ACT Math Flashcards: Real Numbers

What this deck covers

This deck focuses on Real Numbers, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the associative property of addition for real numbers $a,b,c$?

Answer: $(a+b)+c=a+(b+c)$. Addition grouping doesn't affect the result.

Flashcard 2: Identify the reciprocal of 5.

Answer: $\frac{1}{5}$. Reciprocal means $\frac{1}{\text{original number}}$.

Flashcard 3: What is the reciprocal of $\frac{2}{7}$?

Answer: $\frac{7}{2}$. Reciprocal means flipping the numerator and denominator.

Flashcard 4: State the property: $a + b = b + a$ for all real numbers.

Answer: Commutative Property of Addition. Addition order doesn't affect the sum.

Flashcard 5: Which number is an example of an irrational number?

Answer: $\sqrt{2}$. Cannot be expressed as a fraction of integers.

Flashcard 6: What is the inverse of $a$ under multiplication, assuming $a \neq 0$?

Answer: $\frac{1}{a}$. The multiplicative inverse produces 1 when multiplied by the original.

Flashcard 7: What is the exponent rule for dividing same-base powers $\frac{a^m}{a^n}$ for $a\neq 0$?

Answer: $a^{m-n}$. Subtract exponents when dividing same bases.

Flashcard 8: What is the sum of $\frac{3}{4}$ and $\frac{1}{4}$?

Answer:

1. Common denominator makes addition straightforward.

Flashcard 9: Which number is greater: $\sqrt{10}$ or 3?

Answer: $\sqrt{10}$. $\sqrt{10} \approx 3.16$, which is greater than 3.

Flashcard 10: Find the sum of $-2.5$ and $2.5$.

Answer:

1. Opposites sum to zero: $-2.5 + 2.5 = 0$.

Flashcard 11: What is the multiplicative identity in real numbers?

Answer:

1. Multiplying any real number by $1$ gives the original number.

Flashcard 12: What is the definition of a whole number set?

Answer: $\{0,1,2,3,\dots\}$. Natural numbers plus zero.

Flashcard 13: What is the sum of the square roots of 4 and 9?

Answer:

1. $\sqrt{4} + \sqrt{9} = 2 + 3 = 5$.

Flashcard 14: What is $2^3$?

Answer:

1. Exponent means repeated multiplication: $2 \times 2 \times 2 = 8$.

Flashcard 15: What is the definition of a terminating decimal in terms of fractions?

Answer: A decimal equal to $\frac{a}{b}$ with $b=2^m5^n$ in lowest terms. Denominator has only factors of 2 and 5.

Flashcard 16: What is the decimal representation of $\frac{1}{2}$?

Answer: 0.5. Dividing 1 by 2 gives 0.5.

Flashcard 17: What is the distance between two real numbers $a$ and $b$ on a number line?

Answer: $|a-b|$. Absolute value formula for distance on number line.

Flashcard 18: What is the value of $a^0$ for any nonzero real number $a$?

Answer: $a^0=1$ for $a\neq 0$. Any nonzero number to zero power equals one.

Flashcard 19: Which property is illustrated by $(a + b) + c = a + (b + c)$?

Answer: Associative Property of Addition. Grouping of addends doesn't affect the sum.

Flashcard 20: What type of number is $\pi$?

Answer: Irrational. Pi cannot be expressed as a fraction of integers.

Flashcard 21: What is the commutative property of multiplication for real numbers $a$ and $b$?

Answer: $ab=ba$. Multiplication order doesn't affect the result.

Flashcard 22: What is the real number result of $-3 \times 4$?

Answer: -12. Multiply: $(-3) \times 4 = -12$.

Flashcard 23: What is the value of $-3^2$ using order of operations?

Answer: $-9$. Exponent applies only to 3, then negative applied.

Flashcard 24: What is the product of any real number and its multiplicative inverse?

Answer:

1. Definition of multiplicative inverse relationship.

Flashcard 25: What is the multiplicative identity property for real numbers?

Answer: $a\cdot 1=a$. One is the multiplicative identity element.

Flashcard 26: Identify the inverse of $a$ under addition.

Answer: $-a$. The additive inverse produces zero when added to the original.

Flashcard 27: What is the value of $|-7|$?

Answer: $7$. Absolute value is always nonnegative.

Flashcard 28: What is the associative property of addition for real numbers $a,b,c$?

Answer: $(a+b)+c=a+(b+c)$. Addition grouping doesn't affect the result.

Flashcard 29: What is the simplified value of $\sqrt{50}$ in the form $a\sqrt{b}$?

Answer: $5\sqrt{2}$. Factor out perfect square: $\sqrt{25 \cdot 2}$.

Flashcard 30: Identify the correct classification of $\sqrt{49}$ within the real number system.

Answer: Rational (it equals $7$). Perfect square root simplifies to integer.

Flashcard 31: What is the result of $7 \times 0$?

Answer:

1. Any number multiplied by zero equals zero.

Flashcard 32: Identify the additive identity in real numbers.

Answer:

1. Adding zero to any real number gives the original number.

Flashcard 33: What is the commutative property of addition for real numbers $a$ and $b$?

Answer: $a+b=b+a$. Addition order doesn't affect the result.

Flashcard 34: What is the product of any real number and its multiplicative inverse?

Answer:

1. Definition of multiplicative inverse relationship.

Flashcard 35: Identify the real number that is neither positive nor negative.

Answer:

1. Zero is the unique real number that is neither positive nor negative.

Flashcard 36: What is the inverse of $a$ under multiplication, assuming $a \neq 0$?

Answer: $\frac{1}{a}$. The multiplicative inverse produces 1 when multiplied by the original.

Flashcard 37: What is the simplification of $\sqrt{a^2}$ for a real number $a$?

Answer: $|a|$. Square root of squared value is absolute value.

Flashcard 38: What is the associative property of multiplication for real numbers $a,b,c$?

Answer: $(ab)c=a(bc)$. Multiplication grouping doesn't affect the result.

Flashcard 39: What is the value of $|-7|$?

Answer: $7$. Absolute value is always nonnegative.

Flashcard 40: Identify the additive inverse of $-7$.

Answer:

1. The additive inverse of $-7$ is the number that adds to zero.

Flashcard 41: State the property: $a \times 1 = a$ for all real numbers.

Answer: Multiplicative Identity Property. Multiplying by $1$ preserves the original value.

Flashcard 42: What is the exponent rule for multiplying same-base powers $a^m\cdot a^n$?

Answer: $a^{m+n}$. Add exponents when multiplying same bases.

Flashcard 43: Find the sum of $\frac{2}{3}$ and $\frac{1}{3}$.

Answer:

1. Add fractions with same denominator: $\frac{2+1}{3} = 1$.

Flashcard 44: What is the absolute value of $0$?

Answer:

1. The distance from zero is zero.

Flashcard 45: Find the product of the real numbers $-4$ and $3$.

Answer: -12. Multiply: $(-4) \times 3 = -12$.

Flashcard 46: What is the definition of a whole number set?

Answer: $\{0,1,2,3,\dots\}$. Natural numbers plus zero.

Flashcard 47: What is the sum of the square roots of 4 and 9?

Answer:

1. $\sqrt{4} + \sqrt{9} = 2 + 3 = 5$.

Flashcard 48: Which property does $a = a$ illustrate?

Answer: Reflexive Property. Every real number equals itself.

Flashcard 49: What is the reciprocal of $\frac{2}{7}$?

Answer: $\frac{7}{2}$. Reciprocal means flipping the numerator and denominator.

Flashcard 50: Find the difference of the real numbers $18$ and $25$.

Answer: -7. Subtract: $18 - 25 = -7$.

Flashcard 51: What is the result of $9 + (-9)$?

Answer:

1. A number plus its opposite equals zero.

Flashcard 52: What is the reciprocal of the real number $5$?

Answer: $\frac{1}{5}$. The reciprocal multiplied by the original number equals $1$.

Flashcard 53: Identify the square root of $16$.

Answer:

1. The positive square root of $16$ is $4$.

Flashcard 54: State the property: $a \times (b \times c) = (a \times b) \times c$.

Answer: Associative Property of Multiplication. Multiplication grouping doesn't affect the product.

Flashcard 55: What is the reciprocal of the real number $5$?

Answer: $\frac{1}{5}$. The reciprocal multiplied by the original number equals $1$.

Flashcard 56: Which property is illustrated by $a \times 1 = a$?

Answer: Multiplicative Identity Property. Multiplying by 1 preserves the original value.

Flashcard 57: Find the sum of $-2.5$ and $2.5$.

Answer:

1. Opposites sum to zero: $-2.5 + 2.5 = 0$.

Flashcard 58: What is the additive identity property for real numbers?

Answer: $a+0=a$. Zero is the additive identity element.

Flashcard 59: What is the definition of an irrational number in terms of rational numbers?

Answer: A real number that is not rational (not expressible as $\frac{a}{b}$). Cannot be written as a fraction of integers.

Flashcard 60: What is the relationship between real numbers, rational numbers, and irrational numbers?

Answer: $\mathbb{R}=\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q})$ and the sets are disjoint. Real numbers are the union of rationals and irrationals.

Flashcard 61: What is the sum of $\frac{3}{4}$ and $\frac{1}{4}$?

Answer:

1. Common denominator makes addition straightforward.

Flashcard 62: What is the power of a power rule $(a^m)^n$?

Answer: $a^{mn}$. Multiply exponents when raising power to power.

Flashcard 63: What is the absolute value of $-9$?

Answer:

1. Absolute value is the distance from zero, always non-negative.

Flashcard 64: Which number is greater: $\sqrt{10}$ or 3?

Answer: $\sqrt{10}$. $\sqrt{10} \approx 3.16$, which is greater than 3.

Flashcard 65: What is the value of $(-3)^2$?

Answer: $9$. Parentheses make negative base, so even power positive.

Flashcard 66: What is the square of 5?

Answer:

1. Multiplying 5 by itself gives 25.

Flashcard 67: What is the opposite of $-11$?

Answer:

1. The opposite changes the sign of a number.

Flashcard 68: What is the sign of the product of two negative real numbers?

Answer: Positive. Negative times negative equals positive.

Flashcard 69: What is the relationship between real numbers, rational numbers, and irrational numbers?

Answer: $\mathbb{R}=\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q})$ and the sets are disjoint. Real numbers are the union of rationals and irrationals.

Flashcard 70: What is the multiplicative inverse property for a nonzero real number $a$?

Answer: $a\cdot\frac{1}{a}=1$ for $a\neq 0$. Every nonzero number has a multiplicative inverse.

Flashcard 71: What is the absolute value definition of $|x|$ using a piecewise statement?

Answer: $|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}$. Returns the number's distance from zero.

Flashcard 72: What is the rule for multiplying two real numbers with different signs?

Answer: The product is negative. Different signs always give negative result.

Flashcard 73: Identify the multiplicative inverse of $-2$.

Answer: $-\frac{1}{2}$. The multiplicative inverse of $-2$ is $-\frac{1}{2}$.

Flashcard 74: What is the result of squaring any real number?

Answer: Non-negative number. Squaring always produces a non-negative result.

Flashcard 75: What is the decimal form of $\frac{3}{5}$?

Answer: 0.6. Dividing 3 by 5 gives 0.6.

Flashcard 76: What is the multiplicative identity in the set of real numbers?

Answer:

1. Multiplying any number by 1 leaves it unchanged.

Flashcard 77: What is the value of $(-1)^5$?

Answer: -1. Odd power of negative number stays negative.

Flashcard 78: State the property: $a(b + c) = ab + ac$ for all real numbers.

Answer: Distributive Property. Multiplication distributes over addition.

Flashcard 79: What is the definition of a negative exponent $a^{-n}$ for $a\neq 0$?

Answer: $a^{-n}=\frac{1}{a^n}$. Negative exponent means reciprocal of positive power.

Flashcard 80: What is the rule for dividing two real numbers with the same sign?

Answer: The quotient is positive. Same signs always give positive result.

Flashcard 81: Which property is used in $a(b + c) = ab + ac$?

Answer: Distributive Property. Multiplication distributes over addition.

Flashcard 82: State the property: $a \times 1 = a$ for all real numbers.

Answer: Multiplicative Identity Property. Multiplying by $1$ preserves the original value.

Flashcard 83: What is the definition of a repeating decimal in terms of rationality?

Answer: Every repeating decimal is rational. Repeating decimals can always be written as fractions.

Flashcard 84: What is the additive identity in the set of real numbers?

Answer: $0$. Adding $0$ to any number leaves it unchanged.

Flashcard 85: What is the decimal representation of $\frac{1}{2}$?

Answer: 0.5. Dividing 1 by 2 gives 0.5.

Flashcard 86: What is the rule for multiplying two real numbers with different signs?

Answer: The product is negative. Different signs always give negative result.

Flashcard 87: What is the sum of the real numbers $5$ and $-3$?

Answer:

1. Add the numbers: $5 + (-3) = 2$.

Flashcard 88: What is the power of a quotient rule $\left(\frac{a}{b}\right)^n$ for $b\neq 0$?

Answer: $\frac{a^n}{b^n}$. Distribute exponent to numerator and denominator.

Flashcard 89: What is the opposite of $-11$?

Answer:

1. The opposite changes the sign of a number.

Flashcard 90: Which property is illustrated by $(ab)c = a(bc)$?

Answer: Associative Property of Multiplication. Grouping of factors doesn't affect the product.

Flashcard 91: What is the definition of a negative exponent $a^{-n}$ for $a\neq 0$?

Answer: $a^{-n}=\frac{1}{a^n}$. Negative exponent means reciprocal of positive power.

Flashcard 92: What is the additive inverse property for a real number $a$?

Answer: $a+(-a)=0$. Every number has an additive inverse (opposite).

Flashcard 93: What is the multiplicative identity property for real numbers?

Answer: $a\cdot 1=a$. One is the multiplicative identity element.

Flashcard 94: What is the square of the real number $6$?

Answer:

1. Square means multiply by itself: $6 \times 6 = 36$.

Flashcard 95: What is the rule for comparing negative numbers on a number line?

Answer: The number closer to $0$ is greater. Less negative means greater on number line.

Flashcard 96: What type of number is $\pi$?

Answer: Irrational. Pi cannot be expressed as a fraction of integers.

Flashcard 97: What is the greatest common divisor of 18 and 24?

Answer:

1. Largest number that divides both 18 and 24.

Flashcard 98: State the property: $a + b = b + a$ for all real numbers.

Answer: Commutative Property of Addition. Addition order doesn't affect the sum.

Flashcard 99: What is the value of $\frac{2^3\cdot 2^4}{2^5}$?

Answer: $2^2=4$. Simplify using exponent rules: $2^{3+4-5} = 2^2$.

Flashcard 100: Identify the opposite of the real number $8$.

Answer: -8. The opposite has the same magnitude but different sign.