Number skills

Adding signed numbers

Same signs: add the absolute values, keep the sign. Different signs: subtract the absolute values, keep the sign of the one farther from 0.

Example: \(-5 + (-6) = -11\) and \(-7 + 3 = -4\).

Subtracting: add the opposite

\(a - b = a + (-b)\). Change the minus to plus and switch the sign of the number after it (its opposite).

Example: \(4 - (-9) = 4 + 9 = 13\).

Watch out: \(-3 - 5 = -8\), not \(-2\). Both moves go left on the number line.

Multiplying and dividing signs

Same signs give a positive. Different signs give a negative. With many factors: an even count of negatives is positive, an odd count is negative.

Example: \((-6)(-4) = 24\), \(\;\dfrac{-18}{3} = -6\), \(\;(-2)(-3)(-1) = -6\).

Simplifying a fraction

Divide the numerator and denominator by their GCF.

Example: \(\dfrac{18}{24} = \dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}\).

Multiplying fractions

\(\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}\). Cancel common factors first to keep numbers small.

Example: \(\dfrac{4}{9} \cdot \dfrac{3}{8} = \dfrac{1}{3} \cdot \dfrac{1}{2} = \dfrac{1}{6}\).

Dividing fractions

Multiply by the reciprocal of the second fraction: \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c}\).

Example: \(\dfrac{2}{3} \div \dfrac{5}{6} = \dfrac{2}{3} \cdot \dfrac{6}{5} = \dfrac{4}{5}\).

Adding and subtracting fractions

Rewrite over the LCD, then add or subtract the numerators. The denominator stays.

Example: \(\dfrac{5}{6} - \dfrac{3}{4} = \dfrac{10}{12} - \dfrac{9}{12} = \dfrac{1}{12}\).

Watch out: never add the denominators: \(\dfrac{1}{2} + \dfrac{1}{3} \ne \dfrac{2}{5}\).

Mixed numbers and negative fractions

\(2\tfrac{3}{5} = \dfrac{2 \cdot 5 + 3}{5} = \dfrac{13}{5}\). A minus sign can sit in three places: \(-\dfrac{a}{b} = \dfrac{-a}{b} = \dfrac{a}{-b}\).

Example: \(-1\tfrac{1}{2} + \dfrac{3}{4} = -\dfrac{6}{4} + \dfrac{3}{4} = -\dfrac{3}{4}\).

Exponents and square roots

\(a^n\) means \(a\) multiplied by itself \(n\) times (see exponent and base). \(\sqrt{a}\) is the nonnegative number whose square is \(a\).

Example: \(2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16\), \(\;\sqrt{81} = 9\).

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.

\(-3^2\) versus \((-3)^2\)

The exponent touches only what is right next to it. \(-3^2 = -(3 \cdot 3) = -9\), but \((-3)^2 = (-3)(-3) = 9\).

Watch out: when you substitute a negative number, always put it in parentheses.

Order of operations

The order of operations: 1. Parentheses (and anything grouped by a fraction bar). 2. Exponents and roots. 3. Multiply and divide, left to right. 4. Add and subtract, left to right.

Example: \(8 - 2 \cdot 3^2 = 8 - 18 = -10\); \(\;12 \div 3 \cdot 2 = 4 \cdot 2 = 8\); \(\;\dfrac{6 + 4}{7 - 2} = \dfrac{10}{5} = 2\).

Evaluating expressions

To evaluate, replace each variable with its number in parentheses, then follow the order of operations.

Example: \(b^2 - 4ac\) with \(a = 2,\ b = -3,\ c = -5\): \((-3)^2 - 4(2)(-5) = 9 + 40 = 49\).

Example: \(\dfrac{y_2 - y_1}{x_2 - x_1}\) for \((1, -2)\) and \((4, 7)\): \(\dfrac{7 - (-2)}{4 - 1} = \dfrac{9}{3} = 3\).

Combining like terms

Like terms have exactly the same variable part. Add their coefficients; the variable part stays the same.

Example: \(3x^2 - 8x^2 + 5x = -5x^2 + 5x\).

Watch out: \(3x + 2x^2\) cannot be combined: \(x\) and \(x^2\) are different.

The distributive property

The distributive property: \(a(b + c) = ab + ac\). Multiply the outside number by every term inside. A minus in front flips every sign: \(-(x - 3) = -x + 3\).

Example: \(3(2x - 5) - 4(x - 1) = 6x - 15 - 4x + 4 = 2x - 11\).

Watch out: the \(-4\) multiplies the \(-1\) too, giving \(+4\).