 to the power of n, not 10 multiplied by nLeaving the coefficient too large
✕\(\displaystyle 2400 = 24 \times 10^{2} \Rightarrow \text{order} = 2\)✓\(\displaystyle 2400 = 2.4 \times 10^{3} \Rightarrow \text{order} = 3\)
Standard form requires a coefficient from 1 up to, but not including, 10, so the exponent must be adjusted with itDifficulty levels
introA small positive whole number is written in standard form and its order is read
standardA standard-form quantity and a derived total are classified by their powers of 10
challengeA product crossing a power-of-ten boundary is followed by an increase or decrease of orders
examA five-option choice tests the order, digit-count and power-of-ten interpretations
Questions about this topic
Why is the order of magnitude of 2400 equal to 3?+
Since 2400 has standard-form coefficient 2.4 (between 1 and 10) and exponent 3, its order of magnitude is 3
How is the order of magnitude related to the number of digits?+
For a positive whole number that is not zero, the number of digits is one more than the order of magnitude
What does an increase of two orders of magnitude do?+
It multiplies the quantity by 10 squared, which is 100. A decrease of two orders divides by 100
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 9: Variation
In this chapter
Direct variation: write the relationshipDirect variation: find the constant and missing valuesDirect variation: solve a practical problemInverse variation: write the relationshipInverse variation: find the constant and missing valuesInverse variation: solve a practical problem9B Inverse variation: Identify direct or inverse variation from tables and graphs9C Data transformations: Identify direct or inverse variation from tables and graphsSquare and reciprocal data transformationsEvaluate logarithms and antilogarithmsOrders of magnitudeCompare quantities on a log scaleLogarithmic data transformationsFit and use square, reciprocal and logarithmic models