# Order of magnitude

An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually ten, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic distributions are common in nature and considering the order of magnitude of values sampled from such a distribution can be more intuitive. When the reference value is ten, the order of magnitude can be understood as the number of digits in the base-10 representation of the value. Similarly, if the reference value is one of certain powers of two, the magnitude can be understood as the amount of computer memory needed to store the exact integer value.

If two numbers have the same order of magnitude, they are about the same size.

But if one would compare the surface of an orange with that of the earth, one would say the surface of the earth is many orders of magnitude larger than that of the orange.

Orders of magnitude are generally used to make very approximate comparisons. It is mainly used while doing scientific notation. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of about 100. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value.

Powers
of ten
Order of
magnitude
0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 −100
0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 −93
0.000000000000000000000000000000000000000000000000000000000000001 −63
0.000000000000000000000000000000001 −33
0.000000000000000000000000000001 −30
0.000000000000000000000000001 −27
0.000000000000000000000001 −24
0.000000000000000000001 −21
0.000000000000000001 −18
0.000000000000001 −15
0.000000000001 −12
0.00000000001 −11
0.0000000001 −10
0.000000001 −9
0.00000001 −8
0.0000001 −7
0.000001 −6
0.00001 −5
0.0001 −4
0.001 −3
0.01 −2
0.1 −1
1 0
10 1
100 2
1,000 3
10,000 4
100,000 5
1,000,000 6
10,000,000 7
100,000,000 8
1,000,000,000 9
10,000,000,000 10
100,000,000,000 11
1,000,000,000,000 12
1,000,000,000,000,000 15
1,000,000,000,000,000,000 18
1,000,000,000,000,000,000,000 21
1,000,000,000,000,000,000,000,000 24
1,000,000,000,000,000,000,000,000,000 27
1,000,000,000,000,000,000,000,000,000,000 30
1,000,000,000,000,000,000,000,000,000,000,000 33
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 63
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 93
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 100

## Uses

Orders of magnitude are used to make approximate comparisons. If numbers differ by one order of magnitude, x is about ten times different in quantity than y. If values differ by two orders of magnitude, they differ by a factor of about 100. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value.

In words
(long scale)
In words
(short scale)
Prefix (Symbol) Decimal Power
of ten
Order of
magnitude
decillionth novemdecillionth icoso- (i) 0.000000000000000000000000000000000000000000000000000000000001 10−60 −60
nonilliardth octodecillionth enneco- (e) 0.000000000000000000000000000000000000000000000000000000001 10−57 −57
nonillionth septendecillionth octeco- (o) 0.000000000000000000000000000000000000000000000000000001 10−54 −54
octilliardth sexdecillionth hepteco- (hp) 0.000000000000000000000000000000000000000000000000001 10−51 −51
octillionth quindecillionth hexeco- (hx) 0.000000000000000000000000000000000000000000000001 10−48 −48
septilliardth quattuordecillionth penteco- (pc) 0.000000000000000000000000000000000000000000001 10−45 −45
septillionth tredecillionth tetreco- (trc) 0.000000000000000000000000000000000000000001 10−42 −42
sextilliardth duodecillionth treco- (tc) 0.000000000000000000000000000000000000001 10−39 −39
sextillionth undecillionth dueco- (dc) 0.000000000000000000000000000000000001 10−36 −36
quintilliardth decillionth meco- (mc) 0.000000000000000000000000000000001 10−33 −33
quintillionth nonillionth veco- (v) 0.000000000000000000000000000001 10−30 −30
quadrilliardth octillionth xono- (x) 0.000000000000000000000000001 10−27 −27
quadrillionth septillionth yocto- (y) 0.000000000000000000000001 10−24 −24
trilliardth sextillionth zepto- (z) 0.000000000000000000001 10−21 −21
trillionth quintillionth atto- (a) 0.000000000000000001 10−18 −18
billiardth quadrillionth femto- (f) 0.000000000000001 10−15 −15
billionth trillionth pico- (p) 0.000000000001 10−12 −12
milliardth billionth nano- (n) 0.000000001 10−9 −9
millionth millionth micro- (µ) 0.000001 10−6 −6
thousandth thousandth milli- (m) 0.001 10−3 −3
hundredth hundredth centi- (c) 0.01 10−2 −2
tenth tenth deci- (d) 0.1 10−1 −1
one one   1 100 0
ten ten deca- (da) 10 101 1
hundred hundred hecto- (h) 100 102 2
thousand thousand kilo- (k) 1000 103 3
million million mega- (M) 1000000 106 6
milliard billion giga- (G) 1000000000 109 9
billion trillion tera- (T) 1000000000000 1012 12
billiard quadrillion peta- (P) 1000000000000000 1015 15
trillion quintillion exa- (E) 1000000000000000000 1018 18
trilliard sextillion zetta- (Z) 1000000000000000000000 1021 21
quadrillion septillion yotta- (Y) 1000000000000000000000000 1024 24
quadrilliard octillion xenna- (X) 1000000000000000000000000000 1027 27
quintillion nonillion daka- (Da) 1000000000000000000000000000000 1030 30
quintillion decillion henda- (H) 1000000000000000000000000000000000 1033 33
quintillion undecillion doka- (Do) 1000000000000000000000000000000000000 1036 36
quintilliard duodecillion tradaka- (Td) 1000000000000000000000000000000000000000 1039 39
sextillion tredecillion tedaka- (Ted) 1000000000000000000000000000000000000000000 1042 42
sextilliard quattuordecillion pedaka- (Pd) 1000000000000000000000000000000000000000000000 1045 45
septillion quindecillion exdaka- (Ed) 1000000000000000000000000000000000000000000000000 1048 48
septilliard sexdecillion zedaka- (Zd) 1000000000000000000000000000000000000000000000000000 1051 51
octillion septendecillion yodaka- (Yd) 1000000000000000000000000000000000000000000000000000000 1054 54
octilliard octodecillion nedaka- (Nd) 1000000000000000000000000000000000000000000000000000000000 1057 57
nonillion novemdecillion ika- (Ik) 1000000000000000000000000000000000000000000000000000000000000 1060 60
In words
(long scale)
In words
(short scale)
Prefix (Symbol) Decimal Power
of ten
Order of
magnitude