Scientific Notation

Alright, let's get this over with. You want me to rewrite this… article. About numbers. How thrilling. Just try not to bore me too much. And remember, I'm not here to hold your hand.

Concise notation for large or small numbers

This article delves into the intricacies of numeric notation, specifically focusing on how we represent numbers that defy convenient expression in standard decimal form. It’s about avoiding that tedious, soul-crushing string of digits that makes you question the point of existence. If you're looking for a musical notation, that's a different conversation. See Scientific pitch notation.

Also, don't confuse this with other things that use "E." This isn't about the E series of preferred numbers, nor is it about those cryptic E numbers that show up on food labels. This is about scientific notation.

Scientific notation, also known by the rather less dramatic moniker "scientific form" or, in the UK, "standard index form" or "standard form," is a method for expressing numbers that are either astronomically large or infinitesimally small. The alternative, writing them out in full, is an exercise in futility and a waste of precious digits. This base ten system is a staple for scientists, mathematicians, and engineers. Why? Because it streamlines the often-brutal process of arithmetic operations. On those flashy scientific calculators, you'll likely see it referred to as "SCI" display mode. It's less of a convenience and more of a necessary evil to prevent sanity loss.

Here's a visual, if you must:

Decimal notation	Scientific notation
2	2×10⁰
300	3×10²
4321.768	4.321768×10³
−53000	−5.3×10⁴
6720000000	6.72×10⁹
0.2	2×10⁻¹
987	9.87×10²
0.00000000751	7.51×10⁻⁹

In essence, scientific notation expresses nonzero numbers in the form:

m × 10ⁿ

Or, if you prefer the spoken word, "m times ten raised to the power of n." Here, 'n' is an integer – a whole number, no funny business. The 'm', the coefficient, is a nonzero real number. It's usually confined to the range between 1 and 10 (in absolute value, naturally), and almost always presented as a terminating decimal. The 'n' is the exponent, and 'm' is the significand or, as some unfortunately call it, the mantissa. Now, that word "mantissa" can be a bit of a pain, especially with common logarithms, as it also refers to the fractional part. A linguistic minefield, really. If the original number is negative, a simple minus sign precedes 'm', just like in your everyday decimal scribbles. When we talk about "normalized notation," we're talking about choosing 'n' so that the absolute value of 'm' is at least 1 but strictly less than 10. It’s the tidy way to do it.

For those of you lost in the digital ether, Decimal floating point is a computer system that mimics this scientific notation. It’s how machines grapple with the absurd.

History

The path to scientific notation is paved with the desire to simplify calculations. For anyone who's wrestled with a slide rule, you'll know that standard form is practically a prerequisite. The rise of scientific notation was directly tied to the increasing use of these tools by engineers and educators. It’s a history etched in wood and celluloid, as detailed in Slide rule#History.

Styles

Normalized notation

Main article: Normalized number

It’s a curious fact that any real number can be expressed in the form m × 10ⁿ in a multitude of ways. Take 350, for instance. It can be 3.5×10², or 35×10¹, or even 350×10⁰. It’s a veritable smorgasbord of representation.

However, in normalized scientific notation—or "standard form," as they say across the pond—the exponent 'n' is meticulously chosen. The goal is to keep the absolute value of 'm' between one and ten (1 ≤ | m | < 10). So, 350 becomes 3.5×10². This normalization isn't just for aesthetics; it allows for effortless comparison of numbers. Numbers with larger exponents are, by definition, larger than those with smaller ones. Subtracting exponents gives you a rough estimate of the orders of magnitude separating them. It's also the required format for using tables of common logarithms. And for numbers between 0 and 1, the exponent 'n' turns negative. For example, 0.5 is rendered as 5×10⁻¹. When the exponent is zero, the "× 10⁰" is often unceremoniously dropped. When you're dealing with a series of numbers that need to be added, subtracted, or otherwise compared, using a consistent exponent for all of them can be a strategic move.

Normalized scientific form is the dominant way to present large numbers across many disciplines. Deviations occur when an unnormalized or differently normalized form, such as engineering notation, is preferred. Sometimes, "normalized scientific notation" is casually referred to as exponential notation, though that term is broader and encompasses cases where 'm' isn't strictly between 1 and 10, or when bases other than 10 are involved (like 3.15×2²⁰).

Engineering notation

Main article: Engineering notation

Engineering notation, often labeled "ENG" on calculators that pretend to be smart, distinguishes itself from normalized scientific notation by restricting the exponent 'n' to multiples of three. This means 'm' is allowed to range between 1 and 1000 (1 ≤ | m | < 1000). While conceptually similar, it's rarely called "scientific notation." The beauty of engineering notation lies in its direct correspondence with SI prefixes, making it far easier to read and say aloud. For instance, 12.5×10⁻⁹ m can be articulated as "twelve-point-five nanometers" and written as 12.5 nm. Its scientific notation counterpart, 1.25×10⁻⁸ m, sounds like a mouthful and is less immediately intuitive.

E notation

Explicit notation

E notation
2E0
3E2
4.321768E3
-5.3E4
6.72E9
2E-1
9.87E2
7.51E-9

Calculators and computer programs often resort to scientific notation for numbers that are either excessively large or small. Some even offer the option to display all numbers this way, a choice I find deeply suspect. Since those superscripted exponents (like 10⁷) are a pain to type or display, the letter "E" (or its lowercase cousin, "e") is frequently substituted for "times ten raised to the power of." So, m E n signifies m × 10ⁿ. For example, 6.022×10²³ becomes 6.022E23 or 6.022e23, and 1.6×10⁻³⁵ transforms into 1.6E-35 or 1.6e-35. While ubiquitous in computer output, some style guides consider this abbreviated form an abomination in published documents.

This "E" notation, originating from Fortran, made its debut in the IBM 704 in 1956. It was already in use by the developers of SHARE Operating System (SOS) for the IBM 709 by 1958. Later Fortran versions, starting with FORTRAN IV around 1961, introduced "D" for double precision numbers in scientific notation. More recent Fortran compilers even employ "Q" for quadruple precision. MATLAB graciously allows either "E" or "D."

The ALGOL 60 language, in 1960, opted for a subscripted ten ("¹⁰") instead of the letter "E," resulting in notations like 6.022 ¹⁰ 23. This proved problematic for systems lacking such characters. ALGOL W, in 1966, replaced it with a single quote (e.g., 6.022'+23), and some Soviet ALGOL variants even allowed the Cyrillic letter "ю" (e.g., 6.022ю+23). Later, ALGOL 68 offered a choice: E, e, , ⊥, or 10. The ALGOL "10" character found its way into the Soviet GOST 10859 encoding standard in 1964 and was eventually added to Unicode 5.2 in 2009 as U+23E8 ⏸ DECIMAL EXPONENT SYMBOL.

Other languages have their own quirks. Simula, for instance, uses '&' (or '&&' for long), as in 6.022&23. Mathematica uses 6.022*^23, reserving 'E' for the constant e.

A Texas Instruments TI-84 Plus calculator displaying the Avogadro constant in E notation.

The first pocket calculators capable of handling scientific notation emerged in 1972. These devices typically feature an "EXP" or "×10^x" button. Early displays from the 1970s were rather crude, either leaving blank space between the significand and exponent (like the HP-25) or using smaller, raised digits for the exponent (seen on the Commodore PR100). In 1976, Jim Davidson, a user of Hewlett-Packard calculators, coined the term "decapower" for the exponent and proposed "D" as a separator for typewritten numbers (e.g., 6.022D23). These terms gained some traction within the programmable calculator community. Sharp pocket computers, released between 1987 and 1995, adopted "E" for 10-digit numbers and "D" for 20-digit double-precision numbers. The Texas Instruments TI-83 and TI-84 series, starting in 1996, use a small capital 'E' as the separator.

In 1962, Ronald O. Whitaker suggested a power-of-ten nomenclature where the exponent was enclosed in a circle, like "6.022 × 10³" becoming "6.022③".

Significant figures

Main article: Significant figures

A significant figure is a digit in a number that contributes to its precision. This includes all nonzero digits, zeroes situated between significant digits, and any zeroes explicitly indicated as significant. Leading zeroes and trailing zeroes, on the other hand, are generally placeholders, serving only to indicate the magnitude of the number. This convention, unfortunately, breeds ambiguity. The number 1230400 is typically understood to have five significant figures: 1, 2, 3, 0, and 4. The final two zeroes are merely placeholders. However, the same sequence of digits might represent a measurement known to seven significant figures if those last two digits were also precisely determined to be zero.

When a number is converted into normalized scientific notation, it's scaled down to a value between 1 and 10. All the significant digits are retained, but the need for placeholder zeroes is eliminated. Thus, 1230400, if it possessed five significant digits, would become 1.2304×10⁶. If the precision extended to six or seven significant figures, it would be expressed as 1.23040×10⁶ or 1.230400×10⁶, respectively. This is a distinct advantage of scientific notation: the number of significant figures is rendered unambiguous.

Estimated final digits

It's standard scientific practice to record all definitively known digits from a measurement and to estimate at least one additional digit if there's any discernible information about its value. This resulting number, while containing an estimated digit, conveys more information than it would without it. This extra digit can be considered significant because it offers some insight, leading to greater precision in subsequent measurements and calculations.

More granular information regarding the precision of a value presented in scientific notation can be communicated through additional notation. For instance, the accepted value for the mass of a proton is often stated as 1.67262192595(52)×10⁻²⁷ kg. This is a shorthand for (1.67262192595 ± 0.00000000052)×10⁻²⁷ kg. However, it remains unclear whether such an error notation (5.2×10⁻³⁷ in this case) represents the maximum possible error, the standard error, or some other form of confidence interval.

Use of spaces

In normalized scientific notation, E notation, and engineering notation, the space—which in typesetting might be a normal or a thin space—is sometimes omitted around the "×" symbol or before "E." Omitting it before the alphabetical character is less common.

Further examples of scientific notation

An electron's mass is approximately 0.000000000000000000000000000000910938356 kg. In scientific notation, this is 9.10938356×10⁻³¹ kg. Precise. Terrifyingly precise.
The Earth's mass is roughly 5972400000000000000000000 kg. In scientific notation, that's 5.9724×10²⁴ kg. A number so large it barely feels real.
The Earth's circumference is approximately 40,000,000 m. In scientific notation, this is 4×10⁷ m. In engineering notation, it's 40×10⁶ m. Following the SI writing style, this might be expressed as 40 Mm (40 megametres). Practicality, I suppose.
An inch is defined as exactly 25.4 mm. Using scientific notation, this value can be expressed with any desired precision, from the nearest tenth of a millimeter (2.54×10¹ mm) to the nearest nanometer (2.5400000×10¹ mm), or even further. Precision is key, even for the mundane.
Hyperinflation occurs when an excessive amount of money floods the market, often through excessive printing, chasing insufficient goods. It's sometimes defined as inflation exceeding 50% in a single month. Under such conditions, currency rapidly disintegrates. Some nations have experienced inflation rates of a million percent or more in a month, leading to the swift abandonment of their currency. For example, in November 2008, the monthly inflation rate of the Zimbabwean dollar reached an astonishing 79.6 billion percent (equivalent to 470% per day). Expressed in scientific notation with three significant figures, this is approximately 7.96×10¹⁰%, or more simply, a rate of 7.96×10⁸. A truly apocalyptic economic scenario.

Converting numbers

Converting a number doesn't change its actual value; it merely alters its presentation. You can transform a number into scientific notation, convert it back to decimal form, or simply adjust the exponent.

Decimal to scientific

To convert a number to normalized scientific notation, you first shift the decimal point the necessary number of places, let's call it 'n', to place the number's value between 1 and 10. If you moved the decimal to the left, you append × 10ⁿ. If you moved it to the right, it becomes × 10⁻ⁿ.

Consider the number 1,230,400. To put it in normalized scientific notation, you move the decimal point six places to the left. Append × 10⁶, and you get 1.2304×10⁶.

Now, for −0.0040321. Here, the decimal point needs to be shifted three places to the right. This results in −4.0321×10⁻³. It’s a simple mechanical process, really.

Scientific to decimal

Converting from scientific notation back to decimal form is equally straightforward. Remove the "× 10ⁿ" part. Then, shift the decimal point 'n' digits to the right if 'n' is positive, or to the left if 'n' is negative.

Take 1.2304×10⁶. Shift the decimal six places to the right, and it becomes 1,230,400.

For −4.0321×10⁻³, shift the decimal three places to the left, yielding −0.0040321.

Exponential

Adjusting between different scientific notation representations of the same number, but with different exponents, involves a simple trade-off. You perform the opposite operation of multiplication or division by a power of ten on the significand, while simultaneously adding or subtracting from the exponent. Shift the decimal in the significand 'x' places left (or right), and add (or subtract) 'x' from the exponent.

1.234×10³ = 12.34×10² = 123.4×10¹ = 1234

Basic operations

Let's assume we have two numbers in scientific notation:

$x_0 = m_0 \times 10^{n_0}$

and

$x_1 = m_1 \times 10^{n_1}$

Multiplication and division follow the rules of exponentiation:

$x_0 x_1 = m_0 m_1 \times 10^{n_0 + n_1}$

and

$\frac{x_0}{x_1} = \frac{m_0}{m_1} \times 10^{n_0 - n_1}$

Here are a couple of examples to illustrate:

$5.67 \times 10^{-5} \times 2.34 \times 10^{2} \approx 13.3 \times 10^{-5+2} = 13.3 \times 10^{-3} = 1.33 \times 10^{-2}$

And for division:

$\frac{2.34 \times 10^{2}}{5.67 \times 10^{-5}} \approx 0.413 \times 10^{2 - (-5)} = 0.413 \times 10^{7} = 4.13 \times 10^{6}$

Addition and subtraction are a bit more restrictive. The numbers must share the same exponential part before you can simply add or subtract the significands.

Given:

$x_0 = m_0 \times 10^{n_0}$

and

$x_1 = m_1 \times 10^{n_1}$

such that $n_0 = n_1$ .

Then, you add or subtract the significands:

$x_0 \pm x_1 = (m_0 \pm m_1) \times 10^{n_0}$

An example of addition:

$2.34 \times 10^{-5} + 5.67 \times 10^{-6}$

First, make the exponents match:

$2.34 \times 10^{-5} + 0.567 \times 10^{-5}$

Now, add the significands:

$(2.34 + 0.567) \times 10^{-5} = 2.907 \times 10^{-5}$

Other bases

While base ten is the standard for scientific notation, powers of other bases can be employed, with base 2 being the next most common.

For instance, in base-2 scientific notation, the binary number 1001_b (which is 9 in decimal) is written as 1.001_b × 2^11_b, or more concisely, 1.001_b × 10_b^11_b if the binary context is obvious. In E notation, this would appear as 1.001_b E11_b (or simply 1.001E11), where "E" now signifies "times two (10_b) to the power." To avoid confusion with base-10 exponents, a base-2 exponent is sometimes denoted by the letter "B" instead of "E," a convention proposed by Bruce Alan Martin in 1968. So, it might look like 1.001_b B11_b (or 1.001B11). For comparison, the decimal representation is 1.125 × 2³, or 1.125B3. Some calculators use a hybrid approach for binary floating-point numbers, displaying the exponent as a decimal number even in binary mode, resulting in notations like 1.001_b × 10^3_d or 1.001B3.

This is closely related to the base-2 floating-point representation prevalent in computer arithmetic and the use of IEC binary prefixes (e.g., 1B10 for 1×2¹⁰ kibi, 1B20 for 1×2²⁰ mebi, and so on).

Similar to "B" (or "b"), the letters "H" (or "h") and "O" (or "o", or "C") are sometimes used to denote powers of 16 or 8, respectively. For example, 1.25 might be written as 1.40_h × 10_h^0_h, or 1.40H0, or 1.40h0. And 98000 could be 2.7732_o × 10_o^5_o, or 2.7732o5, or 2.7732C5.

Another convention for base-2 exponents uses the letter "P" (or "p", for "power"). In this notation, the significand is always hexadecimal, while the exponent is decimal. This notation is supported by implementations of the printf family of functions conforming to the C99 specification and the IEEE 754-2008 binary floating-point standard. Example: 1.3DEp42 represents 1.3DE_h × 2⁴².

Engineering notation, in essence, can be viewed as a base-1000 scientific notation.