Standard Form - Rewrite Any Number as b x 10^n

Standard form rewrites any number as a = b x 10^n, where the mantissa b sits in [1, 10) and the exponent n is an integer that records the order of magnitude. Splitting the value into a significant-figures part and a size part is what makes very large values like the mass of the Earth and very small values like the mass of a helium atom readable at a glance.

• • Rewrite a very large number: Drop the long string of zeros from 5,972,000,000,000,000,000,000,000 (the mass of the Earth in kg) and read it as 5.972 x 10^24.
• • Rewrite a very small number: Pull the significant digits out of 0.0000000000000000000000000066423 (the mass of a helium atom in kg) and read it as 6.6423 x 10^-27.
• • Read a textbook problem in this form: Convert a number given in the b x 10^n form back to plain decimal, or convert a plain decimal into the textbook-style form expected by exam questions.
• • Match the answer style your teacher expects: Switch between the factor form, number form, and E-notation exponential form to match whatever the question is asking for.

Standard form is the same thing as scientific notation in US math and science. The 10^n part records the order of magnitude: positive exponents above 10, negative exponents below 1, and zero in the 1 to 10 range.

Once b is in [1, 10), you can read precision at a glance: 6.674 x 10^-11 is known to four significant figures, while 7 x 10^0 is known to one. Rounding the mantissa is the standard way to control precision without changing the order of magnitude.

If you only need the E-notation shorthand, the exponential notation calculator writes a number as a single coefficient with an e power and a configurable precision.

The calculator reads the input, handles signs and zero, and shifts the decimal point to find the mantissa b and exponent n. It writes the answer in the display style you chose and shows the expanded form as a sum of digit-times-power-of-10 terms.

• a: The number you typed. The page accepts plain decimal values and shorthand like 6.674e-11; shorthand is parsed back to a plain number first.
• b: The mantissa. After shifting the decimal point, b is the number that ends up in [1, 10). For negative inputs the sign rides along with b.
• n: The exponent or order of magnitude. It is the number of decimal places you shifted, positive when you moved the point to the left and negative when you moved it to the right.
• precision: Significant figures kept in b. The default is 4; raising it does not change n, only how many digits of b you see.

For 12,345.6789 the decimal point moves four places to the left, so b is 1.2346 and n is 4. For 0.00000000006674 it moves eleven places to the right, so b is 6.674 and n is -11. The calculator does this in one pass: n is the floor of log10 of the absolute value, and b is a / 10^n.

The expanded form breaks the result into digit-times-power-of-10 terms. The number 154.37 becomes 1 x 10^2 + 5 x 10^1 + 4 x 10^0 + 3 x 10^-1 + 7 x 10^-2, which makes the relationship between the two views explicit.

According to Omni Calculator, the standard form definition is a = b x 10^n, with b between 1 and 10 and n an integer

When the problem is already written in m x 10^n form, the scientific notation equation calculator converts between this form and plain decimal in both directions.

Four short ideas explain why this form is shaped the way it is and how the calculator handles each part of the rewrite.

The relationship between this form and the place-value system is what makes it useful for science. Each place in the decimal system is one power of 10, and the mantissa is just the digits that survive once the trailing zeros have been absorbed into 10^n.

According to Wikipedia, standard form in the United Kingdom is the same normalized scientific notation that requires the absolute value of the mantissa to be at least 1 and less than 10.

Type any number, pick how the answer should be written, and read the mantissa, exponent, and expanded form in one pass.

1. 1 Type the number you want to rewrite: Use the first field. Plain decimal values like 12345.6789 work, and shorthand like 6.674e-11 is parsed back to a plain number first.
2. 2 Pick a display style for the answer: Factor form is the textbook b x 10^n. Number form renders the same thing with a caret-style 10^n. Exponential form is the E-notation shorthand.
3. 3 Set the mantissa precision: Choose how many significant figures to keep in b. The default 4 is fine for most textbook values; drop to 2 for quick estimates and raise to 6 or 8 for precise engineering inputs.
4. 4 Read the mantissa and exponent: The result panel shows the rewritten form, the numeric mantissa, the integer exponent, the expanded form, and 10^n in plain decimal so you can sanity-check the rewrite.
5. 5 Use the reset button to start over: Reset restores the default input and recalculates, which is the fastest way to try a new number without clearing each field by hand.

Try the gravitational force example from the Omni page: type 5,972,000,000,000,000,000,000,000 for the Earth's mass, switch to factor form, and the page returns 5.972 x 10^24. Repeat for the Moon, the Earth-Moon distance, and the gravitational constant, then plug the four values into F = G x M1 x M2 / R^2.

When the problem is a circle rather than a number, the standard equation circle calculator writes the equation in a different standard form (x - A)^2 + (y - B)^2 = C, which uses the same idea of putting the centre and radius in plain sight.

The page is designed for physics, chemistry, and engineering problems, not just for filling in a worksheet.

• • Reads any number, including shorthand: Accepts plain decimals and shorthand scientific notation, so you do not have to convert 6.674e-11 by hand first.
• • Three display styles in one place: Factor form, number form, and exponential form share one dropdown, so it is easy to match the style the question asks for.
• • Configurable mantissa precision: Pick significant figures on the fly, from 1 for rough estimates up to 10 for high-precision engineering inputs.
• • Expanded form on the same page: Shows the digit-times-power-of-10 decomposition alongside the rewritten value, which is what UK-style questions tend to ask for.
• • Real-time update on every keystroke: Updates as you type or change the dropdown, so you can iterate on a value without clicking calculate between edits.
• • Plain-language placeholder text: Each field has a short hint that names the kind of value expected, so a first-time user does not have to read the formula.

Once you have rewritten the inputs as b x 10^n, the multiplying scientific notation calculator multiplies them by adding the exponents and multiplying the mantissas for you.

Three inputs control the rewrite, and three behaviors of the number affect what the calculator can return.

• • The calculator does not handle complex numbers or symbolic inputs. Type a real number, or a real number written in shorthand E-notation, and the page will do the rewrite.
• • Rounding follows the standard half-to-even rule for the mantissa; if you need a strict half-up convention, do the rounding on b after reading the result.
• • Inputs beyond about 1e308 or below 1e-308 are outside the JavaScript number range and the page will return 'Invalid number'. For values at those extremes, use BigInt or arbitrary-precision tools.

According to Math is Fun, scientific notation is a way to write a number as a product of a number between 1 and 10 and a power of 10

For an equation that lands on x squared and a coefficient written in this form, the quadratic formula calculator handles the algebra and returns the roots in the same notation.