SL 1.1 Scientific notation (standard form)

Why standard form exists

Numbers such as the mass of a bacterium or the storage of a data centre have so many zeros that they are hard to read and easy to miscopy. Standard form packs the size of the number into a power of ten and keeps the meaningful digits in front:

\[a \times 10^{k}, \quad 1 \le a < 10, \quad k \in \mathbb{Z}\]

The exponent \(k\) tells you how many places the decimal point moved. A positive \(k\) means a big number, a negative \(k\) means a number between 0 and 1.

• \(4\,720\,000 = 4.72 \times 10^{6}\) (point moved 6 places left)
• \(0.000\,085 = 8.5 \times 10^{-5}\) (point moved 5 places right)

A negative number keeps its sign in front: \(-3.1 \times 10^{4}\) is fine; the rule \(1 \le a < 10\) applies to the size of \(a\).

Calculating in standard form

Multiply or divide the front numbers and the powers of ten separately, then tidy up so the front number is between 1 and 10 again.

Worked example. Find \((3.2 \times 10^{5}) \times (4 \times 10^{-2})\) and \(\dfrac{6 \times 10^{8}}{1.5 \times 10^{3}}\).

• \(3.2 \times 4 = 12.8\) and \(10^{5} \times 10^{-2} = 10^{3}\), giving \(12.8 \times 10^{3}\). This is not yet standard form, so rewrite \(12.8 = 1.28 \times 10^{1}\): the answer is \(1.28 \times 10^{4}\).
• \(6 \div 1.5 = 4\) and \(10^{8} \div 10^{3} = 10^{5}\), giving \(4 \times 10^{5}\).

Addition and subtraction need a common power first, or simply type the full expression into the calculator.

GDC tips

• The calculator shows standard form as E-notation: 2.5E-4 means \(2.5 \times 10^{-4}\). Never copy the E into a written answer.
• Use the dedicated EE / ×10ⁿ key rather than typing "×10^" by hand; it keeps the number as a single value and avoids bracket errors in divisions.
• Many models have a scientific display mode (often called SCI) that forces every answer into standard form with a chosen number of significant figures.

Context and rounding

Questions often ask you to convert units first (for example kilobytes to bytes, or km to m) and then write the result in standard form. Give the front number to three significant figures unless told otherwise, and keep the full calculator value for later parts of the question.

Common mistakes

• Leaving the answer as \(12.8 \times 10^{3}\) or \(0.6 \times 10^{5}\): the front number must be at least 1 and less than 10.
• Getting the sign of the exponent backwards for small numbers.
• Writing "3.4E7" on the exam paper instead of \(3.4 \times 10^{7}\).
• Rounding an intermediate value and carrying the rounding error into later parts.

Exam tips

• "Write your answer in the form \(a \times 10^{k}\) where \(1 \le a < 10\), \(k \in \mathbb{Z}\)" is a direct instruction: an equivalent value in another form loses the mark.
• If a question asks for the answer correct to the nearest whole number first and then in standard form, do both steps in that order.
• Check the size: a speck of dust should not weigh \(10^{4}\) kg.