DB125: Maths for Diploma Studies: Orders of Magnitude

Orders of Magnitude

Increases in orders of magnitude are increases in numbers by the factor of 10. Here’s a table to illustrate the orders of magnitude, their symbols, and titles. You can now impress your friends by telling them you are a trilliard times better than them.

Table: The orders of magnitude

In Words	Prefix (Symbol)	Decimal	Power of Ten	Order of Magnitude
Quadrillionth	yocto- (y)	0.000000000000000000000001	10⁻²⁴	−24
Trilliardth	zepto- (z)	0.000000000000000000001	10⁻²¹	−21
Trillionth	atto- (a)	0.000000000000000001	10⁻¹⁸	−18
Billiardth	femto- (f)	0.000000000000001	10⁻¹⁵	−15
Billionth	pico- (p)	0.000000000001	10⁻¹²	−12
Milliardth	nano- (n)	0.000000001	10⁻⁹	−9
Millionth	micro- (µ)	0.000001	10⁻⁶	−6
Thousandth	milli- (m)	0.001	10⁻³	−3
Hundredth	centi- (c)	0.01	10⁻²	−2
Tenth	deci- (d)	0.1	10⁻¹	−1
One		1	10⁰	0
Ten	deca- (da)	10	10¹	1
Hundred	hecto- (h)	100	10²	2
thousand	kilo- (k)	1,000	10³	3
Million	mega- (M)	1,000,000	10⁶	6
Milliard	giga- (G)	1,000,000,000	10⁹	9
Billion	tera- (T)	1,000,000,000,000	10¹²	12
Billiard	peta- (P)	1,000,000,000,000,000	10¹⁵	15
Trillion	exa- (E)	1,000,000,000,000,000,000	10¹⁸	18
Trilliard	zetta- (Z)	1,000,000,000,000,000,000,000	10²¹	21
Quadrillion	yotta- (Y)	1,000,000,000,000,000,000,000,000	10²⁴	24

In scientific notation, we use this format to show orders of magnitude:

$C o e f f i c i e n t space cross times space B a s e space to the power of E x p o n e n t end exponent$

As an example:

$2.3 space cross times space 10 to the power of 6$

The exponents in orders of magnitude are subject to the same rules for exponents that we met earlier. For example:

$alpha to the power of x space cross times space alpha to the power of y space equals space alpha to the power of space left parenthesis x plus y right parenthesis end exponent$

Using this same logic for exponents:

$left parenthesis alpha space cross times space 10 to the power of x right parenthesis space cross times space left parenthesis b space cross times space 10 to the power of y right parenthesis space equals space left parenthesis alpha space cross times space b right parenthesis space cross times space 10 to the power of left parenthesis x plus y right parenthesis end exponent$

If you multiply numbers with powers of 10, exponents are added. For example:

$4 space cross times space 10 to the power of 9 space cross times space 2 space cross times space 10 cubed space equals space left parenthesis 2 space cross times space 4 right parenthesis space cross times space 10 to the power of space left parenthesis 9 plus 3 right parenthesis end exponent space equals space 8 space cross times space 10 to the power of 12$

If you divide numbers with powers of 10, the numbers with powers are divided, and exponents are subtracted. For example:

$alpha to the power of x over alpha to the power of y space equals space alpha to the power of space left parenthesis x minus y right parenthesis end exponent$

$fraction numerator 4 space cross times space 10 to the power of 9 over denominator 2 space cross times space 10 cubed end fraction space equals space open parentheses 4 over 2 close parentheses space cross times space 10 to the power of left parenthesis 9 minus 3 right parenthesis end exponent space equals space 2 space cross times space 10 to the power of 6$

When adding or subtracting numbers with powers of 10, the exponents remain unchanged.

The rules we saw earlier for adding and subtracting positive and negative numbers apply to the exponent calculations. For example:

$2 space cross times space 10 to the power of space minus 5 end exponent space cross times space 4 space cross times space 10 to the power of space minus 3 end exponent space equals space left parenthesis 2 space cross times space 4 right parenthesis space cross times space 10 to the power of space left parenthesis negative 5 space minus 3 right parenthesis end exponent space equals space 8 space cross times space 10 to the power of negative 8 end exponent$

And

$fraction numerator 4 space cross times space 10 to the power of negative 4 end exponent over denominator 2 space cross times space 10 to the power of space minus 7 end exponent end fraction space equals space open parentheses 4 over 2 close parentheses space cross times space 10 to the power of left parenthesis negative 4 space minus negative 7 right parenthesis end exponent space equals space 2 space cross times space 10 cubed$

And

$fraction numerator 6 space cross times space 10 to the power of negative 4 end exponent over denominator 3 space cross times space 10 to the power of space 7 end exponent end fraction space equals space open parentheses 6 over 3 close parentheses space cross times space 10 to the power of space left parenthesis negative 4 space minus 7 right parenthesis end exponent space equals space 2 space cross times space 10 to the power of negative 11 end exponent$

To help reinforce what we have just learned, on the next page there are some practice questions. Answers and explanations can be revealed by clicking on show answer.

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Maths for Diploma Studies

Orders of Magnitude

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