Standard form: writing very large and very small numbers
Maths · Key Stage 3 · The School
Why we need it
The Sun is about 150,000,000,000 metres from Earth. A red blood cell is about 0.000008 metres wide. Both numbers are awkward to read and easy to miscount the zeros on. STANDARD FORM writes a number as a value between 1 and 10, multiplied by a power of 10. The Sun's distance becomes 1.5 × 10¹¹ metres. Much easier to read, and much easier to compare.
Large numbers: a positive power
To write 45,000 in standard form, move the decimal point until only one non-zero digit sits in front of it: 4.5. Count how many places you moved it: four. So 45,000 = 4.5 × 10⁴. The power tells you how many places the point moved, and for a number bigger than 1 the power is always positive.
Small numbers: a negative power
To write 0.0032 in standard form, move the point the other way until you reach 3.2. It moved three places. Since the original number is smaller than 1, the power is negative: 0.0032 = 3.2 × 10⁻³. Check it makes sense: a negative power should give you back a small number, not a negative one.
Which is bigger: 6 × 10⁵ or 2 × 10⁷? Compare the powers first. 10⁷ is bigger than 10⁵, so 2 × 10⁷ is bigger, even though 6 is bigger than 2. The power dominates. Only when two powers are equal do you compare the front numbers: 4 × 10⁶ is bigger than 3 × 10⁶, because the powers match and 4 beats 3. Always check the power before you look at the front number.