Scientific notation is a way of writing numbers as a value between 1 and 10 multiplied by a power of ten: 150,000,000 becomes 1.5 × 10⁸. The form is a × 10ⁿ where a is between 1 and 10 (your “mantissa”) and n is an integer. It exists so huge and tiny numbers can be written, compared, and computed reliably.

Key Takeaways

Standard form is a × 10ⁿ with 1 ≤ |a| < 10 and integer n.

87,000,000 = 8.7 × 10⁷ and 0.00045 = 4.5 × 10⁻⁴: the exponent counts decimal-point moves.

Multiply mantissas and add exponents; divide mantissas and subtract exponents.

The exponent does not change measured precision, which is why it pairs with significant figures.

What is scientific notation?

A number in scientific notation has exactly two parts: a mantissa, usually one nonzero digit before the decimal point, and an exponent on 10 that moves that decimal point to the right spot. 2.5 × 10³ means start with 2.5 and shift the decimal three places right, giving 2500. 4.5 × 10⁻⁴ shifts four places left, giving 0.00045. The power of ten is just a restatement of the exponents in the exponents explained post, applied to our base-ten number system.

Why scientists and engineers use it

Three reasons make the notation worth the switch:

Size. Avogadro’s number, 6.022 × 10²³, is unreadable as 602,200,000,000,000,000,000,000. The mass of an electron, 9.11 × 10⁻³¹ kg, is even worse as a decimal. Word counts and display widths are the least of it, floating-point arithmetic in computers also behaves better on numbers that stay in this range.
Comparability. The exponent lets you rank magnitudes at a glance: 8.7 × 10⁷ is one digit bigger than 9.2 × 10⁶, no matter what the mantissas say.
Exactness. 1.50 × 10³ states clearly that there are three significant figures, a promise a bare 1500 cannot make. The significant figures guide develops this link in detail.

How to convert a number into scientific notation

Write the mantissa by placing a decimal point after the first nonzero digit, then count how many places you moved it. A move to the left gives a positive exponent; a move to the right gives a negative one.

Worked example 1: large numbers. 87,000,000 → the first nonzero digit is 8, so the mantissa is 8.7, and the decimal moves 7 places left. Result: 8.7 × 10⁷. Back it out on the calculator with 8.7 × 10^7 → 87,000,000.

Worked example 2: fractions of 1. 0.00045 → the first nonzero digit is 4, the mantissa is 4.5, and the decimal moves 4 places right. Result: 4.5 × 10⁻⁴. Check with 4.5 × 10^-4 → 0.00045.

Worked example 3: trailing zeros. 2500 → the mantissa is 2.5 and the exponent is 3, giving 2.5 × 10³. If the zeros are meaningful measurements, keep them in the mantissa instead: 2500 with three significant figures is 2.50 × 10³.

When you are in a hurry, the scientific notation calculator converts either direction and shows the working, and it expands very large powers exactly instead of rounding them.

How to convert scientific notation back to standard form

Reverse the rule: a positive exponent moves the decimal right, a negative one moves it left. Positive exponents stretch the number, negative exponents shrink it.

Worked example 4: back to a normal number. 2.5 × 10³ = 2500 (move right 3). 4.2 × 10⁻⁵ = 0.000042 (move left 5). 1.5 × 10⁸ = 150,000,000. Try 4.2 × 10^-5 on the calculator and confirm it shows 0.000042.

Doing arithmetic in scientific notation

The mantissas and exponents combine with simple rules, which is the payoff for the notation.

Worked example 5: multiplication. (2.0 × 10³) × (3.0 × 10²) → multiply mantissas (2.0 × 3.0 = 6.0), add exponents (3 + 2 = 5), giving 6.0 × 10⁵. The calculator agrees: (2.0 × 10^3) × (3.0 × 10^2) → 600,000.

Worked example 6: division. (8.4 × 10⁴) ÷ (2.0 × 10²) → divide mantissas (8.4 ÷ 2.0 = 4.2), subtract exponents (4 − 2 = 2), giving 4.2 × 10² = 420.

Worked example 7: addition and subtraction. Match the exponents first. (1.5 × 10⁸) + (2.2 × 10⁸) adds directly because the powers match: 1.5 + 2.2 = 3.7, so the sum is 3.7 × 10⁸. Enter it as (1.5 × 10^8) + (2.2 × 10^8) → 370,000,000.

Scientific notation and significant figures

The mantissa carries every meaningful digit, and the exponent only places the decimal point, so the two concepts are inseparable. 3.0 × 10⁸ claims two significant figures; 3.00 × 10⁸ claims three, with identical numeric value. When a calculation returns a result like 6.022e+23 in the calculator’s display, the exponent tells you the order of magnitude while the mantissa keeps the precision honest. The scientific notation calculator and the significant figures guide both put the mantissa and exponent through their paces.

Using SCI mode and calculator notation

Scientific calculators, including The Scientific Calculator, can display numbers in this form automatically. Pressing the SCI toggle reformats the answer: enter 123456789 and flip SCI on, and the display reads 1.23456789e+8, where the e+8 stands for “× 10⁸”. The value has not changed, only the presentation. Rules of thumb for entering notation on the keypad: use the ^ key for the power (6.022 × 10^23), 10 alone is 10^1, and a negative exponent needs the sign key or a - (9.11 × 10^-31). The exponent calculator runs the underlying power arithmetic while you figure out which exponent you actually have.

Frequently asked questions

Why is scientific notation sometimes written with an “e”?

The e stands for “exponent”: 1.23456789e+8 means 1.23456789 × 10⁸. Displays and programming languages use it because ordinary text cannot always render a superscript.

What is the difference between scientific notation and standard form?

They are the same number in different clothes. Standard form spells out every digit (87,000,000); scientific notation compresses it to a mantissa and a power (8.7 × 10⁷). The rule for a × 10ⁿ is that a stays between 1 and 10.

How do I enter a number like `6.022 × 10²³` on the calculator?

Type 6.022 × 10^23 and press =. The result displays as 6.022e+23. For 9.11 × 10⁻³¹, type 9.11 × 10^-31.

How many significant figures does `2.5 × 10³` have?

Two: the 2 and the 5. Zeros that merely locate the decimal point belong to the exponent, not the mantissa, which is exactly why 2.50 × 10³ exists as a way to say “three significant figures.”