Mental Math with Signed Numbers, Fractions, Decimals, and Percentages

Hello, and welcome to the mathematics foundation for your AI/ML path. This first module rebuilds the arithmetic and algebra habits that later support vectors, probability, calculus, and optimization. We will work by hand first, with attention to why each rule works—not just memorizing a calculator procedure.

This lesson develops reliable arithmetic with signed numbers, fractions, decimals, and percentages. These are different notations for quantities, and moving comfortably among them is essential: for example, $0.125$, $12.5\%$, and $\frac{1}{8}$ all represent the same number.

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1. Signed numbers: value, direction, and opposites

A positive number represents an amount above a reference point; a negative number represents an amount below it. Depending on context, the reference point might be zero money, sea level, a baseline model error, or no change.

The opposite of a number is the number at the same distance from zero on the other side of zero. Thus the opposite of $7$ is $-7$, and the opposite of $-7$ is $7$. Their sum is zero:

$$7+(-7)=0$$

This is the central structural fact behind signed arithmetic. A positive and its negative form a zero pair, also called an additive inverse pair.

Adding signed numbers

For two numbers with the same sign, add their magnitudes and keep the common sign:

$$(-8)+(-5)=-(8+5)=-13$$ $$6+9=15$$

For two numbers with different signs, subtract the smaller magnitude from the larger magnitude, then use the sign attached to the larger magnitude:

$$(-17)+29=12$$

Here, the positive amount $29$ exceeds the negative amount $17$ by $12$, so the result is positive.

A useful way to calculate a longer sum is to collect positive and negative contributions mentally:

$$-17+29-(-6)$$

First deal with the subtraction of a negative, which we will justify in the next section:

$$-17+29+6=18$$

Subtraction means “add the opposite”

The expression $a-b$ means: start with $a$, then add the opposite of $b$.

$$a-b=a+(-b)$$

This is not merely a mnemonic. If $x=a-b$, then $x$ is the number that must be added to $b$ to recover $a$. Now check $a+(-b)$:

$$b+[a+(-b)]=a+[b+(-b)]=a+0=a$$

So $a+(-b)$ is exactly the required difference.

Be careful to distinguish a negative sign from a subtraction operation. Parentheses make the structure visible:

$$8-(-3)=8+3=11$$ $$-8-3=-8+(-3)=-11$$

Multiplication and division with signs

For multiplication and division, first determine the sign, then work with magnitudes.

Signs of two nonzero numbers	Product or quotient
same signs	positive
different signs	negative

Examples:

$$(-3)(-7)=21$$ $$(-42)÷6=-7$$ $$(-42)÷(-6)=7$$

There is a reason that two negatives multiply to a positive. Since $a+(-a)=0$, multiplying both sides by $b$ gives:

$$[a+(-a)]b=0$$

Using distributivity,

$$ab+(-a)b=0$$

Therefore $(-a)b$ must be the opposite of $ab$:

$$(-a)b=-(ab)$$

Applying the same idea a second time shows that $(-a)(-b)=ab$. The sign rules preserve the ordinary distributive law, which is why they are mathematically necessary.

A compact signed-number routine

When a calculation includes several signs:

1. Rewrite every subtraction as addition of an opposite.
2. Simplify signs inside parentheses.
3. Perform multiplication and division before addition and subtraction.
4. For additions, combine values by cancellation or common sign.

For instance:

$$-4+3(-5)-(-7)$$

First multiply and rewrite the subtraction:

$$-4-15+7$$

Then combine the negative contributions and the positive contribution:

$$-19+7=-12$$

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2. Fractions: equal parts and exact arithmetic

A fraction

$$\frac{a}{b}$$

means $a$ divided by $b$, where $b\ne 0$. The numerator $a$ counts parts; the denominator $b$ specifies the size of each part.

Two fractions are equivalent when they name the same number. Multiplying both numerator and denominator by the same nonzero number does not change the value:

$$\frac{3}{4}=\frac{3\cdot 2}{4\cdot 2}=\frac{6}{8}$$

This principle is the foundation of common denominators and decimal division.

Adding and subtracting fractions

You may only add fractions directly when their denominators are the same:

$$\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$$

The denominator tells you the size of the pieces. Three eighths plus two eighths gives five eighths. In contrast, $\frac{1}{3}+\frac{1}{4}$ cannot be called “two sevenths,” because thirds and fourths are different-sized pieces.

To add or subtract unlike denominators, rewrite both fractions using a common denominator, ideally the least common denominator.

$$-\frac{3}{4}+\frac{5}{6}$$

The least common denominator of $4$ and $6$ is $12$:

$$-\frac{3}{4}=-\frac{9}{12}$$ $$\frac{5}{6}=\frac{10}{12}$$

Now add the numerators:

$$-\frac{9}{12}+\frac{10}{12}=\frac{1}{12}$$

The sign rules are exactly the same as for integers; the common denominator lets you see the signed quantities as comparable pieces.

Multiplying fractions

To multiply fractions, multiply numerators and denominators:

$$\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$$

Reduce factors before multiplying whenever possible. This keeps arithmetic small and reduces errors:

$$\left(-\frac{14}{15}\right)\left(-\frac{9}{28}\right)$$

Cancel $14$ with $28$, and $9$ with $15$:

$$\left(-\frac{1}{5}\right)\left(-\frac{3}{2}\right)=\frac{3}{10}$$

Notice that cancellation is valid only among factors connected by multiplication. You cannot cancel parts of a sum, such as in

$$\frac{3+5}{5}$$

because $3+5$ is not a factorization with a common factor of $5$.

Dividing fractions

Dividing by a nonzero fraction means multiplying by its reciprocal:

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$$

The reciprocal of $\frac{c}{d}$ is $\frac{d}{c}$, because their product is $1$:

$$\frac{c}{d}\cdot \frac{d}{c}=1$$

Example:

$$-\frac{5}{12}÷\frac{15}{8}$$

Rewrite division as multiplication by the reciprocal:

$$-\frac{5}{12}\cdot \frac{8}{15}$$

Reduce before multiplying:

$$-\frac{1}{3}\cdot \frac{2}{3}=-\frac{2}{9}$$

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3. Decimals: place value gives the rules

Decimals are another notation for fractions whose denominators are powers of $10$:

$$0.4=\frac{4}{10}$$ $$0.037=\frac{37}{1000}$$

Trailing zeros do not change a decimal’s value:

$$2.5=2.50=2.500$$

Adding zeros is useful because it lets place values line up clearly.

Adding and subtracting decimals

Align decimal points, not the final digits. This automatically aligns ones with ones, tenths with tenths, and so on.

$$\begin{array}{r}-4.75\\ +2.60\\ -2.15\end{array}$$

The signs can be handled separately using the signed-number rules; the decimal alignment handles place value.

Multiplying decimals

For multiplication, multiply as though the numbers were whole numbers, then place the decimal using the total number of decimal places in the factors.

$$3.6\cdot 0.25$$

Compute the whole-number product:

$$36\cdot 25=900$$

There are three decimal places in total: one in $3.6$ and two in $0.25$. Therefore:

$$3.6\cdot 0.25=0.900=0.9$$

With a negative factor:

$$(-3.6)(0.25)=-0.9$$

This works because removing decimals temporarily multiplies each factor by a power of $10$, so the final product must be scaled back by the combined amount.

Dividing decimals

If the divisor is a whole number, carry the decimal point directly into the quotient:

$$12.64÷4=3.16$$

If the divisor contains a decimal, multiply both dividend and divisor by the same power of $10$ until the divisor becomes whole. This preserves the quotient:

$$6.45÷(-1.5)$$

Multiply both quantities by $10$:

$$64.5÷(-15)=-4.3$$

The sign is negative because the numbers have different signs.

Estimation: your error detector

Before trusting a decimal answer, estimate its magnitude.

For example, since $3.6$ is a little less than $4$ and $0.25$ is one quarter, their product should be a little less than $1$. Thus $0.9$ is plausible, while $9$ or $0.09$ should immediately look suspicious.

This habit is valuable later in ML work: a plausible-looking numerical output can still be wrong by a factor of $10$, $100$, or more.

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4. Percentages: “per hundred”

The symbol $\%$ means “per $100$.” Thus:

$$18\%=\frac{18}{100}=0.18$$ $$0.7=70\%$$ $$1.25=125\%$$

A percentage can exceed $100\%$. For example, $125\%$ means $1.25$ times the original amount.

Finding a percentage of a quantity

“$p\%$ of $x$” means multiplication:

$$p\%\text{ of }x=\frac{p}{100}\cdot x$$

For example, find $18\%$ of $250$:

$$0.18\cdot 250$$ $$=\frac{18}{100}\cdot 250$$ $$=18\cdot 2.5=45$$

So $18\%$ of $250$ is $45$.

Percentage increase and decrease

A percentage change is not the same as the resulting percentage.

If a quantity is increased by $18\%$, the new quantity is:

$$\text{original}+0.18(\text{original})$$ $$=1.18(\text{original})$$

If $250$ decreases by $18\%$, subtract $45$:

$$250-45=205$$

Equivalently:

$$250(1-0.18)=250(0.82)=205$$

This distinction matters:

• “$18\%$ of $250$” is $45$.
• “Decrease $250$ by $18\%$” is $205$.
• “$205$ is what percent of $250$?” is a different question, answered by a ratio.

Converting between forms

A number can often be made easier to calculate by choosing the most useful form:

$$37.5\%=0.375=\frac{375}{1000}=\frac{3}{8}$$

Therefore, $37.5\%$ of $64$ can be computed as:

$$\frac{3}{8}\cdot 64=24$$

This flexibility is one reason fractions, decimals, and percentages should be understood as connected representations rather than separate topics.

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5. A reliable hand-calculation workflow

When an expression mixes the forms studied today, use this workflow:

1. Make signs explicit. Put negative numbers in parentheses when needed.
2. Choose a convenient representation. Fractions are often best for exact values; decimals are often best for place-value arithmetic; percentages should usually become fractions or decimals.
3. Respect operation order. Compute parentheses first, then multiplication and division from left to right, then addition and subtraction from left to right.
4. Reduce along the way. Simplify fraction factors before multiplying.
5. Estimate and check the sign. A negative factor should produce a negative product when the other factor is positive; a percentage of a positive quantity should have a sensible size.

For example:

$$(-12.5\%)\cdot 80+\frac{3}{4}\cdot 16$$

Convert the percentage:

$$-0.125\cdot 80+\frac{3}{4}\cdot 16$$

Since $0.125=\frac{1}{8}$,

$$-\frac{1}{8}\cdot 80+\frac{3}{4}\cdot 16$$ $$=-10+12=2$$

The answer is positive because the positive contribution of $12$ exceeds the negative contribution of $10$.

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You now have the core arithmetic language needed for the rest of the course:

• Opposite numbers sum to zero, and subtraction is addition of an opposite.
• Sign rules follow from the distributive law.
• Fractions require common denominators for addition and reciprocals for division.
• Decimal procedures come from place value and equivalent scaling.
• A percentage is a fraction with denominator $100$.
• Estimation and sign checks catch many errors before they spread.

Next, we will build on this arithmetic foundation by using laws of exponents and radicals to simplify numerical expressions.