Scientific notation for positive values
The exponent is an integer. Multiplying back checks the value; checking the coefficient checks the form
- Normalise a positive value The coefficient is at least 1 and less than 10. Zero is written as 0; these conversion questions use positive values \(\displaystyle N=a\times10^n,\quad1\le a<10\)
- Use the exponent sign For a positive value below 1, the exponent is negative. Each step up in the exponent multiplies the value by 10. \(\displaystyle 0.00037=3.7\times10^{-4}\)
- Equivalent value is not always standard form A coefficient of 10 or more must be reduced while the exponent increases \(\displaystyle 37\times10^4=3.7\times10^5\)
- Compare carefully For normalised positive values, compare exponents first. When exponents match, compare coefficients \(\displaystyle 2.8\times10^{-3}>9.1\times10^{-4}\)
Worked example
Write 0.0000486 in scientific notation
\[0.0000486\]
- \(\displaystyle a=4.86\)Place the decimal point after the first nonzero digit
- \(\displaystyle n=-5\)The point moves five places right, so the exponent is negative
- \(\displaystyle 0.0000486=4.86\times10^{-5}\)Multiplying back gives the original value
Common mistakes
Reversing the exponent sign
✕\(\displaystyle 0.00072=7.2\times10^4\)✓\(\displaystyle 0.00072=7.2\times10^{-4}\)
The original positive value is below 1.Using a non-standard coefficient
✕\(\displaystyle 52\times10^3\)✓\(\displaystyle 5.2\times10^4\)
The expressions have equal value. Only the second is normalisedDifficulty levels
introEnter the coefficient and positive exponent for a large numeral
standardConvert in either direction with positive or negative exponents
challengeFind the smallest or largest expression that is also normalised
examChoose the correct conversion or comparison result
Questions about this topic
Can two equal expressions differ in whether they are scientific notation?+
Yes. A non-standard coefficient can preserve the value, but it does not meet the required form
What does exponent zero mean?+
Multiplication by 10⁰ means multiplication by 1. For example, 4.2 × 10⁰ = 4.2
How are negative numbers written?+
Keep the minus sign outside the normalised positive magnitude, for example −3.2 × 10⁴. This practice topic concentrates on positive measurements
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 6: Indices and surds
In this chapter
Work with index notationExpress a number in prime-factor index formSimplify using index laws 1 and 2Simplify using index law 3 and the zero powerExpand using index laws 4 and 5Work with negative indicesSolve an exponential equation by matching basesCombine positive and negative integer indicesScientific notationMultiply and divide in scientific notationSI prefixes and scientific timescalesRound to decimal places and significant figuresRecord measurements and compare instrumentsRelate fractional indices to roots (optional extension)Locate exact square roots on the real number lineSimplify, compare and operate with surds (optional practice)