Grade 9 Mathematics – Number System & Related Concepts

Flashcards cover definitions, theorems, rules, and computation techniques from Grade 9 Number System unit: natural & integer sets, prime factorization, GCF/LCM, rational vs irrational numbers, exponents & radicals, approximation, scientific notation, rationalization, and Euclid’s division algorithm.

What set is described by N = {1, 2, 3,…}?

The set of natural numbers.

Does the set of integers include zero?

Yes. Z = {…, –3, –2, –1, 0, 1, 2, 3,…}.

Define a composite number.

A natural number with more than two distinct factors.

What is the fundamental theorem of arithmetic?

Every composite number can be expressed uniquely as a product of primes, apart from the order of the factors.

State the divisibility rule for 3.

A number is divisible by 3 if the sum of its digits is divisible by 3.

What is a prime factor?

A factor of a composite number that is itself prime.

Give the prime factorization of 180.

2² × 3² × 5.

Define ‘multiple’.

For natural numbers m and p, m is a multiple of p if there exists a natural number q such that m = p × q.

When are two numbers relatively prime?

When their greatest common factor (GCF) is 1.

How is the GCF found using prime factorization?

Multiply the common prime factors, each raised to the smallest power that appears in any factorization.

State the relationship between GCF and LCM of two natural numbers a and b.

GCF(a, b) × LCM(a, b) = a × b.

Define a rational number.

A number that can be written in the form a⁄b where a and b are integers and b ≠ 0.

What type of decimal representation do rational numbers have?

Either terminating or repeating decimals.

Convert 0.7̅ to a fraction.

7⁄9.

Is √2 rational or irrational? Why?

Irrational; its decimal expansion neither terminates nor repeats.

Define ‘real number’.

Any number that is either rational or irrational.

What is the trichotomy property of order on real numbers?

For any two real numbers a and b, exactly one is true: a < b, a = b, or a > b.

Explain the index and radicand in √[n]{a}.

n is the index (root), a is the radicand.

Give the principal square root of 25.

5.

State the rule for even index radicals of a².

√[n]{a²} = |a| when n is even.

Simplify (8 × 27)¹ᐟ³.

6.

Rewrite 5¹ᐟ³ using radical notation.

³√5.

State the law (aᵐ)ⁿ = ?

aᵐⁿ.

When are radicals ‘like radicals’?

When they have the same index and the same radicand.

Add √2 + √8.

3√2.

Give the scientific notation form.

a × 10ᵏ where 1 ≤ a < 10 and k is an integer.

Convert 0.00083 to scientific notation.

8.3 × 10⁻⁴.

What is meant by ‘round to 2 d.p.’?

Round to the nearest hundredth (two digits after the decimal point).

Lower and upper bounds when a length is given as 5.4 cm (to 1 d.p.)?

5.35 cm < length < 5.45 cm.

Define significant figures.

Digits counted from the first non-zero digit on the left; they measure precision of a number.

Rationalize 1⁄√2.

√2⁄2.

Provide a rationalizing factor for 1⁄(3 – √5).

3 + √5.

Write Euclid’s Division Algorithm.

For integers a ≥ 0 and b > 0, there exist unique q, r with 0 ≤ r < b such that a = bq + r.

According to Euclid’s algorithm, GCF(a, b) equals what?

GCF(b, r) where r is the remainder when a is divided by b.

Show 47 using division algorithm with divisor 7.

47 = 7·6 + 5 (q = 6, r = 5).

What is the LCM of 8 and 9?

72.

State the divisibility rule for 8.

A number is divisible by 8 if its last three digits form a number divisible by 8.

Explain ‘terminating decimal’.

A decimal expansion that ends with a remainder of zero.

Convert 1.3456 to a fraction in simplest form.

13456⁄10000 = 841⁄625.

How do you locate √5 on a number line geometrically?

Construct a right triangle with legs 1 unit and 2 units, hypotenuse √5, then transfer that length to the number line.

What is a ‘repetend’?

The repeating digit or block of digits in a repeating decimal.

Give the product rule for radicals (n identical): √[n]{a}·√[n]{b} = ?

√[n]{ab}.

State closure property for real numbers under multiplication.

The product of any two real numbers is a real number.

Is the set of irrational numbers closed under addition?

No; the sum of two irrationals can be rational or irrational.

What is the principal cube root of –27?

–3.

Express 64¹ᐟ³.

4.

Rule for comparing non-negative a and b via squares:

If a² < b², then a < b.

Simplify 2√3 × 3√2.

6√6.

Define rounding ‘to the nearest thousand’.

Replace a number by the closest multiple of 1,000.

What is the additive inverse of √2 + 1?

–(√2 + 1).