LEMMA

Algebra basics

Scientific notation

ALEKS placement

One nonzero digit before the decimal point, times a power of ten.

What this covers

a×10n,1≤∣a∣<10
  • Move the decimal until exactly one nonzero digit sits in front of it.
  • Each move to the left makes the number ten times smaller, so add one to the exponent to pay it back.
  • Each move to the right makes it ten times bigger, so the exponent drops by one — into the negatives for small numbers.
  • Multiplying adds the exponents; dividing subtracts them.
  • Only one digit in front. 27 × 10³ still has a ten hiding inside, so move once more: 2.7 × 10⁴.
  • Read it back as a size check: a negative exponent should look small, a positive one large.

Worked example

Worked example

Multiply (4.5 × 10⁵)(6 × 10⁻²).

  1. 27×103Multiply 4.5 · 6 and add 5 + (-2).
  2. 2.7×10427 is out of range, so move one place and bump the exponent.
Another worked example

Write 0.00045 in normalized scientific notation.

  1. 0.00045=4.5⋅10−4Moving the coefficient four places right requires the factor 10⁻⁴ to preserve the number.
  2. 1≤∣4.5∣<10The coefficient satisfies the normalized range.

A common mistake

A mistake Lemma catches

From 0.00045

0.00045=4.5⋅1040.00045=4.5⋅10−4

“0.00045 is smaller than 1, so its power of 10 is negative. Reaching 4.5 moves the decimal point 4 places right, which multiplies by 10⁴ — so 0.00045 = 4.5 ÷ 10⁴ = 4.5 × 10⁻⁴. With 10⁴ the number would be 45,000.”

Try one

Sample problem

Write as an ordinary number.

6×103

Practice scientific notation free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this