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Scientific Notation

Some numbers are far too big or far too small to write out in full. Scientific notation, also called standard form, gives you a compact, tidy way to handle both.

MYP 4Standard MathsNumberCriteria A · C~9 min read

Written by Fayek Jawad Haider · MYP student

What standard form is

Scientific notation writes any number in the form

\[ a \times 10^{n} \]

where \(a\) is a number between 1 and 10 (it can equal 1 but must be less than 10), and \(n\) is a whole number. The power of ten does the heavy lifting: a positive \(n\) makes the number large, a negative \(n\) makes it small.

Standard form
A number written as \(a \times 10^n\) with \(1 \le a < 10\) and \(n\) an integer. Also called scientific notation.

Check the front number

The value of \(a\) must sit between 1 and 10. So \(45 \times 10^3\) is not in standard form; tidy it to \(4.5 \times 10^4\).

Converting both ways

Going into standard form, count how many places you move the decimal point to leave one non-zero digit in front of it. That count is the power. Move left for a big number (positive power), move right for a small number (negative power).

Worked example

Write \(45300\) and \(0.0067\) in standard form.

1
For \(45300\), move the point so only the 4 sits in front: \(4.5300\). That is 4 places to the left, so the power is \(+4\): \(4.53 \times 10^4\).
2
For \(0.0067\), move the point right until it is just after the 6: \(6.7\). That is 3 places to the right, so the power is \(-3\): \(6.7 \times 10^{-3}\).
\(45300 = 4.53 \times 10^{4}\), \(\quad 0.0067 = 6.7 \times 10^{-3}\)

Going out of standard form, just do the opposite: a power of \(+4\) moves the point 4 places right (filling with zeros), a power of \(-3\) moves it 3 places left.

Simple calculations

To multiply or divide numbers in standard form, handle the front numbers and the powers of ten separately. Multiply or divide the \(a\) values, then use the index laws (add the powers when multiplying, subtract when dividing).

Worked example

Work out \((2 \times 10^{5}) \times (4 \times 10^{3})\).

1
Multiply the front numbers: \(2 \times 4 = 8\).
2
Add the powers of ten: \(10^{5} \times 10^{3} = 10^{5+3} = 10^{8}\).
\(8 \times 10^{8}\)

Where this is assessed

Standard form draws on Criterion A and Criterion C (correct notation). If a calculation leaves the front number outside 1 to 10, for example \(60 \times 10^6\), fix it: \(6 \times 10^7\). Answers are only counted as standard form when \(1 \le a < 10\).

Check yourself

1. Write \(92000\) in standard form. +

Move the point so only the 9 is in front: \(9.2\), a shift of 4 places left. So \(92000 = \mathbf{9.2 \times 10^{4}}\).

2. Write \(3.1 \times 10^{-2}\) as an ordinary number. +

A power of \(-2\) moves the point 2 places left: \(3.1 \rightarrow 0.031\). So the number is \(\mathbf{0.031}\).

3. Work out \((6 \times 10^{7}) \div (3 \times 10^{2})\). +

Divide the front numbers: \(6 \div 3 = 2\). Subtract the powers: \(10^{7} \div 10^{2} = 10^{5}\). So the answer is \(\mathbf{2 \times 10^{5}}\).

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