Getting Reliable Answers From a Scientific Calculator

Check the angle mode first

Trigonometry functions such as sin, cos and tan need to know whether the angle is measured in degrees or radians. A full circle is 360 degrees or 2π radians. If the mode is wrong, the answer is wrong, with no error message.

Example: sin(30).

In degree mode: 0.5.
In radian mode: sin of 30 radians is about −0.988.

School geometry and most everyday problems use degrees. Calculus, physics formulas and programming usually use radians. If you get an unexpected value, the first check should be the DEG or RAD indicator. To convert by hand: radians = degrees × π ÷ 180.

Order of operations

A calculator applies the standard order: brackets first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right.

Expression	Steps	Result
6 + 2 × 3^2	3^2 = 9, 2 × 9 = 18, 6 + 18	24
(6 + 2) × 3^2	8 × 9	72
−3^2	3^2 = 9, then negate	−9
(−3)^2	bracket first	9

The last two rows trip up many people. In most calculators and in standard maths, the power binds more tightly than the minus sign, so −3^2 means −(3^2). Use brackets whenever you want the negative number itself squared.

When in doubt, add brackets. Extra brackets never hurt, while missing ones are a major source of wrong answers. For a fraction written as one line, wrap both the top and the bottom: write (3 + 5) ÷ (2 × 4), not 3 + 5 ÷ 2 × 4. The first gives 1, and the second gives 13.

Powers, roots, logarithms and factorials

Powers: 2^10 means 2 multiplied by itself 10 times, giving 1,024. A negative exponent gives a reciprocal: 2^−1 = 0.5.
Square roots: √25 = 5. Pythagoras gives √(3^2 + 4^2) = √25 = 5, so the long side of a 3-4-5 triangle is 5.
Other roots: use a fractional power. The cube root of 27 is 27^(1/3) = 3. Keep the fraction in brackets, because 27^1/3 means 27 to the power 1, divided by 3, which is 9.
Logarithms: log(10000) = 4, because 10^4 = 10,000. The natural log ln uses base e, so ln(e) = 1.
Factorials: n! multiplies all whole numbers from 1 to n. 6! = 720 and 10! = 3,628,800. They grow extremely quickly: 20! is about 2.43E18.

For repeated multiplication, such as growth over several years, the power key is the quickest route. See the percent change guide for an example.

Scientific notation and the E symbol

Very large and very small numbers are written as a number between 1 and 10 times a power of ten. On a calculator screen, the capital letter E stands for "times ten to the power of".

6.674E-11 means 6.674 × 10^−11, which is 0.00000000006674.
3E8 means 3 × 10^8, or 300,000,000.

The E is part of the number, not the mathematical constant e. Calculations work as you would expect: 3E8 × 2E-3 = 6E5, which is 600,000. And 4.5E3 ÷ 1.5E-2 = 3E5. When multiplying, add the exponents; when dividing, subtract them.

Example: multiply 6.674E-11 by 5.972E24. The numbers multiply to 6.674 × 5.972 ≈ 39.857, and the exponents add: −11 + 24 = 13. That gives 39.857E13, which is the same as about 3.986E14 once you move the decimal point to put one digit before it.

Check your answer with an estimate

Before trusting any result, round the inputs and estimate. This catches most keying errors in seconds.

Circle with radius 7.5: the circumference is 2 × π × 7.5. A quick estimate with π ≈ 3 is 2 × 3 × 7.5 = 45, so the exact answer of 47.12 looks right. The area is π × 7.5^2 ≈ 3 × 56 = 168, against the exact 176.71.
(2 + 3)^2 ÷ 5!: the top is 25 and 5! is 120, so the result should be a little over 0.2. The calculator gives 0.208333.
Degrees to radians: 180 degrees is π radians (3.14159), so 90 degrees is half of that, 1.5708 radians.

If the estimate and the calculator disagree by a lot, re-enter the expression with extra brackets and watch the angle mode and the E exponents. Most "calculator errors" are really entry errors.

Typical mistakes and how to avoid them

Wrong angle mode. Look at DEG or RAD before every trig calculation.
Missing brackets. Bracket every numerator, denominator and exponent that has more than one term.
Confusing log and ln. log is base 10; ln is base e. They give different answers: log(100) = 2, but ln(100) is about 4.605.
Typing a minus instead of a negative sign. Some models have a separate negative key, so check that yours behaves as you expect.
Trusting a rounded display. Keep a few extra digits during a multi-step problem and round only at the end.
Relying on implied brackets. Calculators read an expression such as 1/2π in different ways. Write 1/(2π) when you mean one divided by two pi, which is about 0.159155.

You can try every example above in the scientific calculator.

FAQ

Why does tan(90) give an error in degree mode?
The tangent of 90 degrees is undefined, because the value grows without limit as the angle approaches 90.

What does an answer like 1.2E-5 mean?
It is 1.2 × 10^−5, or 0.000012. A negative exponent moves the decimal point to the left.

How do I get an angle from a ratio?
Use the inverse functions. If sin(angle) = 0.5, then the inverse sine of 0.5 is 30 in degree mode.

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