Use this tool in calculator mode to perform arithmetic operations with scientific numbers using the e-notation (add, subtract, multiply and divide exponential notation numbers like 1.25e+6). Use it in converter mode to easily convert a number from a scientific notation to a decimal notation real number, or from a real number to a scientific notation (decimal to scientific and scientific to decimal converter).
Quick navigation:
- What is scientific notation?
- E-notation
- Engineering notation
- Decimal to scientific
- Scientific to decimal
What is scientific notation?
The scientific notation is also called E-notation, scientific form of a number, standard index form, or just standard form, is a way of writing numbers deemed too big or too small to be conveniently used in their decimal form. On many calculators it is designated as the "SCI" mode. Even with the improvement of computing power, the scientific notation remains popular among engineers, scientists and mathematicians, who, alongside students, are the primary users of this scientific notation calculator / converter.
The form of a number in scientific notation is m x 10n - m times 10 raised to the power of n. The exponent n is always an integer (whole number) while the coefficient m (mantissa, significand) can be any real number. The most convenient form, called a normalized notation, is one in which the exponent is chosen in such a way that the coefficient is between 1 and 9. The sign of the number is written the usual way.
Numbers in scientific notation allow for easier comparison of orders-of-magnitude. For example, the mass of a proton is 0.000 000 000 000 000 000 000 001 6726 g and the mass of an electron is 0.000 000 000 000 000 000 000 000 000 910 938 356 g. It is hard to compare the two this way, but when we write them as 1.6726 x 10-24 and 9.109 383 56 x 10-28 we can just compare the exponents, and since -24 is larger than -28, the proton is about 4 orders of magnitude (10,000 times) more massive than the electron.
E-notation
Calculators and commonly used programming languages like Java, Python, JavaScript, PHP, C++ use the E-notation (exponential notation) to display numbers too large for their display or the allocated space on a monitor. For example, 3.05 x 1021 would be written as 3.05e21 or 3.05e+21, while 3.05 x 10-21 would be displayed as 3.05e-21.
You need to use the exponent notation when using our scientific notation calculator mode, since it is not convenient to enter numbers in the classic " x 10n" form, due to the exponent requiring a separate field to be filled.
Engineering notation
The engineering notation is the same as the scientific notation, but the exponent is restricted to multiples of 3, resulting in larger values for the significand m - between 1 and 1,000, instead of 1 and 10. This is done so that the numbers can match units as defined by the International System of Units (SI).
Converting to and from scientific notation
Converting numbers to and from scientific notation does not change the number itself, it just changes its form. For people not used to the E-notation, it can be confusing to understand what the number actually is, so it is useful to know how to convert both ways. Using our scientific notation calculator, you can do both types of conversions quickly and easily.
Decimal to scientific
The first step is to move the decimal point enough places so that the number is smaller than 10 and larger than 1. If you moved the decimal point to the left, append "x 10n" to the number, where n is the number of positions you moved the point. If you moved it to the right, append "x 10-n", using the same logic. For example, the number 10,550,000 in normalized scientific notation would be 1.055 x 107 and 1.055e7 or 1.055e+7 in e notation.
If using our scientific notation converter, you just enter the decimal number and click "Convert". The result will be displayed in both e-notation and standard "x 10n" notation.
Scientific to decimal
If using our scientific notation calculator, you have two options. You can either enter a single e-notation number in the first field, and press "Convert", or you can enter the significand in the first field and the exponent in the "Exponent" field, then press the button. The result will be displayed in decimal notation.
If you prefer doing the calculation by hand, you need to remove the "x 10n" part or "e±n" part, then shift the decimal point n digits to the right if n is positive, or to the left if n is negative. For example, the number 1.5678 x 106 will have its decimal separator shifted 6 digits to the right: 1,567,800 in decimal form. The number 8.7651 x 10-6 will have its decimal point moved 6 digits to the left to become 0.000 008 7651.
Arithmetic operations with scientific numbers
Using our tool in scientific notation calculator mode you can perform addition, subtraction, multiplication and division of numbers expressed in scientific notation. In order to do the calculations yourself, you need to follow two steps:
- First, convert the number with a lower exponent to one expressed with the exponent of the number with a larger exponent
- Use normal algebra for the significands of both numbers, then append the exponential notation of your choice (scientific or e-notation)
In the easiest scenario both numbers will be of the same exponent, so conversion won't be necessary and you can simply proceed with step 2.
