Algebra
How To Use A Scientific Calculator
A scientific calculator is only as reliable as the assumptions you make about it. Precedence and angle units cause more wrong answers than the arithmetic does.
Quick Answer
brackets first, then powers, then multiply and divide, then add and subtract
- DEG
- Angle unit where a full turn is 360
- RAD
- Angle unit where a full turn is 2 pi
2+3x4 is 14, not 20, because multiplication runs before addition. Adding brackets gives 20. Powers bind to the right, so 2^3^2 is 512, and sin(90) is 1 only when the angle unit is set to degrees.
What Is How To Use A Scientific Calculator?
A scientific calculator differs from a basic one in the operations it offers rather than in the arithmetic it performs. Trigonometric, logarithmic, exponential and power functions sit alongside the four basic operations, and every one of them follows the same precedence rules.
Order of operations is the first thing to get right. Brackets are evaluated first, then powers, then multiplication and division working left to right, and finally addition and subtraction, also left to right. Nothing about the calculator is optional here.
The rule is why 2+3x4 gives 14 rather than 20. Multiplication binds more tightly than addition, so three times four is resolved first and two is added afterwards. Typing the expression exactly as written is all that is required.
Brackets override precedence rather than reorder it. Wrapping the addition gives (2+3)x4 and 20, and the same thirty cents of typing is the only thing standing between a right answer and a wrong one.
Powers bind to the right, which surprises almost everyone. 2^3^2 is read as 2^(3^2), so the inner power is resolved first and the answer is 512 rather than 64. Forcing the other grouping needs brackets: (2^3)^2 is 64.
Unary minus is a sign rather than an operation applied after a power, so -3^2 is -9 and not 9. Brackets are needed to square a negative number, and (-3)^2 is the expression that gives 9.
The angle unit is the single largest source of wrong answers. Degrees treat a full turn as 360 and radians treat it as two pi, and the same key presses give completely different results in the two modes.
sin(90) is 1 in degrees and -0.988 in radians, because 90 radians is not a quarter turn but a little over fourteen of them. The mode is a setting rather than part of the expression, so nothing in the number itself signals which convention is in force.
Radians are the natural unit for calculus, which is why they are the default in so much software. Fractions of pi are the convenient way to write them, and sin(pi/2) is the radians expression that equals sin(90) in degrees.
Inverse trigonometric functions return an angle rather than a ratio, so they inherit the angle unit too. asin(1) is 90 in degrees and about 1.571 in radians, which is pi divided by two in disguise.
Logarithms come in families that differ only by the base. log is base ten in most conventions, ln is base e, and log2 appears on calculators that cater to computing. Mixing them up changes an answer by a constant factor.
Factorials multiply a non-negative integer by every smaller positive integer, and zero factorial is defined as 1. They grow extremely quickly, and 170 is roughly where a floating point number runs out of room.
The square root of two is irrational, so squaring it does not return exactly two. The calculator reports a value that is correct to many more digits than the display shows, and the residue is a floating point artefact rather than an error.
This calculator takes an expression typed exactly as it would be written and evaluates it with a real parser rather than by guessing at the input. Precedence, brackets, factorials and the angle unit are all handled the way a physical scientific calculator handles them.
Formula
brackets, then powers, then multiply and divide, then add and subtract
Multiplication and division resolve before addition and subtraction, and powers resolve before both. Brackets are the only way to force a different grouping.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| a | First operand | value | The value added last. |
| b | Second operand | value | Multiplied before the addition happens. |
| c | Third operand | value | Multiplied with the second operand. |
a ^ b ^ c = a ^ (b ^ c)
Powers bind to the right, so the inner exponent is resolved first. This is the opposite of the intuition most people carry over from addition and multiplication.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| a | Base | value | The value being raised to a power. |
| b | Inner exponent | value | Resolved before it is applied to the base. |
| c | Outer exponent | value | Applied to the whole inner power. |
radians = degrees x pi / 180
The angle unit converts an angle between the two conventions. A trigonometric function applies its argument in whichever unit the calculator is currently set to.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| deg | Angle in degrees | degrees | The angle measured in a 360 part turn. |
| rad | Angle in radians | radians | The same angle measured in a two pi turn. |
How To Calculate How To Use A Scientific Calculator
- 1
Check the angle unit before anything else
The mode is a setting rather than part of the expression, so nothing in the number tells you which convention is live. Set degrees or radians before typing any trigonometric function.
- 2
Type the expression the way it is written
The parser applies standard precedence, so a bare expression needs no extra brackets. Add brackets only when you want to force a grouping that precedence would not give you.
- 3
Use powers with care
Powers bind to the right, so 2^3^2 is 512 rather than 64. A negative base that needs squaring has to be bracketed, because -3^2 is -9 and (-3)^2 is 9.
- 4
Read the result and the error together
A well formed expression shows its value, and a malformed one shows a message explaining where it stopped. An empty expression, a dangling operator and an unclosed bracket all produce distinct messages.
- 5
Store a result in the history
Pressing equals or the enter key puts the current expression and its value into the history, which is the quickest way to keep a few intermediate values without retyping them.
Examples
Example 1: Order of operations on a bare expression
- Expression
- 2+3x4
- Angle unit
- Degrees
| Step | Calculation | Result |
|---|---|---|
| Multiplication binds first | 3 x 4 | 12 |
| Then the addition runs | 2 + 12 | 14 |
| The same expression with brackets added | (2 + 3) x 4 | 20 |
Result: Multiplication runs before addition, so 2+3x4 evaluates to 14 and not to 20. Brackets are the only way to force the other grouping, and adding them around the addition turns the 14 into a 20, which is the answer most people expect. Nothing about the expression needs rewriting, because the precedence rules do the work.
Example 2: Powers bind to the right
- Expression
- 2^3^2
- Angle unit
- Degrees
| Step | Calculation | Result |
|---|---|---|
| The inner power resolves first | 3^2 | 9 |
| Then it is applied to the base | 2^9 | 512 |
| Forcing the other grouping with brackets | (2^3)^2 | 64 |
Result: Powers bind to the right, so 2^3^2 is read as 2^(3^2) and evaluates to 512. Most people expect 64, which is what you get from bracketing the left pair instead. The difference between 512 and 64 is a factor of eight, and the only fix is to write the grouping you actually mean.
Example 3: The same key presses in two angle units
- Expression
- sin(90)
- Angle unit
- Degrees then radians
| Step | Calculation | Result |
|---|---|---|
| With the unit set to degrees | sin(90) | 1 |
| The same expression in radians | sin(90) | -0.9880316240928618 |
| The radians expression that means a quarter turn | sin(pi/2) | 1 |
Result: The angle unit is a setting rather than part of the expression, so sin(90) returns 1 in degrees and -0.9880316240928618 in radians from identical key presses. Ninety radians is a little over fourteen full turns, which is why it lands on a value nowhere near one. Writing the quarter turn as sin(pi/2) is the radians expression that returns 1.
Calculator
Prefer a full-width tool? Open the How To Use A Scientific Calculator calculator page.
Common Mistakes
Assuming addition runs before multiplication
Precedence puts multiplication and division ahead of addition and subtraction, so 2+3x4 is 14. A basic calculator that evaluates keystrokes in order will disagree, which is how the habit forms in the first place.
Leaving the angle unit in radians
Nothing in the number signals which convention is live, so sin(90) quietly returns a value near minus one. Check the mode before every trigonometric expression rather than after an answer looks wrong.
Expecting powers to bind to the left
2^3^2 is 512 rather than 64, because the inner exponent resolves first. Assuming left association turns a right answer into a wrong one by a factor of eight.
Squaring a negative number without brackets
Unary minus is a sign rather than an operation applied after a power, so -3^2 is -9. Brackets around the base are required, and (-3)^2 is the expression that gives 9.
Mixing up log and ln
log is base ten and ln is base e, and the two differ by a constant factor of roughly 2.303. Picking the wrong one changes an answer without looking obviously wrong.
Expecting the square root of two to square back to exactly two
The value is irrational, so squaring it returns a number that differs from two in the last few digits. The difference is a floating point artefact rather than a mistake in the calculator.
Reading a factorial as available for any number
Factorials are defined for non-negative integers only, and zero factorial is 1. A negative or fractional argument is an error rather than a value, and large arguments overflow well before a thousand.
FAQ
Why is 2+3x4 equal to 14 and not 20?
Because multiplication binds more tightly than addition. The calculator resolves three times four first and then adds two, which gives 14. Brackets around the addition are what produce 20.
Is 2^3^2 equal to 512 or 64?
512. Powers bind to the right, so the expression is read as 2 raised to the power of 3 squared, which is 2 to the ninth. Bracketing the left pair gives 64.
Why does sin(90) not give 1?
Because the angle unit is set to radians. In radians the argument is measured in fractions of a full turn of two pi, so 90 radians is many turns rather than a quarter of one. Switch to degrees, or write sin(pi/2).
What is the difference between log and ln?
log is the base ten logarithm and ln is the natural logarithm with base e. They differ by a constant factor, so substituting one for the other changes an answer without looking obviously wrong.
Can I use a minus sign to square a negative number?
Not without brackets. -3^2 is -9, because the power is applied before the sign is considered. (-3)^2 is the expression that gives 9.
What does the calculator do with an expression it cannot parse?
It reports a message giving the position where parsing stopped, rather than guessing at what was meant. A dangling operator, an unclosed bracket and an unknown function each produce a distinct message.
Why does sqrt(2) squared not give exactly 2?
Because the square root of two is irrational and cannot be held exactly in a floating point number. The result is correct to far more digits than the display shows, and the tiny residue is a property of binary arithmetic.
References
- [1]Wikipedia, Order of operations — https://en.wikipedia.org/wiki/Order_of_operations
- [2]Wikipedia, Trigonometric functions and their arguments — https://en.wikipedia.org/wiki/Trigonometric_functions
- [3]IEEE, IEEE 754 floating point arithmetic — https://standards.ieee.org/ieee/754/