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Add Percentage of a value to study sheet

Arithmetic and ratios

Percentage of a value

A=V\frac{p}{100}

When it applies: $p$ is the numeric percentage, so enter $25$ for twenty-five percent. $A$ and $V$ use the same measurement unit. Signed inputs follow ordinary multiplication.

$A$: Resulting part
$V$: Original value
$p$: Numeric percentage

Step-by-step example: Percentage of a value

1. Identify the whole and the requested percentage.
$V=240,\quad p=15$
2. Convert the percentage before multiplying the whole.
$A=240\cdot\frac{15}{100}$
3. Simplify the product in the original measurement unit.
$A=36$

Open percentage calculator OpenStax: fractions, ratios and percentages

Add Signed relative percentage change to study sheet

Arithmetic and ratios

Signed relative percentage change

\Delta=\frac{V_1-V_0}{V_0}\cdot100\%

When it applies: $V_0\ne0$ . This definition divides by the signed original value, not its magnitude. A negative baseline can reverse the everyday interpretation of an increase.

$V_0$: Original value
$V_1$: New value
$\Delta$: Signed relative change

Step-by-step example: Signed relative percentage change

1. Start with the original baseline and the new value.
$V_0=80,\quad V_1=100$
2. Subtract the baseline before dividing by it.
$\Delta=\frac{100-80}{80}\cdot100\%$
3. Express the relative difference as a percentage.
$\Delta=25\%$

Open percentage calculator OpenStax: fractions, ratios and percentages

Add Fraction addition and subtraction to study sheet

Arithmetic and ratios

Fraction addition and subtraction

\frac ab\pm\frac cd=\frac{ad\pm bc}{bd}

When it applies: $b\ne0$ and $d\ne0$ . Use matching signs in the numerator. The denominator product is valid even when a smaller common denominator is available.

$a,c$: Numerators
$b,d$: Nonzero denominators

Step-by-step example: Fraction addition and subtraction

1. Use the plus operation for these two source fractions.
$\frac23+\frac14$
2. Cross-multiply the numerators over the denominator product.
$\frac{2\cdot4+1\cdot3}{3\cdot4}$
3. Add the integer numerator terms and reduce if needed.
$\frac{11}{12}$

Open fraction calculator OpenStax: fractions, ratios and percentages

Add Division of fractions to study sheet

Arithmetic and ratios

Division of fractions

\frac ab\div\frac cd=\frac ab\cdot\frac dc=\frac{ad}{bc}

When it applies: $b,d,c\ne0$ . A divisor fraction must not equal zero. Multiply by the reciprocal of the divisor, not by a reciprocal of the original dividend.

$a,b$: Dividend numerator and denominator
$c,d$: Divisor numerator and denominator

Step-by-step example: Division of fractions

1. Identify the fraction being divided and the nonzero divisor.
$\frac23\div\frac45$
2. Invert only the divisor and multiply.
$\frac23\cdot\frac54$
3. Reduce the resulting product to lowest terms.
$\frac{10}{12}=\frac56$

Open fraction calculator OpenStax: fractions, ratios and percentages

Add Equivalent ratio scaling to study sheet

Arithmetic and ratios

Equivalent ratio scaling

b_{\mathrm{new}}=b\frac{a_{\mathrm{new}}}{a}

When it applies: $a\ne0$ . Keep the two ratio positions and their units consistent. For counts or recipe quantities, also require physically meaningful nonnegative inputs.

$a,b$: Original paired quantities
$a_{\mathrm{new}}$: New first quantity
$b_{\mathrm{new}}$: Scaled second quantity

Step-by-step example: Equivalent ratio scaling

1. Keep the correspondence between the two original positions.
$a:b=3:5,\quad a_{\mathrm{new}}=12$
2. Compute the common scale factor from the first position.
$k=\frac{12}{3}=4$
3. Apply that same factor to the second position.
$b_{\mathrm{new}}=5\cdot4=20$

Open ratio calculator OpenStax: fractions, ratios and percentages

Add One-variable linear equation to study sheet

Algebra and powers

One-variable linear equation

ax+b=c\quad\Longrightarrow\quad x=\frac{c-b}{a}

When it applies: $a\ne0$ for this division rule. If $a=0$ , the equation has every real value as a solution when $b=c$ , and no solution otherwise.

$a$: Nonzero coefficient
$b,c$: Constant terms
$x$: Unknown real value

Step-by-step example: One-variable linear equation

1. Identify the coefficient and both constant terms.
$3x+2=14$
2. Subtract the same constant from both sides.
$3x=12$
3. Divide both sides by the nonzero coefficient.
$x=4$

Open linear equation solver OpenStax: algebraic expressions and equations

Add Quadratic roots to study sheet

Algebra and powers

Quadratic roots

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

When it applies: $a\ne0$ in $ax^2+bx+c=0$ . With real coefficients, real roots require $b^2-4ac\ge0$ . A zero discriminant gives one repeated real root.

$a,b,c$: Quadratic coefficients
$x$: Root of the equation

Step-by-step example: Quadratic roots

1. Read all three coefficients, including their signs.
$x^2-3x+2=0,\quad a=1,\ b=-3,\ c=2$
2. Evaluate the discriminant before taking its square root.
$D=(-3)^2-4\cdot1\cdot2=1$
3. Substitute into the two branches of the root formula.
$x=\frac{3\pm1}{2}\quad\Longrightarrow\quad x\in\{1,2\}$

OpenStax: algebraic expressions and equations

Add Difference of squares to study sheet

Algebra and powers

Difference of squares

a^2-b^2=(a-b)(a+b)

When it applies: This identity holds for real or complex $a,b$ . It factors a difference, not the sum $a^2+b^2$ . Keep both binomial factors.

$a,b$: Quantities being squared

Step-by-step example: Difference of squares

1. Recognize both terms as squares with a subtraction sign.
$9x^2-16=(3x)^2-4^2$
2. Form one difference and one sum of the square roots.
$(3x-4)(3x+4)$
3. Check a numerical substitution in both equivalent forms.
$x=2:\quad36-16=2\cdot10=20$

OpenStax: algebraic expressions and equations

Add Product and quotient of integer powers to study sheet

Algebra and powers

Product and quotient of integer powers

a^m a^n=a^{m+n},\qquad \frac{a^m}{a^n}=a^{m-n}

When it applies: $a\ne0$ and $m,n\in\mathbb Z$ for this combined statement. The bases must match; exponents do not combine this way when unrelated bases are added.

$a$: Common nonzero base
$m,n$: Integer exponents

Step-by-step example: Product and quotient of integer powers

1. Notice the identical nonzero base in the product.
$2^3\cdot2^4$
2. Add the exponents for multiplication.
$2^{3+4}=2^7$
3. Evaluate the resulting integer power.
$2^7=128$

OpenStax: algebraic expressions and equations

Add Logarithm of a product to study sheet

Algebra and powers

Logarithm of a product

\log_b(uv)=\log_b u+\log_b v

When it applies: For real logarithms require $u,v>0$ , $b>0$ and $b\ne1$ . This is a product rule; it does not turn the logarithm of a sum into a sum of logarithms.

$b$: Logarithm base
$u,v$: Positive factors

Step-by-step example: Logarithm of a product

1. Identify positive factors and a valid logarithm base.
$\log_2(8\cdot4)$
2. Separate the product into two logarithms of the same base.
$\log_2 8+\log_2 4=3+2$
3. Add the exponents and verify the original product.
$3+2=5,\qquad2^5=32$

OpenStax: algebraic expressions and equations

Add Slope between two points to study sheet

Coordinate geometry

Slope between two points

m=\frac{y_2-y_1}{x_2-x_1}

When it applies: $x_2\ne x_1$ . A vertical line has no finite slope. Two identical points do not determine a unique line, even though both coordinate differences vanish.

$x_1,y_1$: First point coordinates
$x_2,y_2$: Second point coordinates
$m$: Change in vertical units per horizontal unit

Step-by-step example: Slope between two points

1. Keep each horizontal coordinate paired with its vertical coordinate.
$(x_1,y_1)=(1,2),\quad(x_2,y_2)=(5,10)$
2. Use the same point order in both differences.
$m=\frac{10-2}{5-1}=\frac84$
3. Reduce the ratio of vertical to horizontal change.
$m=2$

Open slope calculator OpenStax: algebraic expressions and equations

Add Point-slope line equation to study sheet

Coordinate geometry

Point-slope line equation

y-y_1=m(x-x_1)

When it applies: The slope $m$ must be finite and the line must pass through $(x_1,y_1)$ . Use $x=x_1$ instead for a vertical line; this finite-slope form does not represent it.

$m$: Finite slope
$x_1,y_1$: Known point on the line
$x,y$: Coordinates of an arbitrary point on the line

Step-by-step example: Point-slope line equation

1. Use a known point and a finite slope.
$m=2,\quad(x_1,y_1)=(1,3)$
2. Substitute into the point-slope relationship.
$y-3=2(x-1)$
3. Expand and isolate the vertical coordinate.
$y=2x+1$

Open slope calculator OpenStax: algebraic expressions and equations

Add Midpoint of a segment to study sheet

Coordinate geometry

Midpoint of a segment

M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

When it applies: Use Cartesian coordinates in a consistent coordinate system. Average matching coordinate positions separately. This is an arithmetic midpoint, not a curved-surface geographic midpoint.

$x_1,y_1$: First endpoint
$x_2,y_2$: Second endpoint
$M$: Segment midpoint

Step-by-step example: Midpoint of a segment

1. Read the two endpoints in horizontal-vertical order.
$(x_1,y_1)=(2,1),\quad(x_2,y_2)=(8,5)$
2. Average each matching coordinate pair.
$M=\left(\frac{2+8}{2},\frac{1+5}{2}\right)$
3. Simplify both coordinate averages.
$M=(5,3)$

Open midpoint calculator OpenStax: algebraic expressions and equations

Add Euclidean distance in a plane to study sheet

Coordinate geometry

Euclidean distance in a plane

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

When it applies: Use a flat Cartesian coordinate system with compatible length scales on both axes. The nonnegative square root gives distance; longitude and latitude require a different geographical model.

$x_1,y_1$: First point
$x_2,y_2$: Second point
$d$: Nonnegative straight-line distance

Step-by-step example: Euclidean distance in a plane

1. Use compatible coordinate units for the two points.
$(x_1,y_1)=(1,1),\quad(x_2,y_2)=(4,5)$
2. Square the coordinate differences before adding.
$d=\sqrt{(4-1)^2+(5-1)^2}=\sqrt{9+16}$
3. Take the nonnegative square root of the total.
$d=5$

Open distance formula calculator OpenStax: algebraic expressions and equations

Add Rectangle area and perimeter to study sheet

Plane geometry

Rectangle area and perimeter

A=lw,\qquad P=2(l+w)

When it applies: $l,w>0$ for a nondegenerate rectangle, with both dimensions expressed in the same length unit. Area uses squared units; perimeter uses the original length unit.

$l,w$: Adjacent side lengths
$A$: Area
$P$: Perimeter

Step-by-step example: Rectangle area and perimeter

1. Identify the two adjacent side lengths.
$l=8,\quad w=3$
2. Multiply the lengths for area and add them for the half-perimeter.
$A=8\cdot3,\quad P=2(8+3)$
3. Keep area and boundary length as different quantities.
$A=24,\quad P=22$

Open area calculator OpenStax: geometric formulas

Add Square area and perimeter to study sheet

Plane geometry

Square area and perimeter

A=s^2,\qquad P=4s

When it applies: $s>0$ is a side length of a nondegenerate square. All four sides must be equal and adjacent sides perpendicular. Do not substitute a diagonal for the side without converting it.

$s$: Side length
$A$: Area in squared length units
$P$: Boundary length

Step-by-step example: Square area and perimeter

1. Use the length of a side rather than the diagonal.
$s=5$
2. Square the side for area and multiply it by four for the boundary.
$A=5^2,\quad P=4\cdot5$
3. Evaluate the two measures in their appropriate units.
$A=25,\quad P=20$

Open perimeter calculator OpenStax: geometric formulas

Add Circle area to study sheet

Plane geometry

Circle area

A=\pi r^2

When it applies: $r>0$ is the radius, not the diameter. Use $r=d/2$ if a diameter is supplied. The resulting area is measured in squared length units.

$r$: Radius
$d$: Diameter
$A$: Enclosed area

Step-by-step example: Circle area

1. Take half the supplied diameter to obtain the radius.
$d=10,\quad r=5$
2. Square the radius before multiplying by the circle constant.
$A=\pi\cdot5^2$
3. Retain the exact constant instead of prematurely rounding it.
$A=25\pi$

Open circle calculator OpenStax: geometric formulas

Add Circle circumference to study sheet

Plane geometry

Circle circumference

C=2\pi r=\pi d

When it applies: $r>0$ and $d=2r$ for a nondegenerate circle. Circumference measures boundary length and uses length units rather than squared area units.

$r$: Radius
$d$: Diameter
$C$: Circumference

Step-by-step example: Circle circumference

1. Read the radius from the supplied measurement.
$r=3$
2. Multiply the radius by twice the circle constant.
$C=2\pi\cdot3$
3. Simplify while retaining the exact constant.
$C=6\pi$

Open circle calculator OpenStax: geometric formulas

Add Triangle area from base and height to study sheet

Plane geometry

Triangle area from base and height

A=\frac{bh}{2}

When it applies: $b,h>0$ with $h$ perpendicular to the chosen base line. A sloping side is not automatically the corresponding height. The altitude may lie outside an obtuse triangle.

$b$: Chosen base length
$h$: Perpendicular height to that base
$A$: Triangle area

Step-by-step example: Triangle area from base and height

1. Use the base and its perpendicular height.
$b=10,\quad h=6$
2. Multiply those dimensions and divide by two.
$A=\frac{10\cdot6}{2}$
3. Evaluate the area in squared length units.
$A=30$

Open triangle calculator OpenStax: geometric formulas

Add Trapezoid area to study sheet

Plane geometry

Trapezoid area

A=\frac{(a+b)h}{2}

When it applies: $a,b,h>0$ . The lengths $a,b$ are parallel bases and $h$ is their perpendicular separation. Unrelated side lengths cannot replace the parallel bases.

$a,b$: Lengths of the parallel bases
$h$: Perpendicular separation
$A$: Enclosed area

Step-by-step example: Trapezoid area

1. Identify the parallel sides and their perpendicular separation.
$a=4,\quad b=8,\quad h=3$
2. Average the base lengths and multiply by the height.
$A=\frac{(4+8)\cdot3}{2}$
3. Simplify the resulting area.
$A=18$

Open area calculator OpenStax: geometric formulas

Add Pythagorean theorem to study sheet

Plane geometry

Pythagorean theorem

a^2+b^2=c^2

When it applies: Only for a right triangle: $a,b>0$ are its perpendicular legs and $c>0$ is the hypotenuse opposite the right angle. Take the positive root when finding a length.

$a,b$: Perpendicular leg lengths
$c$: Hypotenuse

Step-by-step example: Pythagorean theorem

1. Confirm the known sides are perpendicular legs.
$a=5,\quad b=12$
2. Add their squared lengths.
$c^2=5^2+12^2=169$
3. Use the positive root for the geometric length.
$c=13$

Open Pythagorean theorem calculator OpenStax: geometric formulas

Add Degrees and radians to study sheet

Trigonometry

Degrees and radians

\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}

When it applies: Degrees and radians are different angle scales. A full turn is $360^\circ=2\pi$ radians. Keep the conversion direction consistent when substituting into a trigonometric formula.

$\theta_{\mathrm{deg}}$: Angle measured in degrees
$\theta_{\mathrm{rad}}$: Same angle measured in radians

Step-by-step example: Degrees and radians

1. Start with the angle on the degree scale.
$\theta_{\mathrm{deg}}=60$
2. Multiply by the radian conversion factor.
$\theta_{\mathrm{rad}}=60\cdot\frac{\pi}{180}$
3. Reduce the rational coefficient of the circle constant.
$\theta_{\mathrm{rad}}=\frac{\pi}{3}$

Open degree-radian converter NIST DLMF: trigonometric identities

Add Right-triangle trigonometric ratios to study sheet

Trigonometry

Right-triangle trigonometric ratios

\sin\theta=\frac{o}{h},\quad\cos\theta=\frac{a}{h},\quad\tan\theta=\frac{o}{a}

When it applies: For an acute angle of a right triangle, $o$ is opposite, $a$ is the adjacent leg and $h$ is the hypotenuse. These positive geometric side ratios do not use an arbitrary adjacent sloping side.

$o$: Leg opposite the selected angle
$a$: Leg adjacent to the selected angle
$h$: Hypotenuse length

Step-by-step example: Right-triangle trigonometric ratios

1. Identify opposite, adjacent and hypotenuse relative to one acute angle.
$o=3,\quad a=4,\quad h=5$
2. Form each ratio using its stated denominator.
$\sin\theta=\frac35,\quad\cos\theta=\frac45,\quad\tan\theta=\frac34$
3. Check that both legs reproduce the given hypotenuse.
$3^2+4^2=5^2$

Open right triangle calculator NIST DLMF: trigonometric identities

Add Pythagorean trigonometric identity to study sheet

Trigonometry

Pythagorean trigonometric identity

\sin^2\theta+\cos^2\theta=1

When it applies: This identity holds for real angles on a consistent angle scale. A squared sine means $(\sin\theta)^2$ , not a sine with its angle squared.

$\theta$: Real angle
$\sin^2\theta,\cos^2\theta$: Squared trigonometric function values

Step-by-step example: Pythagorean trigonometric identity

1. Choose an angle with familiar exact unit-circle values.
$\theta=\frac{\pi}{6},\quad\sin\theta=\frac12,\quad\cos\theta=\frac{\sqrt3}{2}$
2. Square both function values before adding.
$\left(\frac12\right)^2+\left(\frac{\sqrt3}{2}\right)^2$
3. Simplify the two squares to verify the identity.
$\frac14+\frac34=1$

Open unit circle calculator NIST DLMF: trigonometric identities

Add Law of cosines to study sheet

Trigonometry

Law of cosines

c^2=a^2+b^2-2ab\cos C

When it applies: $a,b>0$ with included angle $0<C<\pi$ for a nondegenerate triangle; $c$ is opposite $C$ . Convert degrees to the angle scale your calculator expects.

$a,b$: Two known side lengths
$C$: Included angle
$c$: Side opposite the included angle

Step-by-step example: Law of cosines

1. Keep the included angle between the two known sides.
$a=5,\quad b=7,\quad C=60^\circ$
2. Use the exact cosine of that angle.
$c^2=25+49-2\cdot5\cdot7\cdot\frac12=39$
3. Take the positive square root for the opposite side.
$c=\sqrt{39}$

Open triangle calculator NIST DLMF: trigonometric identities

Add Arithmetic mean to study sheet

Statistics

Arithmetic mean

\bar{x}=\frac1n\sum_{i=1}^{n}x_i

When it applies: $n\ge1$ counts observations rather than distinct values. Repeated observations contribute repeatedly. Every value must use a compatible numerical scale.

$x_i$: Individual observation
$n$: Observation count
$\bar{x}$: Arithmetic mean

Step-by-step example: Arithmetic mean

1. Retain all three observations in the count.
$x_1=2,\quad x_2=4,\quad x_3=9,\quad n=3$
2. Add before dividing by the total count.
$\bar{x}=\frac{2+4+9}{3}$
3. Reduce the resulting exact average.
$\bar{x}=5$

Open mean median mode calculator NIST: variance and standard deviation

Add Sample variance to study sheet

Statistics

Sample variance

s^2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}

When it applies: $n\ge2$ . This sample-variance convention uses the sample mean and denominator $n-1$ . Its usual unbiased-estimation interpretation needs appropriate sampling assumptions and finite population variance.

$x_i$: Individual observation
$\bar{x}$: Sample arithmetic mean
$n$: Sample observation count
$s^2$: Sample variance in squared units

Step-by-step example: Sample variance

1. Find the common mean of the three observations.
$(x_1,x_2,x_3)=(2,4,9),\quad\bar{x}=5$
2. Square each deviation from that same mean and add.
$S=(2-5)^2+(4-5)^2+(9-5)^2=26$
3. Divide the squared total by one less than the count.
$s^2=\frac{26}{3-1}=13$

Open standard deviation calculator NIST: variance and standard deviation

Add Population variance to study sheet

Statistics

Population variance

\sigma^2=\frac1n\sum_{i=1}^{n}(x_i-\mu)^2

When it applies: $n\ge1$ , with $\mu$ the arithmetic mean of the complete supplied population. This descriptive denominator is $n$ , unlike the common sample-variance denominator.

$x_i$: Population observation
$\mu$: Population arithmetic mean
$n$: Population count
$\sigma^2$: Population variance in squared units

Step-by-step example: Population variance

1. Treat the stated observations as the complete population of interest.
$(x_1,x_2,x_3)=(2,4,9),\quad\mu=5$
2. Use the common population mean in every squared deviation.
$S=(2-5)^2+(4-5)^2+(9-5)^2=26$
3. Divide the same total by the full population count.
$\sigma^2=\frac{26}{3}$

Open standard deviation calculator NIST: variance and standard deviation

Add Standard deviation from variance to study sheet

Statistics

Standard deviation from variance

s=\sqrt{s^2},\qquad\sigma=\sqrt{\sigma^2}

When it applies: Use a defined nonnegative variance and its corresponding sample or population convention. The square root is nonnegative and restores the original measurement unit from squared units.

$s,\sigma$: Sample and population standard deviations
$s^2,\sigma^2$: Corresponding nonnegative variances

Step-by-step example: Standard deviation from variance

1. Start with the already defined sample variance.
$s^2=13$
2. Take its nonnegative square root.
$s=\sqrt{13}$
3. Keep the exact root alongside a separately rounded value.
$s\approx3.605551$

Open standard deviation calculator NIST: variance and standard deviation

Add Type 7 percentile threshold to study sheet

Statistics

Type 7 percentile threshold

\begin{aligned}h&=1+(n-1)p,\quad j=\lfloor h\rfloor,\quad g=h-j\\Q(p)&=(1-g)x_{(j)}+g\,x_{(\min(j+1,n))}\end{aligned}

When it applies: Sort all $n\ge1$ observations and use a probability $0\le p\le1$ , not a numeric percentage. The upper index is clamped at the last observation. This is one quantile convention, not a score percentile-rank formula.

$x_{(j)}$: Observation at sorted one-based rank $j$
$p$: Probability corresponding to the requested percentile
$h,j,g$: Interpolation position, floor and fractional part
$Q(p)$: Type 7 measurement threshold

Step-by-step example: Type 7 percentile threshold

1. Sort the four observations and convert the requested percentage.
$(x_{(1)},x_{(2)},x_{(3)},x_{(4)})=(1,3,5,7),\quad p=\frac14$
2. Calculate the interpolation position and its fractional part.
$h=1+3\cdot\frac14=\frac74,\quad j=1,\quad g=\frac34$
3. Interpolate between the two contributing ordered observations.
$Q\left(\frac14\right)=\frac14\cdot1+\frac34\cdot3=\frac52$

Open percentile calculator R documentation: sample quantile conventions

Add Empirical percentile rank and ties to study sheet

Statistics

Empirical percentile rank and ties

\begin{aligned}R_s&=\frac{100B}{n}\%,&R_w&=\frac{100(B+E)}{n}\%\\R_m&=\frac{100(B+E/2)}{n}\%,&R_a&=\frac{100(B+(E+\mathbf1_{E>0})/2)}{n}\%\end{aligned}

When it applies: $n\ge1$ . Count all observations below and equal to the evaluated score, including repetitions. Strict, weak, mean and average-matching-rank conventions differ when ties exist.

$B,E$: Counts below and equal to the score
$n$: Reference observation count
$R_s,R_w,R_m,R_a$: Strict, weak, mean and average-matching ranks
$\mathbf1_{E>0}$: One when a match exists and zero otherwise

Step-by-step example: Empirical percentile rank and ties

1. Compare the evaluated score with every observation in the reference set.
$(1,2,3,3,4),\quad v=3$
2. Count repetitions below and equal to the score.
$B=2,\quad E=2,\quad n=5$
3. Apply each stated tie convention rather than substituting an interpolation position.
$R_s=40\%,\quad R_w=80\%,\quad R_m=60\%,\quad R_a=70\%$

Open percentile calculator SciPy documentation: percentile ranks and ties

Add Standardized z-score to study sheet

Statistics

Standardized z-score

z=\frac{x-\mu}{\sigma}

When it applies: $\sigma>0$ , with value, mean and standard deviation on compatible scales. Standardization alone does not require normality and does not by itself determine an empirical percentile or a test p-value.

$x$: Raw value
$\mu$: Reference mean
$\sigma$: Positive reference standard deviation
$z$: Unitless standardized distance

Step-by-step example: Standardized z-score

1. Identify the value, reference center and positive reference scale.
$x=82,\quad\mu=75,\quad\sigma=8$
2. Subtract the mean before dividing by the scale.
$z=\frac{82-75}{8}$
3. Retain the exact fraction and its decimal view.
$z=\frac78=0.875$

Open z-score calculator NIST: variance and standard deviation

Add Normalized scientific notation to study sheet

Notation and number theory

Normalized scientific notation

x=a\cdot10^k,\qquad1\le|a|<10,\quad k\in\mathbb Z

When it applies: For a nonzero real value use the stated normalized mantissa range. Zero has no nonzero normalized mantissa and is handled separately. The sign stays with the mantissa.

$x$: Original nonzero value
$a$: Signed normalized mantissa
$k$: Integer decimal exponent

Step-by-step example: Normalized scientific notation

1. Identify the first nonzero decimal digit.
$x=0.0042$
2. Shift the mantissa so its magnitude is in the normalized range.
$a=4.2,\quad k=-3$
3. Check that the power of ten restores the original magnitude.
$x=4.2\cdot10^{-3}$

Open scientific notation calculator OpenStax: prime factorization and common multiples

Add Prime-power factorization to study sheet

Notation and number theory

Prime-power factorization

n=\prod_{i=1}^{r}p_i^{a_i}

When it applies: $n\ge2$ is an integer, the $p_i$ are distinct primes and $a_i$ positive integers. One has an empty product, zero has no finite prime factorization, and a negative sign is recorded separately.

$n$: Positive integer being factored
$p_i$: Distinct prime factors
$a_i$: Positive prime exponents

Step-by-step example: Prime-power factorization

1. Begin with the positive integer to be decomposed.
$n=360$
2. Divide out prime factors until the remaining factor is one.
$360=2\cdot2\cdot2\cdot3\cdot3\cdot5$
3. Group repeated primes into powers.
$360=2^3\cdot3^2\cdot5$

Open prime factorization calculator OpenStax: prime factorization and common multiples

Add Greatest common divisor by prime powers to study sheet

Notation and number theory

Greatest common divisor by prime powers

\gcd(a,b)=\prod_p p^{\min(\alpha_p,\beta_p)}

When it applies: $a,b$ are positive integers, factored over the union of their prime factors. Use exponent zero for an absent prime. Zero-input conventions are outside this prime-factor representation.

$\alpha_p,\beta_p$: Exponents of prime $p$ in $a,b$
$a,b$: Positive integers
$\gcd(a,b)$: Greatest positive common divisor

Step-by-step example: Greatest common divisor by prime powers

1. Express both positive integers using the same prime bases.
$72=2^3\cdot3^2,\quad120=2^3\cdot3\cdot5$
2. Choose the smaller exponent at every prime, including zero.
$\gcd(72,120)=2^{\min(3,3)}3^{\min(2,1)}5^{\min(0,1)}$
3. Multiply the retained prime powers.
$\gcd(72,120)=24$

Open GCD and LCM calculator OpenStax: prime factorization and common multiples

Add Least common multiple by prime powers to study sheet

Notation and number theory

Least common multiple by prime powers

\operatorname{lcm}(a,b)=\prod_p p^{\max(\alpha_p,\beta_p)}

When it applies: $a,b$ are positive integers using the union of their prime bases. Include exponent zero for an absent prime. The result is the least positive integer divisible by both inputs.

$\alpha_p,\beta_p$: Exponents of prime $p$ in $a,b$
$a,b$: Positive integers
$\operatorname{lcm}(a,b)$: Least positive common multiple

Step-by-step example: Least common multiple by prime powers

1. Use the factorizations over the union of prime bases.
$72=2^3\cdot3^2,\quad120=2^3\cdot3\cdot5$
2. Choose the larger exponent for each prime.
$\operatorname{lcm}(72,120)=2^3\cdot3^2\cdot5$
3. Evaluate and check that both integers divide the result.
$\operatorname{lcm}(72,120)=360$

Open GCD and LCM calculator OpenStax: prime factorization and common multiples

What Can You Create?

Build a reference that explains its assumptions

A focused formula lookup

Find a relationship by topic, familiar name, a symbol alias or a relevant condition. The cards put a formula beside its domain and variable legend, so the notation is connected to a question instead of presented as an isolated string. Search can combine with topic and study-sheet filters without changing the original formulas or hiding an important applicability restriction inside a separate calculator.

A personal study selection

Collect related entries without copying an entire topic table. A local sheet can combine coordinate geometry with a trigonometric identity or put sample and population variance beside each other for comparison. The chosen identifiers remain on this device when storage is available. Shared configurations preserve a deliberate filtered view, while a reset returns to the full catalog without deleting the local selection.

A bridge to an interactive calculation

Read a three-stage worked example before opening the matching calculator where one exists. The example distinguishes substitution from simplification and keeps exact fractions, constants and square roots visible. A calculator link supports new values and additional checks; it does not mean that every reference identity has a dedicated solver or that the formula alone validates the assumptions of a real measurement problem.

Read the conditions before substituting numbers

Editorial formula diagram showing percentage, slope, circle area, and mean relationships from the math formula library. — A compact map of representative arithmetic, coordinate, geometry, and statistics formulas in the library.

A zero denominator changes the question

For a linear equation $ax+b=c$ , division by $a$ is valid only when $a\ne0$ . If that coefficient vanishes, inspect the remaining constants rather than forcing a quotient. Likewise, a vertical line has no finite slope, while two identical points do not determine one line. These cases explain why the domain is part of the reference rather than an optional warning after a numerical answer.

0x+3=3\quad\Longrightarrow\quad x\in\mathbb R

0x+3=5\quad\Longrightarrow\quad\text{no solution}

A method name is part of a statistical answer

The same squared-deviation total can describe a complete supplied population or estimate a wider population variance from a sample. Those interpretations use different denominators. Similarly, a quantile interpolation position is not an empirical score percentile rank. Matching the data and number of observations is insufficient if the method changes between reports. The library names these conventions directly and links to the relevant comparison tools.

S=26,\quad n=3:\qquad\sigma^2=\frac{26}{3},\quad s^2=\frac{26}{2}=13

Why Users Love This Tool

Why a useful formula reference needs more than equations

The mathematical scope remains visible

Denominator and sample-size restrictions appear on the same card as the relationship.
A radius, perpendicular height and angle convention keep their distinct meanings.
Exact constants and roots remain separate from rounded numerical views.
Quantile thresholds, tie-aware score ranks and normal-model interpretations are distinguished.

The reference remains practical

Every formula and example is server-rendered with accessible mathematical markup.
Native disclosures let readers open examples without requiring JavaScript.
Search and local-sheet filters combine without modifying the underlying catalog.
Copies and downloads retain conditions and examples even when the screen view is compact.

Perfect For

For revision, teaching and checking a calculation

Students choosing a method

Read the stated variables before substituting familiar numbers. Compare the structure of a worked example with your own problem, then confirm that the same restrictions apply. The local sheet can keep a few easily confused formulas together, while native example disclosures let you revisit the intermediate stages instead of memorizing only an unexplained final expression.

Teachers assembling a topic sheet

Filter to a topic, select a small comparison set and choose whether to print the worked stages. A compact sheet keeps domains and symbol meanings beside the equations, making it easier to discuss why a standard formula does not cover a vertical line, an invalid fraction divisor or a singleton sample variance. The printed content reflects the current visible set.

People reviewing a numerical answer

Use the reference to verify the relationship, variable roles and output units before inspecting a result. A symbolic identity is not evidence that the correct measurements or statistical convention were supplied. When a matching calculator is linked, use it to check new values and boundary cases while keeping the original assumptions with any copied or exported answer.

How It Works

How it works in three quick steps.

Find the relationship that matches your question

Search a formula name, symbol alias or condition, or choose a topic. Search and topic filters work together. Read the applicability statement before substituting numbers. Check whether the question concerns a measurement threshold, a score rank, a length, an area or a sample estimate, because similar-looking expressions may answer different questions.

Read the symbols and follow the worked example

Use the variable legend to identify every quantity, its role and any required unit convention. Open the native worked-example disclosure to follow three stages from the chosen values to the simplified result. Preserve parentheses, distinguish exact constants from numerical approximations and check whether a denominator, angle convention or sample-size condition rules out the intended substitution.

Build a focused study sheet and check a live calculation

Add relevant formulas to a study sheet stored on your device. Combine its filter with a topic or search query to make a focused visible set. Copy or download that set with complete examples, or print compact cards and optionally include the worked steps. Open a matching calculator where one is linked for interactive values and additional boundary-case explanations.

Download & Print

Copy, download or print the current reference view

Copy source or a complete explanation

Each card provides a formula-source copy for a LaTeX-compatible destination and a readable explanation for ordinary notes. The toolbar copies all visible formulas with their conditions, variable definitions and worked examples. Mathematical source is converted only when the destination needs plain text; visible formulas remain properly typeset. Clipboard failures produce feedback and leave the reference available for downloading.

Choose a suitable study-sheet file

The LaTeX option produces Markdown with inline or display math delimiters, while text is suitable for a destination without mathematical rendering. CSV keeps a rectangular set of title, topic, formula source, conditions, variables, example and calculator fields. These formats preserve the currently visible formulas, so a saved selection hidden by another topic filter is not unexpectedly included in the export.

Print compact cards or full examples

Compact printing keeps the formula, its applicability notes and variable legend. Enable worked examples to include the three ordered stages as well. The browser's before-and-after-print handling restores the earlier disclosure state when the print dialog closes. Search and selection controls are removed from the worksheet, while the printed-view description states which topic and filters determined the retained entries.

FAQ

Frequently Asked Questions

Does this page contain every formula in mathematics?

No finite reference page can cover every mathematical field or convention. This library contains thirty-six selected relationships across seven topics, emphasizing the arithmetic, algebra, geometry, trigonometry, statistics and number-theory questions used by Toolarithm's math tools. Each card states its scope and gives a worked example. Use the listed primary references or a course-specific source when you need a definition, theorem or specialized formula beyond this selection.

Why do formulas include conditions and symbol definitions?

A correct expression can still produce an invalid answer when its domain or variable meanings are ignored. A slope quotient requires different horizontal coordinates, a fraction denominator cannot be zero, a geometric height must be perpendicular to the chosen base and a sample variance needs an appropriate observation count. The library keeps those restrictions beside the formula so the conditions remain available in copied explanations, downloads and compact printed sheets.

How are percentile thresholds different from percentile ranks?

A percentile threshold uses an ordered dataset and a requested probability to produce a measurement value under a selected quantile convention. A percentile rank starts with a score and compares it with the reference observations. The Type 7 interpolation-position expression is not itself a score-rank formula. Separate cards explain Type 7 thresholds and the strict, weak, mean and average-matching rank conventions so a familiar rank-looking expression is not used for the wrong question.

Do the geometry formulas use length units or squared units?

Boundary lengths and distances use the original length unit, while planar areas use the square of that unit. All dimensions must first be converted to a compatible scale. The circle-area card needs a radius rather than an unconverted diameter, and triangle and trapezoid heights are perpendicular separations rather than arbitrary sloping sides. A unit label does not perform a conversion or establish that the supplied dimensions describe the required shape.

Does a z-score automatically give a percentile or p-value?

A standardized score expresses a difference from a reference mean in standard-deviation units. A normal-model area is an additional distributional assumption, and an empirical percentile rank instead depends on the actual reference observations and tie convention. A p-value also needs a justified test and sampling distribution. The z-score card therefore explains the algebraic transformation without claiming that the transformation alone establishes normality, a percentile or a valid hypothesis test.

How do search and the local study sheet work together?

The search query, selected topic and study-sheet-only switch are combined. A formula must satisfy every active filter to remain visible. Adding or removing an entry updates the identifier list stored on this device; storage failures leave the current visit usable and produce feedback. Opening a shared selection does not overwrite an existing local sheet until you edit the selection. Resetting the view clears filters while preserving the selected formula identifiers.

What is included in copies and downloads?

The toolbar exports the formulas currently visible after all filters, rather than silently exporting every saved entry or the whole catalog. Text and CSV retain conditions, symbol meanings and the complete three-step examples. The LaTeX sheet is a Markdown file containing real math delimiters and can be used in a compatible editor; it is not a standalone TeX document. Individual cards also support copying the formula source or a readable complete explanation.

How do I print a compact sheet or include worked examples?

Use the download-and-print panel to print the current visible set, then choose your browser's paper and PDF destination. Compact cards retain the relationship, applicability notes and variable legend. Enable the worked-example option when you want the three stages included; the previous open or closed example state is restored after printing. Empty filtered views cannot be exported. For a shorter sheet, narrow the topic or select a few related formulas first.

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About This Tool

About the Toolarithm math formulas library

This math formulas library is a reference for choosing and interpreting a relationship before using it in a calculation. Its thirty-six entries cover seven selected topics rather than claiming to be an exhaustive encyclopedia. Each entry combines a typeset relationship with applicability conditions, a symbol legend and an original worked example. That structure is intended to make a formula traceable: readers can identify the quantities, check the domain, substitute a small example and understand what the resulting number describes.

The library keeps several distinctions that are easy to lose in a short formula table. A signed percentage change depends on its baseline convention. A slope quotient does not represent a vertical line. Planar areas use squared units, and the height in a geometric area formula is perpendicular to its base. Sample and population variances use different denominators, while an interpolation position for a quantile answers a different question from the percentile rank of a score. These distinctions remain attached to the formulas when content is copied or exported.

Interactive controls provide a convenient view of the reference without making the mathematical content depend on them. The complete initial catalog and worked steps are rendered on the server, and native disclosures remain readable without JavaScript. Search, topic filters and a local study selection can then narrow the visible set. Downloads and printed sheets preserve the context needed for later revision, while linked calculators support new numerical inputs where a suitable tool exists. A reference relationship does not certify a measurement, establish statistical assumptions or replace checking a course-specific convention.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

The editorial team checks definitions, examples, linked tools, and cited references. Dates shown here come from the page's maintained publication record.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

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