Number Concepts: Place Value, Integers, Exponents & Roots

Number Concepts: Place Value, Integers, Exponents & Roots

Numbers and numeration is one of the content strands the HSPT Mathematics subtest names explicitly, and it is the layer everything else rests on. Fraction arithmetic, algebra, and geometry all break down when place value, sign rules, or exponent rules are shaky. These ideas also show up directly in the 24 Concepts items, which ask you to name the principle behind a problem rather than to compute an answer.

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Place Value and Number Names

Rounding rule: look at the digit immediately to the right of the target place. If it is 5 or greater, round up; otherwise leave the target digit alone. Digits further right never matter.

• 3,472.586 to the nearest tenth: the hundredths digit is 8, so round up to 3,472.6.
• 4.4999 to the nearest whole number: the tenths digit is 4, so round down to 4 - the trailing nines are irrelevant.

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Number Sets

Two facts the Concepts items like: every integer is rational (7 = 7/1), and a decimal that terminates or repeats is rational, while one that neither terminates nor repeats is irrational.

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Integers, Signs and Absolute Value

• Adding same signs: add the absolute values, keep the sign. -6 + (-5) = -11.
• Adding different signs: subtract the absolute values, take the sign of the larger one. -9 + 4 = -5.
• Subtracting a negative: becomes addition. 7 - (-3) = 7 + 3 = 10.
• Multiplying or dividing: same signs give a positive, different signs give a negative. (-8)(-3) = 24; -20 / 5 = -4.

Absolute value is distance from zero, so it is never negative: the absolute value of -12 is 12, and the absolute value of 12 is also 12. Inside an expression, absolute-value bars act as grouping symbols - evaluate what is inside first.

• The absolute value of (3 - 11) equals the absolute value of -8, which is 8.

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Odd, Even, Prime and Composite

Special cases worth memorizing: 1 is neither prime nor composite, and 2 is the only even prime.

Sign and parity behavior appears constantly in Concepts items:

• odd + odd = even; even + even = even; odd + even = odd
• odd x odd = odd; any number x even = even
• A negative raised to an even power is positive; raised to an odd power it stays negative.

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Exponents

The parentheses rule, which decides more HSPT items than any other exponent fact: (-4) squared = (-4)(-4) = 16, but -4 squared = -(4 x 4) = -16. The sign is part of the base only when it sits inside the parentheses.

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Roots

Memorize these and most root questions become recall.

Estimating a non-perfect root: trap it between two perfect squares. The square root of 50 lies between the square root of 49 (which is 7) and the square root of 64 (which is 8), and because 50 is very close to 49, the value is just above 7.

Simplifying: pull out perfect-square factors. The square root of 72 equals the square root of 36 times the square root of 2, which is 6 times the square root of 2.

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Scientific Notation

A number in scientific notation is written as a value between 1 and 10 multiplied by a power of 10.

• 47,000 = 4.7 x 10 to the fourth (decimal moved 4 places left, exponent positive).
• 0.0032 = 3.2 x 10 to the negative third (decimal moved 3 places right, exponent negative).

The direction rule in one line: a large number gets a positive exponent, a small decimal gets a negative one. If your exponent's sign disagrees with the size of the number, you moved the decimal the wrong way.

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Ordering Rational Numbers

To place mixed forms on a number line, convert everything to decimals, then order left to right.

Order: -1/2, 0.75, -0.8, 2/3, 0

Decimals: -0.5, 0.75, -0.8, 0.667, 0. Sorted: -0.8, -1/2, 0, 2/3, 0.75.

Negative ordering trap: with negative numbers, the value with the larger absolute value is the smaller number. -0.8 sits to the left of -0.5, even though 8 is bigger than 5.

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Traps to Rehearse

• Absolute value of a difference. The absolute value of (4 - 9) is 5, not -5.
• Zero. Zero is even, is neither positive nor negative, and is a whole number but not a natural number.
• The exponent-parenthesis slip. Copying -3 squared as (-3) squared changes 9 into -9 or the reverse.
• Rounding chains. Round once, from the original number. Rounding 4.46 to tenths then to ones gives 5; rounding 4.46 straight to ones gives the correct 4.