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 10 Mathematics, chapter 3: Algebra and indices
In this chapter
Identify the parts of an algebraic expressionForm and evaluate algebraic expressionsSubstitute values into a formulaEvaluate an integer expressionIdentify and collect like termsMultiply and divide algebraic termsExpand brackets and simplifyFind the HCF and factorise algebraic expressionsSimplify, multiply and divide algebraic fractionsAdd and subtract algebraic fractionsIndex notation and index laws 1 and 2Index laws 3 to 5 and the zero powerNegative indices and combined index lawsScientific notationRound to decimal places or significant figuresScientific notation with significant figuresExponential growth, decay and compound interestModel a compound-interest investment or loan with a recurrence relationFind term n of a geometric sequence using the ruleInterpret and use logarithmic scalesImplement algorithms with lists